Gromov Hyperbolicity of Periodic Graphs

Abstract

Gromov hyperbolicity grasps the essence of both negatively curved spaces and discrete spaces. The hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it; hence, characterizing hyperbolic graphs is a main problem in the theory of hyperbolicity. Since this is a very ambitious goal, a more achievable problem is to characterize hyperbolic graphs in particular classes of graphs. The main result in this paper is a characterization of the hyperbolicity of periodic graphs.

1 Introduction

Gromov hyperbolicity grasps the essence of both negatively curved spaces and discrete spaces. As observed in [5, Section 1.3], the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. Characterizing hyperbolic graphs is a main problem in the theory of hyperbolicity; since this is a very ambitious goal, a more achievable (yet very difficult) problem is to characterize hyperbolic graphs in particular classes of graphs. The papers [2, 4, 7–9, 11, 12, 25, 27, 30–32, 37, 39] study the hyperbolicity of complement of graphs, chordal graphs, periodic planar graphs, planar graphs, strong product graphs, line graphs, Cartesian product graphs, cubic graphs, short graphs, median graphs, and different generalizations of chordal graphs; however, characterizations of the hyperbolicity in the corresponding classes are obtained only in a few of them. In a previous work, [8], periodic planar graphs were considered. In this work, we shall study how hyperbolicity is affected when considering general periodic graphs, not necessarily planar; a simple characterization of the hyperbolic periodic graphs will be obtained. The key ingredient will be the speed at which points and their images under an isometry separate. The general setting is much more complicated than the planar one and the characterization obtained is totally unexpected. X is a geodesic metric space if for every $x,y\in X$ there exists a geodesic joining x and y; denote by [xy] any of such geodesics (since uniqueness of geodesics is not required, this notation is ambiguous, but convenient). It is clear that every geodesic metric space is path-connected. If the metric space X is a graph, [u, v] denotes the edge joining the vertices u and v.

In order to consider a graph G as a geodesic metric space, one must identify any edge $[u,v]\in E(G)$ with the real interval [0, l] $\big (\mathrm{if} l:=L([u,v])\big )$; therefore, any point in the interior of any edge is a point of G and, if the edge [u, v] is considered as a graph with just one edge, then it is isometric to [0, l]. A connected graph G is naturally equipped with a distance defined on its points, induced by taking shortest paths in G, inducing in G the structure of a metric graph. Note that edges can have arbitrary lengths. As usual, the set of vertices of a graph G will be denoted by V(G).

Let $(X,d_X)$ and $(Y,d_Y)$ be two metric spaces. A map $f: X\longrightarrow Y$ is said to be an $(\alpha , \beta )$-quasi-isometric embedding, with constants $\alpha \ge 1,\ \beta \ge 0$ if, for every $x, y\in X$:

$$\begin{aligned} \alpha ^{-1}d_X(x,y)-\beta \le d_Y\big (f(x),f(y)\big )\le \alpha d_X(x,y)+\beta . \end{aligned}$$

The function f is $\varepsilon$-full if for each $y \in Y$ there exists $x\in X$ with $d_Y\big (f(x),y\big )\le \varepsilon$.

A quasi-isometry from X to Y is a map $f: X\longrightarrow Y$ that is an $\varepsilon$-full $(\alpha , \beta )$-quasi-isometric embedding for some $\alpha \ge 1$ and $\beta , \varepsilon \ge 0$. Two metric spaces X and Y are quasi-isometric if there exists a quasi-isometry $f:X\longrightarrow Y$. Quasi-isometry is an equivalence relation on metric spaces.

An $(\alpha ,\beta )$-quasigeodesic of a metric space X is an $(\alpha ,\beta )$-quasi-isometric embedding $\gamma : I\longrightarrow X$, where I is an interval of $\mathbb R.$ A quasigeodesic is an $(\alpha ,\beta )$-quasigeodesic for some $\alpha \ge 1,\ \beta \ge 0$. Note that a (1, 0)-quasigeodesic is a geodesic. A geodesic line is a geodesic with domain $\mathbb {R}$.

This work deals with periodic graphs. A graph G is periodic if there exist a geodesic line $\gamma _0$ and an isometry T of G with the following properties:

• (1) $T\gamma _0\cap \gamma _0= \emptyset$,

• (2) $G{\setminus } \gamma _0$ has two connected components,

• (3) $G{\setminus } \{\gamma _0\cup T\gamma _0\}$ has at least three connected components, two of them, $G_1$ and $G_2$, satisfy $\partial G_1 \subset \gamma _0$ and $\partial G_2 \subset T\gamma _0$, and the subgraph $G^*:= G{\setminus } \{G_1\cup G_2\}$ is connected and $\cup _{n\in \mathbb {Z}} T^n(G^*)=G$.

Such subgraph $G^*$ is a period graph of G.

In what follows and throughout the paper, G will denote a periodic graph and $G^*$ a period graph of G. In fact, given a periodic graph G, we will fix a geodesic line $\gamma _0$, an isometry T and their corresponding period graph $G^*$. By $\eta _0$, we will denote an arc-length parametrization of $\gamma _0$ in G. Let $\eta _k:=T^k \eta _0$ be a parametrization of $T^k \gamma _0$ for any $k\in \mathbb {Z}$. Also, for any function $f:G\rightarrow \mathbb {R}$ denote by $\limsup _{z\rightarrow +\infty , z\in \gamma _0} f(z)$, the limit

$$\begin{aligned} \limsup _{z\rightarrow +\infty , z\in \gamma _0} f(z):= \limsup _{t\rightarrow +\infty } f(\eta _0(t)), \end{aligned}$$

and analogously for any other limit along the curve.

Our main result is the following:

Theorem 1.1

Let G be a periodic graph.

• If  $\inf _{z\in \gamma _0} d_G(z,Tz)>0$, then G is hyperbolic if only if  $G^*$ is hyperbolic and $\displaystyle \lim _{|z| \rightarrow \infty , z \in \gamma _0} d_G(z,Tz) = \infty$.

• If  $\inf _{z\in \gamma _0} d_{G}(z,Tz)=0$, then G is hyperbolic if and only if  $G^*$ is hyperbolic and G has quasi-exponential decay.

For the definition of quasi-exponential decay, let G be a periodic graph with $\inf _{z\in \gamma _0} d_{G}(z,Tz)=0$, let $\eta _0(t)$ be a parametrization of $\gamma _0$ and define $\Phi _{\eta _0}(t)$ as the greatest non-increasing minorant of F(t), where $F(t):=d_G\big (\eta _0(t),T\eta _0(t)\big )$ on $[0,\infty )$. The graph G has quasi-exponential decay if there exist a parametrization $\eta _0(t)$ for which $\lim _{t \rightarrow -\infty } d_G\big (\eta _0(t),T\eta _0(t)\big ) = \infty$ and

$$\begin{aligned} \sup _{s_2 \ge s_1 \ge 0} (s_2-s_1)\frac{\Phi _{\eta _0}(s_2)}{\Phi _{\eta _0}(s_1)} < \infty . \end{aligned}$$

In what follows, we will write $\Phi _{\eta _0}(t)$ as $\Phi (t)$.

Note that such condition is satisfied by any exponential function $\Phi (t)=e^{-at}$. Also, on the other hand, if a positive function $\Phi (t)$ satisfies this condition, then $\Phi (t) \le k e^{-at}$ on $[0,\infty )$ for some $k,a>0$. Consequently, if G has quasi-exponential decay, then $\lim _{t\rightarrow \infty } \Phi (t)=0$ and $\liminf _{t\rightarrow \infty } F(t)=0$. We obtain an equivalent definition of quasi-exponential decay if we replace $\eta _0(t)$ by $\eta _0(t-t_0)$, i.e., if one considers $t\ge t_0$ instead of $t\ge 0$, for any fixed $t_0$.

The outline of the paper is as follows. Section 2 states some definitions and background used throughout the paper. In Sect. 3, some technical and basic results on periodic graphs are presented. Section 4 is devoted to the proof of the first part of Theorem 1.1. Finally, the proof of the second part is shown in Sect. 5.

2 Definitions and Background

If X is a geodesic metric space and $J=\{J_1,J_2,\dots ,J_n\}$ is a polygon, with sides $J_j\subseteq X$, the polygon J is $\delta$-thin if for every $x\in J_i$ the distance $d(x,\cup _{j\ne i}J_{j})\le \delta$. Denote by $\delta (J)$ the sharp thin constant of J, i.e., $\delta (J):=\inf \{\delta : \, J \, \text { is }\delta \text {-thin}\,\}\,.$ If $x_1,x_2,x_3\in X$, a geodesic triangle $\mathcal {T}=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$. The space X is $\delta$-hyperbolic if every geodesic triangle in X is $\delta$-thin. Denote by $\delta (X)$ the sharp hyperbolicity constant of X, i.e., $\delta (X):=\sup \{\delta (\mathcal {T}): \, \mathcal {T} \, \text { is a geodesic triangle in }\,X\,\}.$ The space X is hyperbolic if X is $\delta$-hyperbolic for some $\delta$. Note that if X is $\delta$-hyperbolic, then every geodesic polygon with n sides is $(n-2)\delta$-thin; in particular, every geodesic quadrilateral is $2\delta$-thin. In the classical references on this subject (see, e.g., [5, 17]) appear several different definitions of Gromov hyperbolicity, which are equivalent in the sense that if X is $\delta$-hyperbolic with respect to one definition, then it is $\delta ^{\prime }$-hyperbolic with respect to another definition (for some $\delta ^{\prime }$ related to $\delta$), see for example Theorem A in Sect. 5. The definition that we have chosen has a deep geometric meaning (see, e.g., [17]).

Let X be a metric space, Y a non-empty subset of X and $\varepsilon$ a positive number. The $\varepsilon$-neighborhood of Y in X, denoted by $\mathcal {V}_{\varepsilon }(Y)$ is the set $\{x\in X: d_X(x,Y)\le \varepsilon \}$. The Hausdorff distance between two non-empty subsets Y and Z of X, denoted by $\mathcal {H}_{X}(Y,Z)$ or $\mathcal {H}(Y,Z)$, is the number defined by:

$$\begin{aligned} \inf \{\varepsilon >0: Y\subset \mathcal {V}_{\varepsilon }(Z)\ and \ Z\subset \mathcal {V}_{\varepsilon }(Y)\}. \end{aligned}$$

A useful property of hyperbolic spaces is the invariance of hyperbolicity. Namely, if $f:X\longrightarrow Y$ is an $(\alpha ,\beta )$-quasi-isometric embedding between the geodesic metric spaces X and Y, and if Y is $\delta$-hyperbolic, then X is $\delta ^{\prime }$-hyperbolic, where $\delta ^{\prime }$ is a constant which just depends on $\delta$, $\alpha$, and $\beta$. Besides, if f is $\varepsilon$-full for some $\varepsilon \ge 0$ (a quasi-isometry), then X is hyperbolic if and only if $\,Y$ is hyperbolic. Furthermore, if X is $\delta ^{\prime }$-hyperbolic, then Y is $\delta$-hyperbolic, where $\delta$ is a constant which just depends on $\delta ^{\prime }$, $\alpha$, $\beta$, and $\varepsilon$.

Given a geodesic metric space X and a closed connected subset $X_0\subset X$, the inner distance $d_{X_0}$ is defined by minimizing $d_X$-length of paths contained in $X_0$.

A subspace $X_0$ of a geodesic metric space X is an isometric subspace if the inner distance $d_{X_0}$ satisfies that $d_{X_0}(x,y)=d_X(x,y)$ for all $x,y\in X_0$. If $X_0$ is an isometric subspace of X then every geodesic in $X_0$ is also a geodesic in X, and therefore $\delta (X_0) \le \delta (X)$.

The following lemma shows that in order to prove the hyperbolicity of a geodesic metric space it suffices to consider geodesic triangles verifying a useful property (see [34, Lemma 2.1]):

Lemma A

In any geodesic metric space X,

$$\begin{aligned} \delta (X)= \sup \big \{ \delta (\mathcal {T}): \, \mathcal {T} \,\text { is a geodesic triangle that is a simple closed curve}\,\big \} . \end{aligned}$$

Another fundamental property of hyperbolic spaces is their geodesic stability: if X is a $\delta$-hyperbolic geodesic metric space ($\delta \ge 0$), and $\alpha \ge 1$ and $\beta \ge 0$ are given constants, there exists a constant $H=H(\delta ,\alpha , \beta )$ such that for any pair of $(\alpha ,\beta )$-quasigeodesics g, h with the same endpoints, $\mathcal {H}(g,h)\le H$.

In view of this stability, one can extend the thinness to quasigeodesic polygons:

Lemma 2.1

Let X be a $\delta$-hyperbolic geodesic metric space and P an $(\alpha ,\beta )$-quasigeodesic polygon with n sides in X. Then P is $\Delta$-thin, where $\Delta$ depends only on $n, \delta ,\alpha , \beta$.

Proof

Let $P^{\prime }$ be a geodesic polygon in X with the same vertices as P. By geodesic stability, the Hausdorff distance between a quasigeodesic side in P and its corresponding geodesic side in $P^{\prime }$ is less than or equal to the constant $H=H(\delta ,\alpha , \beta )$. By splitting $P^{\prime }$ in $n-2$ geodesic triangles, one can check that $P^{\prime }$ is $(n-2)\delta$-thin. If p belongs to a side of P, then there exists a point $p^{\prime }$ on its corresponding geodesic side on $P^{\prime }$ at distance from p less than or equal to H; since $P^{\prime }$ is a geodesic polygon with n sides, there exists a point $q^{\prime }$ on the union of the other $n-1$ geodesic sides in $P^{\prime }$ at distance from $p^{\prime }$ less than or equal to $(n-2)\delta$; then, there exists a point q in the union of the corresponding $n-1$ quasigeodesic sides in P at distance from $q^{\prime }$ less than or equal to H, and $d_G(p,q)\le (n-2)\delta +2H$. Hence, P is $\big ((n-2)\delta +2H\big )$-thin. $\square$

3 Technical Results on Periodic Graphs

In this section, some definitions and results which will be used throughout the paper are stated.

The following lemmas will be of use in the proof of Theorem 1.1 (see [8, Lemma 3.9] and the proof of [8, Lemma 3.10]):

Lemma B

Let G be a graph and let $\gamma _0$ be a geodesic line in G such that $G{\setminus } \gamma _0$ has two connected components $G_1^{\prime },G_2^{\prime }$. Define $G_1:=G_1^{\prime }\cup \gamma _0$ and $G_2:=G_2^{\prime }\cup \gamma _0$. If G is $\delta$-hyperbolic, then $G_1,G_2$ are $\delta$-hyperbolic. If $G_1,G_2$ are $\delta$-hyperbolic, then G is $120\delta$-hyperbolic.

A geodesic $\gamma =[xy]$ with $x\in T^jG^*$, $y\in T^kG^*$ and $j\le k$ is a straight geodesic if $\gamma \cap T^i G^*$ is a connected set for every $j\le i\le k$; note that then $\gamma \subset \cup _{i=j}^k T^iG^*$.

The proof of [8, Lemma 3.11] gives:

Lemma C

Let G be a periodic graph such that $G^*$ is $\delta ^*$-hyperbolic and $\lim _{|z| \rightarrow \infty , z \in \gamma _0} d_G(z,Tz) = \infty$. Assume also that there exists $z_0\in \gamma _0$ with $[z_0,Tz_0]\in E(G)$ and $L\big ([z_0,Tz_0]\big )= d_G(\gamma _0,T\gamma _0)>0$. Denote by $\gamma$ a geodesic joining $x\in T^j G^*$ and $y\in T^k G^*$, $j\le k$. Then:

(1) There exists a constant M that depends only on $G^*$ and a straight geodesic $\gamma ^{\prime }$ joining x and y such that $\mathcal {H}(\gamma ,\gamma ^{\prime })\le M$.

(2) There exists a constant N that depends only on $G^*$ such that if $\sigma :=\cup _{n\in \mathbb {Z}} [T^nz_0, T^{n+1}z_0]$ and $j+2\le k$, for each $j<i<k$ there exists a point $z_i \in \gamma ^{\prime }$ with $d_{T^i G^*}(z_i,\sigma \cap T^iG^*) \le N$.

A geometric consequence of the previous lemma is that two geodesics that start at the same copy of $G^*$ and end at the same copy of $G^*$ are at bounded distance in the intermediate copies of $G^*$. Namely,

Lemma 3.1

Under the hypotheses of Lemma C, consider two geodesics $\gamma , \tilde{\gamma }$ in G from points $x, \tilde{x}\in T^jG^*$ to points $y, \tilde{y}\in T^kG^*$, respectively, where $k-j\ge 4$. If $p\in T^iG^*\cap \gamma$ and $q\in T^iG^*\cap \tilde{\gamma }$ with $j+2\le i\le k- 2$, then $d_G(p,q)\le 2M+6N+5d_1$, where $d_1= L\big ([z_0,Tz_0]\big )= d_G(\gamma _0,T\gamma _0)$ and M, N are the constants in Lemma C. Furthermore, if $\gamma$ and $\tilde{\gamma }$ are straight geodesics, then $d_G(p,q)\le 6N+5d_1$.

Proof

By part (1) in Lemma C, it suffices to prove $d_G(p,q)\le 6N+5d_1$ when $\gamma$ and $\tilde{\gamma }$ are straight geodesics. By Lemma C, there exist points $z_i \in T^iG^*\cap \gamma$ and $\tilde{z}_i\in T^iG^*\cap \tilde{\gamma }$ so that

$$\begin{aligned} d_{T^iG^*}(z_i, \sigma \cap T^iG^*), \, d_{T^iG^*}(\tilde{z}_i, \sigma \cap T^iG^*) \le N \end{aligned}$$

for $j+1\le i\le k-1$.

Consider $p\in T^iG^*\cap \gamma$ and $q\in T^iG^*\cap \tilde{\gamma }$, with $j+2\le i\le k-2$. Then,

$$\begin{aligned} d_G(p,z_i) \le \max \big \{d_G(z_{i-1},z_i), d_G(z_i, z_{i+1})\big \} \le 2N+2d_1. \end{aligned}$$

And, identically, $d_G(q,\tilde{z}_i) \le 2N + 2d_1$. Since $d_G(z_i, \tilde{z}_i) \le 2N+d_1$, one gets the desired result. $\square$

The following two lemmas will relate distances among points on $\gamma _0$ and $T\gamma _0$.

Lemma 3.2

Let G be a periodic graph. Assume that there exist $a^{\prime }\in \gamma _0$, $b^{\prime }\in T\gamma _0$ such that

$$\begin{aligned} d_G(a^{\prime },b^{\prime }) \le \eta _1^{-1}(b^{\prime }) - \eta _0^{-1}(a^{\prime }) = d_G(b^{\prime }, Ta^{\prime }). \end{aligned}$$

If $a \in \gamma _0$ so that $\eta _0^{-1}(a)\le \eta _0^{-1}(a^{\prime })$ then, for every $b\in T\gamma _0$

$$\begin{aligned} d_{G}(a,b) \ge \eta _0^{-1}(a) - \eta _1^{-1}(b). \end{aligned}$$

Furthermore, if $\eta _1^{-1}(b) \le \eta _0^{-1}(a)$, then $d_G(a, b) \ge d_G(a, Ta)/2$.

Remark

By symmetry, if $d_G(a^{\prime },b^{\prime }) \le \eta _0^{-1}(a^{\prime }) - \eta _1^{-1}(b^{\prime })$ and if $b \in T\gamma _0$ is so that $\eta _1^{-1}(b)\le \eta _1^{-1}(b^{\prime })$ then $d_{G}(a,b) \ge \eta _1^{-1}(b) - \eta _0^{-1}(a)$ for any $a\in \gamma _0$.

Proof

Seeking for a contradiction assume that there exist $a \in \gamma _0$ and $b \in T\gamma _0$ with $\eta _0^{-1}(a)- \eta _1^{-1}(b)> d_{G}(a,b)$ and $\eta _0^{-1}(a)\le \eta _0^{-1}(a^{\prime })$. Then

$$\begin{aligned} \begin{aligned} d_{G}(b,b^{\prime })&\le d_{G}(b,a) + d_{G}(a,a^{\prime }) + d_{G}(a^{\prime },b^{\prime }) \\&< \eta _0^{-1}(a)- \eta _1^{-1}(b) + \eta _0^{-1}(a^{\prime }) - \eta _0^{-1}(a) + \eta _1^{-1}(b^{\prime })- \eta _0^{-1}(a^{\prime }) \\&= \eta _1^{-1}(b^{\prime }) - \eta _1^{-1}(b) = d_{G}(b,b^{\prime }) \,, \end{aligned} \end{aligned}$$

which is a contradiction. Thus, $\eta _0^{-1}(a)- \eta _1^{-1}(b)\le d_{G}(a,b)$.

If $\eta _1^{-1}(b)\le \eta _0^{-1}(a)$, notice that $d_{G}(b,Ta)= \eta _0^{-1}(a)- \eta _1^{-1}(b)\le d_{G}(a,b)$. Hence, $d_{G}(a,Ta)\le d_{G}(a,b) + d_{G}(b,Ta) \le 2 d_{G}(a,b)$. $\square$

The second lemma relating distances among points on the “boundary” of $G^*$ states:

Lemma 3.3

Let G be a periodic graph and assume that there exist an unbounded sequence $\{\zeta _n\} \subset \gamma _0$ and some constant $c_0$ with $d_{G}(\zeta _n,T\zeta _n) \le c_0$ for every $n\in \mathbb {N}$. Then $d_{G}(z_1,z_2)\le d_{G}(z_1,Tz_2) + c_0$ for every $z_1,z_2\in \gamma _0$. Furthermore, $d_{G}(z_1,Tz_1)\le 2 d_{G}(z_1,Tz_2) + c_0$ and $d_{G}(z_1,T\gamma _0) \le d_{G}(z_1,Tz_1)\le 2 d_{G}(z_1,T\gamma _0) + c_0$.

Proof

Fix $z_1,z_2\in \gamma _0$. Let $\eta _0$ be a fixed arc-length parametrization of $\gamma _0$ with $\eta _0^{-1}(z_1)\ge \eta _0^{-1}(z_2)$. By hypothesis, there exists $n\in \mathbb {N}$ with either $\eta _0^{-1}(\zeta _n)> \eta _0^{-1}(z_1)$ or $\eta _0^{-1}(\zeta _n)< \eta _0^{-1}(z_2)$. Assume that $\eta _0^{-1}(\zeta _n)> \eta _0^{-1}(z_1)$ (the case $\eta _0^{-1}(\zeta _n)< \eta _0^{-1}(z_2)$ is similar). Hence

$$\begin{aligned} d_{G}(Tz_2,Tz_1) + d_{G}(Tz_1,T\zeta _n)= & {} d_{G}(Tz_2,T\zeta _n) \le d_{G}(Tz_2,z_1) + d_{G}(z_1,\zeta _n) \\&+ d_{G}(\zeta _n,T\zeta _n) \,, \end{aligned}$$

and, since T is an isometry and $T\gamma _0$ is a geodesic,

$$\begin{aligned} d_{G}(z_1,z_2) \le d_{G}(z_1,Tz_2) + c_0 \,. \end{aligned}$$

Moreover, $d_{G}(z_1,Tz_1)\le d_{G}(z_1,Tz_2) + d_{G}(Tz_1,Tz_2) \le 2 d_{G}(z_1,Tz_2) + c_0$. $\square$

This last result has two corollaries which will be useful in the proof of the second part of Theorem 1.1. Both give more specific quantitative relations between distances among points. Namely,

Corollary 3.4

Let G be a periodic graph with $\inf _{z\in \gamma _0} d_{G}(z,Tz)=0$. Then $d_{G}(z_1,z_2)\le d_{G}(z_1,Tz_2)$ for every $z_1,z_2\in \gamma _0$. Furthermore, $d_{G}(z_1,Tz_1)\le 2 d_{G}(z_1,Tz_2)$, $d_{G}(z_1,T\gamma _0) \le d_{G}(z_1,Tz_1)\le 2 d_{G}(z_1,T\gamma _0)$ and

$$\begin{aligned} \frac{1}{3} \big (d_{G}(z_1,z_2)+ \max _{i=1,2}\{ d_{G}(z_i,Tz_i)\} \big ) \le d_{G}(z_1,Tz_2) \le d_{G}(z_1,z_2)+ \min _{i=1,2}\{ d_{G}(z_i,Tz_i)\}. \end{aligned}$$

(3.1)

Proof

In order to prove the inequalities previous to (3.1), it suffices to apply Lemma 3.3 for any $c_0>0$ and take the limit as $c_0\rightarrow 0^+$.

The right hand side of (3.1) follows from the triangle inequality and the fact $d_{G}(Tz_1,Tz_2) = d_{G}(z_1,z_2)$. The left hand side follows by symmetry and the previous inequalities. $\square$

Some notation is needed for the second corollary. Given $z \in T^m \gamma _0, w \in T^n \gamma _0$, define $D_G(z,w)$ as follows: if $m= n$, set $D_G(z,w) := d_G(z,w)$; if $m< n$, then

$$\begin{aligned} D_G(z,w) := \inf \left\{ \, \sum _{j=m}^{n-1} \left( d_G(x_j, T^{-1}x_{j+1}) + d_G(T^{-1}x_{j+1}, x_{j+1})\right) + d_G(x_{n}, w) \right\} , \end{aligned}$$

where the infimum is taken among all sets of points $\{x_j\}_{j=m}^n$ with $x_j \in T^j \gamma _0$ and $x_m=z$; finally, if $m>n$ define $D_G(z,w) := D_G(w,z)$. (One can check that the infimum above is in fact a minimum; see, e.g., [6, p.  24]).

Corollary 3.5

Let G be a periodic graph with $\inf _{z\in \gamma _0} d_{G}(z,Tz)=0$. Then $d_{G}(z_1,z_2)\le d_{G}(z_1,T^nz_2)$ and $D_{G}(z_1,T^nz_2)/3 \le d_{G}(z_1,T^nz_2) \le D_{G}(z_1,T^nz_2)$ for every $z_1,z_2\in \gamma _0$ and $n\in \mathbb {Z}$.

Lemma 3.6

Let G be a periodic graph. Assume that there exist an unbounded sequence $\{\zeta _n\} \subset \gamma _0$ and some constant $c_0$ with $d_{G}(\zeta _n,T\zeta _n) \le c_0$ for every $n\in \mathbb {N}$. Then, for each arc-length parametrization $\eta _0$ of $\gamma _0$ one of the following situations holds:

• (1) There exists $R\in \mathbb {R}$ such that if $a\in \gamma _0$, $b\in T^m\gamma _0$ ($m\in \mathbb Z$) with $\eta _0^{-1}(a),\eta _{m}^{-1}(b) \ge R$ then $d_{G}(a,b) \ge \eta _{m}^{-1}(b)- \eta _0^{-1}(a) - c_0.$

• (2) For any $m\ge 0$, $a\in \gamma _0$, $b\in T^m\gamma _0$ then $d_{G}(a,b) \ge \eta _{m}^{-1}(b)- \eta _0^{-1}(a).$

• (3) For any $m\le 0$, $a\in \gamma _0$, $b\in T^m\gamma _0$ then $d_{G}(a,b) \ge \eta _{m}^{-1}(b)- \eta _0^{-1}(a).$

  (Recall the notation $\eta _m=T^m\circ \eta _0$ for a parametrization of $T^m\gamma _0$.)

Proof

Case 1. Suppose that there exists $R\in \mathbb {R}$ so that

$$\begin{aligned} d_{G}(z,w)\ge |\eta _0^{-1}(z)- \eta _1^{-1}(w)| \end{aligned}$$

(3.2)

for all $z\in \eta _0([R,\infty ))$ and $w\in \eta _1([R,\infty ))$.

Let $a\in \gamma _0$ and $b\in T^m\gamma _0$ with $\eta _m^{-1}(b) \ge \eta _0^{-1}(a) \ge R$ and $m\ge 0$ (if $\eta _m^{-1}(b) < \eta _0^{-1}(a)$, then $d_{G}(a,b) \ge 0 > \eta _{m}^{-1}(b)- \eta _0^{-1}(a) - c_0$). Let g be a straight geodesic joining a to b and choose points $u_j\in g\cap T^j\gamma _0$, for $0\le j \le m$, with $a= u_0$ and $b=u_m$. If $\eta _j^{-1}(u_j)\ge R$ for $0\le j\le m$ then by (3.2),

$$\begin{aligned} d_{G}(a,b)= & {} \displaystyle \sum _{j=0}^{m-1} d_{G}(u_j,u_{j+1})\ge \displaystyle \sum _{j=0}^{m-1}\big (\eta _{j+1}^{-1}(u_{j+1}) - \eta _{j}^{-1}(u_{j})\big )\\= & {} \eta _{m}^{-1}(u_{m}) - \eta _0^{-1}(u_0)=\eta _m^{-1}(b) - \eta _0^{-1}(a)\,. \end{aligned}$$

Otherwise, there exists $0<j_0<m$ such that $\eta _j^{-1}(u_j)\ge R$ for all $j_0< j\le m$ and $\eta ^{-1}_{j_0}(u_{j_0}) < R$. Then,

$$\begin{aligned} d_{G}(a,b) = \sum _{j=0}^{m-1} d_{G}(u_j,u_{j+1}) \ge \sum _{j=j_0}^{m-1} d_{G}(u_j,u_{j+1})\,. \end{aligned}$$

By Lemma 3.3,

$$\begin{aligned} d_{G}(u_{j_0},u_{j_0+1}) \ge \eta _{j_0+1}^{-1}(u_{j_0+1}) - \eta _{j_0}^{-1}(u_{j_0}) - c_0 \, , \end{aligned}$$

and by (3.2),

$$\begin{aligned} d_{G}(u_{j},u_{j+1}) \ge \eta _{j+1}^{-1}(u_{j+1}) - \eta _{j}^{-1}(u_{j})\, , \quad j_0< j \le m-1 \, . \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned} d_{G}(a,b)&\ge \eta _{j_0+1}^{-1}(u_{j_0+1}) - \eta _{j_0}^{-1}(u_{j_0}) - c_0 + \sum _{j=j_0+1}^{m-1} \left( \eta _{j+1}^{-1}(u_{j+1}) - \eta _{j}^{-1}(u_j) \right) \\&= \eta _{m}^{-1}(u_{m}) - \eta _{j_0}^{-1}(u_{j_0}) - c_0 \ge \eta _m^{-1}(b) - \eta _0^{-1}(a) - c_0 \, , \end{aligned} \end{aligned}$$

where the last inequality follows from the fact that $\eta ^{-1}_{j_0}(u_{j_0}) < R \le \eta ^{-1}_{0} (a)$. The same argument works when $m<0$.

Case 2. Suppose that there exist a sequence $R_k\nearrow \infty$ and sequences $z_k\in \eta _0([R_k, \infty ))$, $w_k\in \eta _1([R_k, \infty ))$ so that $d(z_k, w_k) < \eta _0^{-1}(z_k) - \eta _1^{-1}(w_k)$.

As above, let g be a straight geodesic joining a to b and choose points $u_j\in g\cap T^j\gamma _0$, for $0\le j \le m$, with $a= u_0$ and $b=u_m$. There exists k such that $\eta _j^{-1}(u_j) < R_k$ for every $0\le j\le m$. By (remark after) Lemma 3.2,

$$\begin{aligned} d_G(u_j, u_{j+1}) \ge \eta _{j+1}^{-1}(u_{j+1}) - \eta _{j}^{-1}(u_j) \end{aligned}$$

and thus,

$$\begin{aligned} d_{G}(a,b)= & {} \sum _{j=0}^{m-1} d_G(u_j, u_{j+1}) \ge \sum _{j=0}^{m-1} \left( \eta _{j+1}^{-1}(u_{j+1}) - \eta _{j}^{-1}(u_j) \right) \\= & {} \eta _{m}^{-1}(u_{m}) - \eta _{0}^{-1}(u_{0}) = \eta _{m}^{-1}(b) - \eta _{0}^{-1}(a)\,. \end{aligned}$$

Case 3. Suppose that there exist a sequence $R_k\nearrow \infty$, and sequences $z_k\in \eta _0([R_k, \infty ))$, $w_k\in \eta _1([R_k, \infty ))$ such that $d(z_k, w_k) < \eta _1^{-1}(w_k)- \eta _0^{-1}(z_k)$. Let g be the straight geodesic from a to b and define points $u_j:=g\cap T^{-j}\gamma _0$, for $0\le j \le |m|$, with $a= u_0$ and $b=u_{|m|}$. There exists k such that $\eta _{-j}^{-1}(u_{j}) < R_k$ for every $0\le j\le |m|$. By Lemma 3.2,

$$\begin{aligned} d_G(u_j, u_{j+1}) \ge \eta _{-j-1}^{-1}(u_{j+1}) - \eta _{-j}^{-1}(u_j) \end{aligned}$$

and thus,

$$\begin{aligned} d_{G}(a,b)= & {} \sum _{j=0}^{|m|-1} d_G(u_j, u_{j+1}) \ge \sum _{j=0}^{|m|-1} \left( \eta _{-j-1}^{-1}(u_{j+1}) - \eta _{-j}^{-1}(u_j) \right) \\= & {} \eta _{m}^{-1}(u_{|m|}) - \eta _{0}^{-1}(u_{0}) = \eta _{m}^{-1}(b) - \eta _{0}^{-1}(a)\,. \end{aligned}$$

$\square$

4 Proof of the First Part of Theorem 1.1

This section is devoted to the proof of the first part of Theorem 1.1. For clarity’s sake, we shall begin by stating some lemmas and claims which will be used along the proof.

The first lemma introduces a new graph, $G^{\prime }$ (quasi-isometric to G) which will guarantee the existence of a transversal geodesic.

Lemma 4.1

Let G be a periodic graph such that $d_G(\gamma _0,T\gamma _0) =:d_1 > 0$. Fix $z_0\in \gamma _0$ and define $G^{\prime }$ by adding to G the edges $\big \{ [T^{n}z_0,T^{n+1}z_0]\big \}_{n\in \mathbb {Z}}$ with $L\big ([T^{n}z_0,T^{n+1}z_0]\big )=d_1$ for every $n\in \mathbb {Z}$. Then, the graphs $G^{\prime }$ and G are quasi-isometric and, moreover, $\cup _{n\in \mathbb {Z}} [T^{n}z_0,T^{n+1}z_0]$ is a geodesic in $G^{\prime }$.

Proof

It is clear that $\cup _{n\in \mathbb {Z}} [T^{n}z_0,T^{n+1}z_0]$ is a geodesic in $G^{\prime }$. It will be shown that the inclusion $i:G \rightarrow G^{\prime }$ is a quasi-isometry. Clearly, the inequality $d_{G^{\prime }}(x,y) \le d_G(x,y)$ holds for every $x,y\in G$.

Consider $x,y\in G$. If x, y are so that $d_{G^{\prime }}(x,y) = d_G(x,y)$, then there is nothing to prove. If $d_{G^{\prime }}(x,y) < d_G(x,y)$, then there exist $m,n\in \mathbb {Z}$ such that $d_{G^{\prime }}(x,y) = d_G(x,T^m z_0)+ d_{G^{\prime }}(T^m z_0,T^n z_0)+ d_G(T^n z_0,y)$. Hence,

$$\begin{aligned} \begin{aligned} d_{G}(x,y)&\le d_G(x,T^m z_0)+ d_{G}(T^m z_0,T^n z_0)+ d_G(T^n z_0,y) \le d_G(x,T^m z_0)\\&\quad + |m-n|d_{G}(z_0,T z_0)+ d_G(T^n z_0,y) \\&\le \frac{d_{G}(z_0,T z_0)}{d_1} \big ( d_G(x,T^m z_0)+ |m-n|d_1+ d_G(T^n z_0,y) \big ) \\&= \frac{d_{G}(z_0,T z_0)}{d_1} \big ( d_G(x,T^m z_0)+ d_{G^{\prime }}(T^m z_0,T^n z_0)+ d_G(T^n z_0,y) \big )\\&= \frac{d_{G}(z_0,T z_0)}{d_1} \, d_{G^{\prime }}(x,y)\,. \end{aligned} \end{aligned}$$

Since $L([T^{n}z_0,T^{n+1}z_0])=d_1$ for every $n\in \mathbb {Z}$, the map i is $(d_1/2)$-full, and we conclude that $G^{\prime }$ and G are quasi-isometric. $\square$

The next lemma will show that a certain curve on the graph G is a quasi-geodesic.

Lemma 4.2

Let G be a periodic graph such that $\inf _{z\in \gamma _0} d_G(z,Tz) =:d_0 > 0$. Let $\zeta \in \gamma _0$ and let $\sigma$ be a geodesic in $G^*$ joining $\zeta$ and $T\zeta$. Then, for each $m\in \mathbb {N}$ the curve $\sigma ^m:=\bigcup _{j=0}^{m-1} T^j\sigma$ is an $(\alpha _0, \beta _0)$-quasi-geodesic in G, with $\alpha _0, \beta _0$ depending only on $d_G(\zeta , T\zeta )$, $d_0$ and $d_G(\gamma _0, T\gamma _0)$.

In fact, the explicit expressions for $\alpha _0$ and $\beta _0$ will be obtained in the proof of this lemma.

Proof

Notice that $\sigma ^m$ is a continuous curve in G joining $\zeta$ and $T^m \zeta$. Define $c_0:=d_G(\zeta ,T\zeta )$. Fix an arc-length parametrization of $\sigma ^m$ starting at $\zeta$ and $s,t\in \mathbb {R}$ in the domain of $\sigma ^m$ with $s<t$. Clearly $d_G(\sigma ^m(t),\sigma ^m(s)) \le L(\sigma ^m|_{[s,t]})=t-s$. Let $j,r\in \mathbb {N}$ be so that $\sigma ^m(s) \in T^j\sigma$ and $\sigma ^m(t) \in T^{j+r}\sigma$. The following inequality holds

$$\begin{aligned} t-s \le (r+1)L(\sigma ) = (r+1)d_{G}(\zeta ,T\zeta ) = (r+1)c_0 \,. \end{aligned}$$

(4.1)

For the lower bound, notice first that if $d_1:=d_G(\gamma _0, T\gamma _0)>0$,

$$\begin{aligned} d_G(\sigma ^m(t),\sigma ^m(s))\ge (r-1)d_1 =(r+1)d_1-2d_1 \ge \frac{d_1}{c_0} \, (t-s) -2d_1\,. \end{aligned}$$

Assume next that $d_G(\gamma _0, T\gamma _0)=0$. Since $d_0>0$, there exist monotonous unbounded sequences $\{z_n^{\prime }\} \subset \gamma _0$ and $\{w_n^{\prime }\} \subset T\gamma _0$ with $d_{G}(z^{\prime }_n,w^{\prime }_n)<d_0/2$. Fix an arc-length parametrization $\eta _0$ of $\gamma _0$ such that there exists a subsequence $\{z^{\prime }_{n_k}\}$ with $\lim _{k\rightarrow \infty } \eta _0^{-1}(z^{\prime }_{n_k})=\infty$; without loss of generality by replacing $\{z^{\prime }_n\}$ by the subsequence $\{z^{\prime }_{n_k}\}$ if necessary, one can assume that $\lim _{k\rightarrow \infty } \eta _0^{-1}(z^{\prime }_{n})=\infty$. Recall the notation for $\eta _k$.

Assume that $\eta _1^{-1}(w_n^{\prime }) - \eta _0^{-1}(z_n^{\prime }) \ge 0$ for infinitely many $n^{\prime }s$ (otherwise, the argument is symmetric). By choosing a subsequence if necessary, one can assume without loss of generality that $\eta _1^{-1}(w_n^{\prime }) - \eta _0^{-1}(z_n^{\prime }) \ge 0$ for every n. Then,

$$\begin{aligned} \eta _1^{-1}(w_n^{\prime }) - \eta _0^{-1}(z_n^{\prime })= & {} d_{G}(w_n^{\prime },Tz_n^{\prime }) \ge d_{G}(z_n^{\prime },Tz_n^{\prime }) - d_{G}(z_n^{\prime },w_n^{\prime }) \nonumber \\> & {} d_0 - \frac{d_0}{2} = \frac{d_0}{2} \ge d_G(z_n^{\prime }, w_n^{\prime })\, . \end{aligned}$$

(4.2)

Let $s^{\prime }\le s \le t \le t^{\prime }$ such that $\sigma ^m(s^{\prime })$ is the first point of $\sigma ^m$ in $T^j\sigma$ and $\sigma ^m(t^{\prime })$ is the last point of $\sigma ^m$ in $T^{j+r}\sigma$; then $d_G(\sigma ^m(s^{\prime }),\sigma ^m(s))=s-s^{\prime }\le c_0$ and $d_G(\sigma ^m(t^{\prime }),\sigma ^m(t))=t^{\prime }-t\le c_0$. Let $\Gamma$ be a geodesic joining $\sigma ^m(s^{\prime })$ and $\sigma ^m(t^{\prime })$. Define $x_0:=\sigma ^m(s^{\prime })\in T^{j}\gamma _0$, $x_{r+1}:=\sigma ^m(t^{\prime })\in T^{j+r+1}\gamma _0$, and let $x_{i}$ be any point of $\Gamma$ in $T^{j+i}\gamma _0$ for $1\le i \le r$.

Define $N_{1},N_{21},N_{22},$ as the sets of indices

$$\begin{aligned} \begin{aligned} N_{1}&:= \big \{ 0 \le i \le r: \quad \eta _{j+i}^{-1}(x_{i})\ge \eta _{j+i+1}^{-1}(x_{i+1}) \big \}\, , \\ N_{21}&:= \big \{ 0 \le i \le r: \quad \eta _{j+i}^{-1}(x_{i})< \eta _{j+i+1}^{-1}(x_{i+1}) \text { and } d_{G}(x_i,x_{i+1}) \ge d_0/2 \big \}\, , \\ N_{22}&:= \big \{ 0 \le i \le r: \quad \eta _{j+i}^{-1}(x_{i})< \eta _{j+i+1}^{-1}(x_{i+1}) \text { and } d_{G}(x_i,x_{i+1}) < d_0/2 \big \}\, . \end{aligned} \end{aligned}$$

Then ${{\mathrm{card\,}}}N_{1}+{{\mathrm{card\,}}}N_{21}+{{\mathrm{card\,}}}N_{22}=r+1$. For $i\in N_1$, $\eta _{j+i}^{-1}(x_i) \ge \eta _{j+i+1}^{-1}(x_{i+1})$. Take $n\in \mathbb {N}$ so that $\eta _{0}^{-1}(z^{\prime }_n) > \eta _{j+i}^{-1}(x_i)$. Then, by (4.2) the points $x_i$ and $x_{i+1}$ are under the hypothesis of Lemma 3.2, and hence

$$\begin{aligned} d_G(x_i, x_{i+1}) \ge \eta _{j+i}^{-1}(x_i) - \eta _{j+i+1}^{-1}(x_{i+1})= & {} d_G(x_{i+1}, Tx_i ) \ge d_G(x_{i}, Tx_i ) \\ - d_G(x_{i+1},x_i)\ge & {} d_0 - d_G(x_i,x_{i+1}) \end{aligned}$$

and conclude $d_G(x_i, x_{i+1}) \ge d_0/2$.

If ${{\mathrm{card\,}}}N_{1}+{{\mathrm{card\,}}}N_{21} \ge (r+1)/2$, then

$$\begin{aligned} d_{G}(\sigma ^m(s),\sigma ^m(t)) +2c_0 \ge d_{G}\big (\sigma ^m(s^{\prime }),\sigma ^m(t^{\prime })\big ) = \sum _{i=0}^r d_{G}(x_i,x_{i+1}) \ge \frac{d_0}{4}\,(r+1)\,. \end{aligned}$$

Hence, by (4.1),

$$\begin{aligned} d_{G}(\sigma ^m(t),\sigma ^m(s)) \ge \frac{d_0}{4} (r+1) -2 c_0 \ge \frac{d_0}{4 c_0} (t-s) - 2 c_0 \,. \end{aligned}$$

Assume now that ${{\mathrm{card\,}}}N_{22}\ge (r+1)/2$. Note that if $i \in N_{22}$, then

$$\begin{aligned} \eta _{j+i+1}^{-1}(x_{i+1}) - \eta _{j+i}^{-1}(x_{i})= & {} d_G(x_{i+1}, Tx_{i}) \ge d_G(x_{i}, Tx_{i}) - d_G(x_{i+1}, x_{i})\\\ge & {} d_0 - \frac{d_0}{2} = \frac{d_0}{2}, \end{aligned}$$

and therefore

$$\begin{aligned} \sum _{i\in N_{22}} \left( \eta _{j+i+1}^{-1}(x_{i+1}) - \eta _{j+i}^{-1}(x_{i}) \right) \ge \frac{d_0}{2} \,{{\mathrm{card\,}}}N_{22} \ge \frac{d_0}{4} (r+1). \end{aligned}$$

Note that

$$\begin{aligned} \sum _{i\in N_{22}} \left( \eta _{j+i+1}^{-1}(x_{i+1}) - \eta _{j+i}^{-1}(x_{i}) \right)\le & {} \sum _{i\in N_{22}\cup N_{21}} \left( \eta _{j+i+1}^{-1}(x_{i+1}) - \eta _{j+i}^{-1}(x_{i}) \right) \\= & {} \sum _{i\in N_{1}} \left( \eta _{j+i}^{-1}(x_{i}) - \eta _{j+i+1}^{-1}(x_{i+1}) \right) \end{aligned}$$

since $\eta ^{-1}_{j+r+1} (x_{r+1}) = \eta ^{-1}_{j}(x_0)$. Therefore, applying Lemma 3.2,

$$\begin{aligned} \begin{aligned} \sum _{i\in N_{1}} \left( \eta _{j+i}^{-1}(x_{i}) - \eta _{j+i+1}^{-1}(x_{i+1}) \right)&\le \sum _{i\in N_{1}} d_G( x_{i} , x_{i+1} )\\&\le \sum _{i=0}^r d_{G}(x_i,x_{i+1}) = d_{G}(\sigma ^m(s^{\prime }),\sigma ^m(t^{\prime })) \\&\le d_{G}(\sigma ^m(s),\sigma ^m(t)) + 2 c_0\,. \end{aligned} \end{aligned}$$

Hence,

$$\begin{aligned} \begin{aligned} d_{G}\big (\sigma ^m(t),\sigma ^m(s)\big )&\ge \frac{d_0}{4} (r+1) -2 c_0 \ge \frac{d_0}{4 c_0} (t-s) - 2 c_0\, . \end{aligned} \end{aligned}$$

One concludes that $\sigma ^m$ is an $(\alpha _0,\beta _0)$-quasigeodesic (for every m), where $\alpha _0=c_0/d_1$ if $d_1>0$ (note that $c_0 \ge d_0 \ge d_1$), $\alpha _0=4c_0/d_0$ if $d_1=0$, and $\beta _0=\max \{2c_0,2d_1\}$. $\square$

With these previous lemmas established, let us proceed to prove the first part of Theorem 1.1, the main goal of this section.

Proof

(First part of Theorem 1.1 ). Assume first that G is hyperbolic. Since $\gamma _0$ and $T\gamma _0$ are geodesic lines, $G^*$ is an isometric subgraph of G and $\delta (G^*) \le \delta (G)$. Thus, it remains to show that $\lim _{|z|\rightarrow \infty , z\in \gamma _0} d_G(z,Tz) = \infty$.

Assume that there exists an unbounded sequence $\{\zeta _n\}_{n\ge 1} \subset \gamma _0$ and a constant $c_0$ with $d_G(\zeta _n, T\zeta _n) \le c_0$ for every n. Choosing a subsequence of $\{\zeta _n\}_{n\ge 1}$ if it is necessary, one can assume that there exists an arc-length parametrization $\eta _0$ of $\gamma _0$ with $\eta _0^{-1}(\zeta _n)\nearrow \infty$. Let $\sigma _n$ be a geodesic in $G^*$ joining $\zeta _n$ and $T\zeta _n$. Let $\sigma _n^m:=\cup _{k=0}^{m-1} T^k\sigma _n$ and $\gamma _0^n$ be the subcurve of $\gamma _0$ joining $\zeta _{n_0}$ and $\zeta _n$, where $n_0$ is chosen as follows: if (1) in Lemma 3.6 holds, take $n_0$ with $\eta _0^{-1}(\zeta _{n_0}) \ge R$; otherwise, take $n_0=1$. Hence, by Lemma 4.2, $Q_{n,m}:=\{ \gamma _0^n, \sigma _n^m, T^m\gamma _0^n, \sigma _{n_0}^m \}$ is an $(\alpha _0, \beta _0)$-quasigeodesic quadrilateral for every n, m, where $\alpha _0$ and $\beta _0$ do not depend on n and m.

Since G is hyperbolic, by Lemma 2.1, $Q_{n,m}$ is $(2\delta (G)+2H)$-thin, with $H=H(\delta (G),\alpha _0,\beta _0)$ for any n, m. Let M be a constant with $M>2\delta (G)+2H$.

Taking $n\in \mathbb {N}$ large enough, $L(\gamma _0^n) > 2M + 4c_0$, and taking $m=m(n)$ large enough, $d_G\left( \gamma _0^n, T^m\gamma _0^n\right) > M$. Choose a point $p\in \gamma _0^n$ so that,

• (1) $d_G(p,\zeta _{n_0})=\eta _0^{-1}(p)-\eta _0^{-1}(\zeta _{n_0})> M + 2c_0$,

• (2) $d_G(p, \zeta _n) = \eta _0^{-1} (\zeta _n) - \eta _0^{-1} (p) > M + 2c_0$.

We also have $d_G(p, T^m\gamma _0^n)\ge d_G(\gamma _0^n,T^m\gamma _0^n) > M$.

Let us proceed to show that $d_G(p, \sigma _{n_0}^m)>M$. Let $V^m$ be the set of points $V^m:=\left\{ \zeta _{n_0}, T\zeta _{n_0}, T^2\zeta _{n_0}, \dots , T^m\zeta _{n_0} \right\}$. By the triangle inequality, it is enough to show that $d_G(p, V^m) > M + c_0$.

Case I. Assume that (1) in Lemma 3.6 holds. Since $R \le \eta _0^{-1}(\zeta _{n_0}) = \eta _k^{-1}(T^k\zeta _{n_0}) < \eta _0^{-1}(p)$ for $0\le k \le m$, Lemma 3.6 (1) gives,

$$\begin{aligned} d_G(p, T^k\zeta _{n_0}) \ge \eta _0^{-1}(p) - \eta _0^{-1}(\zeta _{n_0})-c_0 > M+c_0 \, , \end{aligned}$$

thus $d_G(p, V^m) > M + c_0$.

Case II. Suppose that (2) in Lemma 3.6 holds. Then,

$$\begin{aligned} d_G(p, T^k\zeta _{n_0}) \ge \eta _0^{-1}(p) - \eta _k^{-1}(T^k\zeta _{n_0}) = \eta _0^{-1}(p) - \eta _0^{-1} (\zeta _{n_0}) > M+ 2c_0 \, , \end{aligned}$$

thus $d_G(p, V^m) > M + 2 c_0 > M + c_0$.

Case III. If (3) in Lemma 3.6 holds, the argument in case II gives the result, taking now $m\le k\le 0$.

A similar argument shows also that $d_G(p,\sigma _n^m)>M$. Hence, $d_G(p,T^m\gamma _0^n \cup \sigma _{n_0}^m \cup \sigma _n^m)>M$. Since $M>2\delta (G)+2H$, the quadrilateral $Q_{n,m}$ is not $(2\delta (G)+2H)$-thin, which is a contradiction. Therefore, G is not hyperbolic.

Let us prove the converse implication to conclude that G is hyperbolic. Since $\lim _{|z|\rightarrow \infty , z\in \gamma _0} d_G(z,Tz) = \infty$, then $d_G(\gamma _0,T\gamma _0)=:d_1>0$. By Lemma 4.1, without loss of generality one can assume that there exists a vertex $z_0 \in V(G)\cap \gamma _0$ such that $[z_0,Tz_0]\in E(G)$, with $L([z_0,Tz_0])= d_G(\gamma _0,T\gamma _0)=d_1$, and so that $\sigma _0:=\cup _{n\in \mathbb {Z}} [T^{n}z_0,T^{n+1}z_0]$ is a geodesic in G. Define $\delta ^*:=\delta (G^*)$ and consider a geodesic triangle $\mathcal {T}=\{x_1,x_2,x_3\}$ with $x_i\in T^{j_i}G^*$ and $j_1\le j_2\le j_3$. By Lemma C, one can assume that the geodesics of $\mathcal {T}$ are straight.

Suppose first that $\max \{j_2-j_1, j_3-j_2\} \le 2$. Then, $\mathcal {T} \subset \cup _{j=j_2-2}^{j_2+2} T^jG^*$ is $\delta _0$-thin, with $\delta _0=(120)^4\delta ^*$ since $T^jG^*$ is $\delta ^*$-hyperbolic (apply at most four times Lemma B). Otherwise, $\mathcal {T} \cap \big ( T^{j_2-1}\gamma _0 \cup T^{j_2+2}\gamma _0\big ) \ne \emptyset$. If $\mathcal {T} \cap \big ( T^{j_2-1}\gamma _0 \big ) \ne \emptyset$, choose $y_1 \in [x_1 x_2]\cap T^{j_2-1}\gamma _0$ and $y_2 \in [x_1 x_3]\cap T^{j_2-1}\gamma _0$. By Lemma 3.1,

$$\begin{aligned} d_G(y_1, y_2) \le 6N+5d_1\,. \end{aligned}$$

(4.3)

Analogously, if $\mathcal {T} \cap \big ( T^{j_2+2}\gamma _0\big ) \ne \emptyset$, let $z_1 \in [x_1 x_3]\cap T^{j_2+2}\gamma _0$ and $z_2 \in [x_2 x_3]\cap T^{j_2+2}\gamma _0$. Again, by Lemma 3.1,

$$\begin{aligned} d_G(z_1, z_2) \le 6N+5d_1\,. \end{aligned}$$

(4.4)

Let $p\in \mathcal {T}$. If $p\in T^jG^*$ with $j\in [j_1+2, j_2-2] \cup [j_2+2, j_3-2]$, apply Lemma 3.1 to find $q\in T^jG^*$ on another side of $\mathcal {T}$ with $d_G(p,q) \le 6N+5d_1$.

If $p\in T^jG^*$ with $j\in [j_2-1, j_2+1]$, let $\mathcal {P} \subset \cup _{j=j_2-1}^{j_2+1} T^jG^*$ be the geodesic polygon formed by $\mathcal {T} \cap \cup _{j=j_2-1}^{j_2+1} T^jG^*$ and $[y_1 y_2] \subset T^{j_2-1}\gamma _0$ and $[z_1 z_2] \subset T^{j_2+2}\gamma _0$ whenever they exist. Thus, $\mathcal {P}$ is either a pentagon or a quadrilateral contained in $\cup _{j=j_2-2}^{j_2+2} T^jG^*$ and therefore it is $3\delta _0-$thin. Therefore, there exists a point $q^{\prime } \in \mathcal {P}$ on another side of $\mathcal {P}$ so that $d_G(p,q^{\prime }) \le 3\delta _0$. If $q^{\prime }\notin \mathcal {T}$, then $q^{\prime }\in [y_1 y_2] \cup [z_1 z_2]$ and equations (4.3) and (4.4) imply that there is $q\in \mathcal {P}\cap \mathcal {T}$ on another side of $\mathcal {T}$ with $d_G(p,q) \le 3\delta _0 + 6N + 5d_1$.

If $p\in T^jG^*$ with $j\in \{j_1,j_1+1,j_3-1,j_3\}$, a similar argument with a triangle (in $T^{j_1} G^*\cup T^{j_1+1}G^*$ or $T^{j_3-1}G^*\cup T^{j_3}G^*$) instead of $\mathcal {P}$ gives $d_G(p,q)\le \delta _0+6N+5d_1$.

Hence, $\delta (\mathcal {T}) \le 3\delta _0 + 6N + 5d_1$ and Lemma C gives $\delta (G)\le 2M+3\delta _0+6N+5d_1$. $\square$

5 Proof of the Second Part of Theorem 1.1

To prove the second part of Theorem 1.1, some auxiliary metric spaces will be defined, and some results relating these new sets with the original one will be given.

Let G be a periodic graph. Sometimes we will require the arc-length parametrization $\eta _0$ of $\gamma _0$ to also satisfy:

$$\begin{aligned} 0=\liminf _{t\rightarrow \infty } d_G(\eta _0(t), T\eta _0(t)) \le \limsup _{t\rightarrow \infty } d_G(\eta _0(t), T\eta _0(t)) < \infty . \end{aligned}$$

(5.1)

Fix $t_0\in \mathbb {R}$ and $\eta _0$. Define $G_1$ as the geodesic metric space given by $G \cup \big ( \cup _{n\in \mathbb {Z}, t\ge t_0} U_{n,t} \big )$, where $U_{n,t}$ is a segment joining $T^n \eta _0(t)$ with $T^{n+1} \eta _0(t)$ of length $d_G(\eta _0(t), T\eta _0(t))$. Set $G_2$ to be the geodesic metric space given by $\big ( \cup _{n\in \mathbb {Z}} T^n \eta _0([t_0,\infty )) \big )\cup \big ( \cup _{n\in \mathbb {Z}, t\ge t_0} U_{n,t} \big )$. The isometry T can be extended to $G_1$ in an obvious way; also denote this extension by T. Define a period graph of $G_1$ as $G_1^*:=G^* \cup \big ( \cup _{t\ge t_0} U_{0,t} \big )$. Below, the constant $t_0$ will be chosen as the constant in Lemma 5.12.

It is clear that $G,G_2$ are contained in $G_1,$ $G \cup G_2=G_1,$ and G is an isometric subspace of $G_1$; thus $\delta (G) \le \delta (G_1)$.

With these definitions in mind, let us state some results on hyperbolicity.

Lemma 5.1

If a periodic graph G is hyperbolic and satisfies (5.1) and $\liminf _{t\rightarrow -\infty } d_G(\eta _0(t), T\eta _0(t))>0$, then $G_2$ is hyperbolic.

Proof

Given any fixed $t_0\in \mathbb {R}$, the hypotheses imply that there exist constants M, m such that $d_G\big (\eta _0(t), T\eta _0(t)\big ) \le M$ for every $t \in [t_0,\infty )$ and $d_G\big (\eta _0(t),T\eta _0(t)\big )\ge m$ for every $t\in (-\infty ,t_0]$; then every segment $U_{n,t}$ has length at most M and $D_G\le d_{G_2}\le (M/m)D_G$ on $\cup _{n\in \mathbb {Z}} T^n \eta _0([t_0,\infty ))$. Consider the map $f:G_2 \rightarrow G$ defined by $f(x)=T^n \eta _0(t)$ for every $x\in U_{n,t} {\setminus } T^{n+1} \eta _0(t)$. By Corollary 3.5, the restriction of f to $\cup _{n\in \mathbb {Z}} T^n \eta _0([t_0,\infty ))$ (the identity map) is a (3M / m, 0)-quasi-isometric embedding. Since $L(U_{n,t}) \le M$ for every $n\in \mathbb {Z}, t\ge t_0,$ f is a quasi-isometric embedding and invariance of hyperbolicity gives the result. $\square$

Lemma 5.2

Consider a periodic graph G satisfying (5.1). Then $G^*$ is hyperbolic if and only if $G_1^*$ is hyperbolic.

Proof

By (5.1), there exists a constant M such that $d_G\big (\eta _0(t), T\eta _0(t)\big ) \le M$ for every $t \in [t_0,\infty )$; then every segment $U_{n,t}$ has length at most M. The inclusion map $i:G^* \rightarrow G_1^*$ is a (M / 2)-full (1, 0)-quasi-isometry, and thus, the invariance of hyperbolicity gives the result. $\square$

Finally, the last auxiliary space will be defined and its hyperbolicity related to that of G will be stated.

Given $t_0\in \mathbb {R}$ and $\eta _0$, define $G_3$ as the geodesic metric space given by $\big ( \cup _{n\in \mathbb {Z}} T^n \eta _0([t_0,\infty )) \big )\cup \big ( \cup _{n\in \mathbb {Z}, t\ge t_0} V_{n,t} \big )$, where $V_{n,t}$ is a segment joining $T^n \eta _0(t)$ with $T^{n+1} \eta _0(t)$ of length $\Phi (t)$, where $\Phi$ is the greatest non-increasing minorant of $d_G\big (\eta _0(t),T\eta _0(t)\big )$ on $[t_0,\infty )$, i.e., $\Phi (t)=\min \big \{ d_G\big (\eta _0(s),T\eta _0(s)\big ):\, s\in [t_0,t] \big \}$.

Lemma 5.3

Let G be a periodic graph satisfying (5.1) and $\sup \big \{t_2-t_1: \, \Phi (t_1)=\Phi (t_2),\, t_2\ge t_1 \ge t_0 \big \} < \infty$. Then $G_2$ and $G_3$ are quasi-isometric.

Proof

Consider the map $f:G_3 \rightarrow G_2$ defined as the identity on $\cup _{n\in \mathbb {Z}} T^n \eta _0([t_0,\infty ))$ and as a dilation on each $V_{n,t}$ with $f(V_{n,t})=U_{n,t}$ for every $n\in \mathbb {Z}, t\ge t_0$.

Clearly, f is 0-full and $d_{G_2}(f(x),f(y)) \ge d_{G_3}(x,y)$ for every $x,y \in G_3$. By (5.1), there exists a constant M such that $L(U_{n,t}) \le M$ for every $n\in \mathbb {Z}, t\ge t_0$. Also $L(V_{n,t}) \le L(U_{n,t}) \le M$ for every $n\in \mathbb {Z}, t\ge t_0$. Define $N:=\sup \big \{t_2-t_1: \, \Phi (t_1)=\Phi (t_2),\, t_2\ge t_1 \ge t_0 \big \} < \infty$.

Given $x_0\in T^m \eta _0([t_0,\infty ))$ and $y_0\in T^n \eta _0([t_0,\infty ))$ with $m\le n$, let $\gamma$ be a geodesic in $G_3$ joining $x_0$ and $y_0$ such that $\gamma = [x_0 \eta _m(t)] \cup V_{m,t} \cup \cdots \cup V_{n-1,t} \cup [\eta _n(t) y_0]$ for some $t\ge t_0$. Let $t^{\prime }\ge t$ be defined as $t^{\prime }:=\sup \big \{s: \, \Phi (s)=\Phi (t),\, s\ge t \big \} \le t+N$; thus $d_{G_3}\big (\eta _0(t^{\prime }),T\eta _0(t^{\prime })\big )=\Phi (t^{\prime })=\Phi (t)$ and $L(V_{k,t}) = L(U_{k,t^{\prime }})$ for every $k\in \mathbb {Z}$. Consider the curve $\gamma _1$ in $G_2$ joining $x_0$ and $y_0$ given by $\gamma _1:= [x_0 \eta _m(t^{\prime })] \cup U_{m,t^{\prime }} \cup \cdots \cup U_{n-1,t^{\prime }} \cup [\eta _n(t^{\prime }) y_0]$; then $d_{G_2}(f(x_0),f(y_0)) \le L(\gamma _1) \le L(\gamma )+2N = d_{G_3}(x_0,y_0)+2N$.

Finally, since $L(V_{n,t}) \le L(U_{n,t}) \le M$ for every $n\in \mathbb {Z}, t\ge t_0,$ given $x,y \in G_3,$ then $d_{G_2}(f(x),f(y))\le d_{G_3}(x,y)+2N+2M$. $\square$

Lemmas 5.1 and 5.3 and the invariance of hyperbolicity, imply the following result.

Lemma 5.4

Let G be a periodic graph satisfying (5.1), $\liminf _{t\rightarrow -\infty } d_G(\eta _0(t),T\eta _0(t))>0$ and $\sup \big \{t_2-t_1: \, \Phi (t_1)=\Phi (t_2),\, t_2\ge t_1 \ge t_0 \big \} < \infty$. If G is hyperbolic, then $G_3$ is hyperbolic.

Recall the definition of quasi-exponential decay given below Theorem 1.1.

Lemma 5.5

Let G be any periodic graph. If G has quasi-exponential decay, then, for any fixed $t_0$, $\sup \{t_2-t_1: \ \Phi (t_1)=\Phi (t_2), \, t_2\ge t_1\ge t_0\}<\infty$ and (5.1) holds.

Proof

Fix $t_0$ and let $K:=\sup _{s_2 \ge s_1 \ge t_0} (s_2-s_1)\Phi (s_2)/\Phi (s_1) < \infty$. If $t_2\ge t_1 \ge t_0$ and $\Phi (t_1)=\Phi (t_2)$, then $t_2-t_1=(t_2-t_1)\Phi (t_2)/\Phi (t_1)\le K$. Recall that $\liminf _{t\rightarrow \infty } F(t)=0$ and that $\Phi (t) \le F(t)$. Given $\varepsilon >0$, take $t_\varepsilon = \inf \{t\in \mathbb {R}: \Phi (s) \le \varepsilon for all s\ge t \}$. Clearly, $F(t_\varepsilon ) = \Phi (t_\varepsilon ) = \varepsilon$. Let $t>t_\varepsilon$. If $F(t) = \Phi (t)$, then $F(t) \le \varepsilon < K+\varepsilon$. Otherwise $F(t) > \Phi (t)$ and there exist $t_1, t_2$ such that $t_\varepsilon \le t_1<t<t_2$ and $F(t_1) =\Phi (t_1) = \Phi (t) = \Phi (t_2) = F(t_2)\le \varepsilon$. Then, $F(t) > F(t_1)$ and, since F is Lipschitz, $F(t) -F(t_1) \le 2(t-t_1)$, $F(t) -F(t_2) \le 2(t_2-t)$, and thus $F(t) \le t_2-t_1+F(t_1) \le t_2-t_1+\varepsilon$. Using that $t_2-t_1 \le K$, one deduces $F(t)\le K+\varepsilon$. Consequently, $\limsup _{t\rightarrow \infty } F(t)\le K < \infty$ and (5.1) holds. $\square$

Given a periodic graph G, a geodesic in $G_3$ is a fundamental geodesic if it is equal to $\cup _{n=n_1}^{n_2} V_{n,t}$ for some $n_1,n_2 \in \mathbb {Z}, t\ge t_0$. Define $\mathfrak {L}(G_3):= \sup \big \{ L(\gamma ):\, \gamma \,\text { is a fundamental geodesic in }\, G_3 \big \}$.

Lemma 5.6

Let G be a periodic graph.

(1) If $\mathfrak {L}(G_3)=\infty$, then $G_3$ is not hyperbolic.

(2) $\mathfrak {L}(G_3)<\infty$ if and only if $\sup _{s_2 \ge s_1 \ge t_0} (s_2-s_1)\Phi (s_2)/\Phi (s_1) < \infty$. In fact, if $\sup _{s_2 \ge s_1 \ge t_0} (s_2-s_1)\Phi (s_2)/\Phi (s_1)=:K < \infty$, then $\mathfrak {L}(G_3)\le 8K$.

Proof

(1) Assume first that $\mathfrak {L}(G_3)=\infty$. Note that if $\cup _{n=n_1}^{n_2} V_{n,t}$ is a fundamental geodesic, then $\cup _{n=n_1+k}^{n_2+k} V_{n,t}$ is also a fundamental geodesic for every $k\in \mathbb {Z}$; hence,

$$\begin{aligned} \mathfrak {L}(G_3)=\sup \{L(\gamma ):\, \gamma = \cup _{n=0}^{n_2} V_{n,t} \text { is a fundamental geodesic in } G_3 \}\,. \end{aligned}$$

Consider any fixed fundamental geodesic $\sigma =\cup _{n=0}^{n_2} V_{n,t}$ for some $n_2 \in \mathbb {N}, t\ge t_0,$ with $L(\sigma )=\ell$. Since $\mathfrak {L}(G_3)=\infty$, one can find $t^{\prime }\ge t+\ell$ such that $\sigma ^{\prime }=\cup _{n=0}^{n_2} V_{n,t^{\prime }}$ is also a fundamental geodesic. Define $\sigma _1:=\eta _{0}([t,t^{\prime }]), \, \sigma _2:=\eta _{n_2+1}([t,t^{\prime }])$ and the geodesic quadrilateral $Q:=\{\sigma ,\sigma _1, \sigma _2, \sigma ^{\prime } \}$.

If $p=\eta _{0}(t+\ell /4)$, then $d_{G_3}(p,\sigma )=\ell /4$, $d_{G_3}(p,\sigma ^{\prime })\ge 3\ell /4$; choose $s\ge 0$ so that $d_{G_3}(p,\sigma _2)=s+(1+n_2)\Phi (s+t+\ell /4)$. If $s > \ell /4$, then $d_{G_3}(p,\sigma _2)\ge s > \ell /4$. If $0\le s \le \ell /4$, then $d_{G_3}(p,\sigma _2)\ge 2(s+ \ell /4)- 3\ell /4 +(1+n_2)\Phi (s+t+\ell /4)$. Since $\sigma$ is a geodesic, $\ell \le 2(s+\ell /4) + (1+n_2)\Phi (s+t+\ell /4)$, and therefore, $d_{G_3}(p, \sigma _2) \ge \ell -3\ell /4 = \ell /4$. Hence, $2\delta (G_3) \ge \delta (Q) \ge \ell /4$ and we conclude that $G_3$ is not hyperbolic, since $\mathfrak {L}(G_3)=\infty$.

(2) Assume now that $l:=\mathfrak {L}(G_3)<\infty$. Let $s_1\ge t_0$ and $n\in \mathbb {N}$ with $n \Phi (s_1)> l$. Therefore, $\cup _{k=0}^{n-1} V_{k,s_1}$ is not a geodesic joining $\eta _{0}(s_1)$ and $\eta _{n}(s_1)$; then there exits $s_{2,n}>s_1$ with $n \Phi (s_1)> 2(s_{2,n} - s_1) + n\Phi (s_{2,n}) = d_{G_3}\big (\eta _{0}(s_1),\eta _{n}(s_1)\big )$. It is possible to choose the sequence $\{s_{2,n}\}$ with $s_{2,n+1} \ge s_{2,n}$. Hence, $2(s_{2,n} - s_1)<n \Phi (s_1)$, $\cup _{k=0}^{n-1} V_{k,s_{2,n}}$ is a fundamental geodesic and $n\Phi (s_{2,n})\le l$. We conclude that $2(s_{2,n} - s_1)\Phi (s_{2,n})/\Phi (s_{1}) < n \Phi (s_1) \Phi (s_{2,n})/\Phi (s_{1}) \le l$.

Furthermore, $d_{G_3}\big (\eta _{0}(s_{2,n}),\eta _{n+1}(s_{2,n})\big )\le (n+1)\Phi (s_{2,n})\le 2n\Phi (s_{2,n}) \le 2l$. Since any sub-arc of a geodesic is again a geodesic, it is clear that $2(s_{2,n+1} - s_{2,n}) < 2(s_{2,n+1} - s_{2,n})+ (n+1)\Phi (s_{2,n+1})\le (n+1)\Phi (s_{2,n}) \le 2l$, and then $s_{2,n+1} < s_{2,n} + l$. If $s_{2} \in [s_{2,n} , s_{2,n+1}]$, then

$$\begin{aligned} (s_2-s_1)\frac{\Phi (s_2)}{\Phi (s_1)} < (s_{2,n}+l-s_1)\frac{\Phi (s_{2,n})}{\Phi (s_1)} \le \frac{l}{2}+ l\,\frac{\Phi (s_{2,n})}{\Phi (s_1)} \le \frac{3l}{2}\,. \end{aligned}$$

Let $n_0$ be the least integer such that $n_0 \Phi (s_1)> l$. Thus, $n_0 \Phi (s_1)= (n_0-1) \Phi (s_1) + \Phi (s_1)\le l + \Phi (t_0)$ and $2(s_{2,n_0} - s_{1}) < 2(s_{2,n_0} - s_{1})+n_0\Phi (s_{2,n_0})\le n_0\Phi (s_{1}) \le l + \Phi (t_0)$. If $s_{2} \in [s_{1} , s_{2,n_0}]$, then

$$\begin{aligned} (s_2-s_1)\frac{\Phi (s_2)}{\Phi (s_1)} \le s_{2,n_0}-s_1 \le \frac{1}{2} \big ( l + \Phi (t_0) \big ), \end{aligned}$$

and we conclude, since $\mathfrak {L}(G_3)<\infty$ implies $\lim _{n\rightarrow \infty } s_{2,n}=\infty$, that

$$\begin{aligned} \sup _{s_2 \ge s_1 \ge t_0} (s_2-s_1)\frac{\Phi (s_2)}{\Phi (s_1)} \le \max \Big \{ \frac{3l}{2}\,, \, \frac{1}{2} \big ( l + \Phi (t_0) \big )\Big \}. \end{aligned}$$

For the reverse implication, let $K:=\sup _{s_2 \ge s_1 \ge t_0} (s_2-s_1)\Phi (s_2)/\Phi (s_1) < \infty$. Then, any fundamental geodesic $\cup _{k_1\le n < k_2}V_{n,s}$ satisfies

$$\begin{aligned} \begin{aligned} (k_2-k_1)\Phi (s)&\le 2K+ (k_2-k_1)\Phi (s+2K) + 2K \le 4K+ (k_2-k_1) K \frac{\Phi (s)}{2K}\,, \\ L\big ( \cup _{k_1\le n < k_2}V_{n,s} \big )&= (k_2-k_1)\Phi (s) \le 8K . \end{aligned} \end{aligned}$$

Notice that this means that for a fixed s, a fundamental geodesic cannot cross arbitrarily many $T^n\gamma _0(s)$. $\square$

Lemma 5.7

Let G be any periodic graph with quasi-exponential decay. Then $G_3$ is hyperbolic.

Proof

It will be enough to show this result for triangles whose sides are certain geodesics which will be introduced below, the canonical geodesics, since any other geodesic of $G_3$ will be close to one of these.

Consider a parametrization $\eta _0$ of $\gamma _0$ satisfying

$$\begin{aligned} \sup _{s_2 \ge s_1 \ge t_0} (s_2-s_1) \Phi (s_2)/\Phi (s_1) =: K< \infty . \end{aligned}$$

(5.2)

Let $x_1,x_2\in \cup _{n\in \mathbb {Z}} T^n \eta _0([t_0,\infty ))$. Without loss of generality, $x_1 = T^{n_1} \eta _0(t_1)$ and $x_2 = T^{n_2} \eta _0(t_2)$ with $n_1 \le n_2$. Define $g(t):=t-t_1 + (n_2-n_1)\Phi (t)+t-t_2$, and let $t^{\prime }$ be such that

$$\begin{aligned} g(t^{\prime }) = \inf \big \{g(t):\, t\ge \max \{t_1,t_2\} \big \} . \end{aligned}$$

Note that this infimum is, in fact, a minimum, and that the curve

$$\begin{aligned} \gamma _{x_1x_2}:=[x_1T^{n_1} \eta _0(t^{\prime })]\cup (\cup _{n_1\le n < n_2}V_{n,t^{\prime }})\cup [T^{n_2} \eta _0(t^{\prime })x_2] \end{aligned}$$

is a geodesic in $G_3$ with $d_{G_3}(x_1,x_2)=L(\gamma _{x_1 x_2})=g(t^{\prime })$, referred to as a canonical geodesic joining $x_1$ and $x_2$. If $n_1 = n_2$, then $\gamma _{x_1x_2}$ is a segment on $T^{n_1}\gamma _0$.

Any other canonical geodesic $\sigma$ in $G_3$ joining $x_1$ and $x_2$ will be at a fixed distance from a canonical geodesic: indeed, if there exists another canonical geodesic with $g(t^{\prime \prime }) = g(t^{\prime })$ (one can assume that $t^{\prime \prime }\ge t^{\prime }$), then $8K \ge (n_2-n_1)\Phi (t^{\prime })=2(t^{\prime \prime }-t^{\prime }) + (n_2-n_1)\Phi (t^{\prime \prime })$ by Lemma 5.6, and hence $t^{\prime \prime }-t^{\prime } \le 4K$.

More generally, if $\sigma$ is any geodesic joining $x_1$ and $x_2$ which contains just one fundamental geodesic, $\cup _{n_1\le n < n_2}V_{n,t}$, for which $t_0\le t<\max \{t_1, t_2\}:=\tau$, then $\Phi (\tau ) = \Phi (t)$ and the curve $\sigma ^{\prime }:= [x_1T^{n_1} \eta _0(\tau )]\cup (\cup _{n_1\le n < n_2}V_{n,\tau })\cup [T^{n_2} \eta _0(\tau )x_2]$ is a canonical geodesic. By (5.2), $\tau - t \le K$; since $t^{\prime }-\tau \le 4K$, $t^{\prime }-t\le 5K$, and thus $\mathcal {H}(\sigma ,\gamma _{x_1x_2}) \le 5K + \Phi (t_0)/2$.

Finally, if $\sigma$ contains at least two fundamental geodesics, applying the same argument one also gets $\mathcal {H}(\sigma ,\gamma _{x_1x_2}) \le 5 K + \Phi (t_0)/2$.

Consider a geodesic triangle $\mathcal {T}=\{x_1,x_2,x_3\}$ in $G_3$ with its vertices lying on $\cup _{n\in \mathbb {Z}} T^n\gamma _0$, concretely, $x_1 = T^{n_1} \eta _0(t_1)$, $x_2 = T^{n_2} \eta _0(t_2)$ and $x_3 = T^{n_3} \eta _0(t_3)$ with $n_1 \le n_2 \le n_3$. Let $\mathcal {T}_0$ be the geodesic triangle in $G_3$ given by $\mathcal {T}_0=\{\gamma _{x_1x_2},\gamma _{x_2x_3},\gamma _{x_1x_3}\}$. If $\mathcal {T}_0$ is $\delta$-thin, then $\mathcal {T}$ is $(\delta +10K + \Phi (t_0))$-thin.

There exist three fundamental geodesics $g_{12}:=\cup _{n_1\le n < n_2}V_{n,s_1} \subseteq \gamma _{x_1x_2}$, $g_{23}:=\cup _{n_2\le n < n_3}V_{n,s_2} \subseteq \gamma _{x_2x_3}$ and $g_{13}:=\cup _{n_1\le n <, n_3}V_{n,s_3} \subseteq \gamma _{x_1x_3}$. Assume that $s_1 \le s_2 \le s_3$ (the other cases are similar). Note that $L(\cup _{n_1\le n < n_2}V_{n,s_2}) \le L(\cup _{n_1\le n < n_2}V_{n,s_1}) = L(g_{12}) \le 8K$; thus $L(\cup _{n_1\le n < n_3}V_{n,s_2}) \le 16K$ and $s_3-s_2 \le 8K$. Clearly, from these estimates, if p lies on one side of $\mathcal {T}_0$, then the distance from p to the union of the other two sides is less than 24K. Any other combination of vertices $x_1$, $x_2$, $x_3$ gives the same estimate.

Hence, $\delta (\mathcal {T}_0)\le 24K$ and $\delta (\mathcal {T})\le 34K + \Phi (t_0)$. Consequently, if H is any geodesic hexagon in $G_3$ with every vertex in $\cup _{n\in \mathbb {Z}} T^n \eta _0([t_0,\infty ))$, then $\delta (H)\le 4(34K + \Phi (t_0))=136K + 4\Phi (t_0)$.

Consider now any fixed geodesic triangle $\mathcal {T}=\{x_1,x_2,x_3\}$ in $G_3$ that is a simple closed curve. Assume that $x_1,x_2,x_3\notin \cup _{n\in \mathbb {Z}} T^n \eta _0([t_0,\infty ))$ (the other cases are similar). For each $x_i$, there exist $n_i\in \mathbb {Z}$ and $t_i \ge 0$ such that $x_i \in V_{n_i, t_i}$; let $x_i^{\prime }$ and $x_i^{\prime \prime }$ be the endpoints of $V_{n_i, t_i}$; since $\mathcal {T}$ is a simple closed curve, $V_{n_i, t_i} \subset \mathcal {T}$. Consider the geodesic hexagon $H=\{x_1^{\prime },x_1^{\prime \prime },x_2^{\prime },x_2^{\prime \prime },x_3^{\prime },x_3^{\prime \prime }\}$. Since the vertices of H lie on $\cup _{n\in \mathbb {Z}} T^n\eta _0([t_0, \infty ))$, $\delta (H)\le 136K + 4\Phi (t_0)$.

Given $p\in T$, denote by $\delta (p)$ the distance from p to the union of the two other sides of T. Assume p lies on a side of H that is contained in a side of T. Then, $\delta (p) \le \delta (H) + L(V_{n_i, t_i})$ for some $i=1, 2, 3$. Since $L(V_{n_i, t_i}) \le \Phi (t_i) \le \Phi (t_0)$, then $\delta (p)\le \delta (H) + \Phi (t_0) \le 136K + 5\Phi (t_0)$.

If p lies on $V_{n_i, t_i}$, ($i=1,2,3$), then $\delta (p) \le L(V_{n_i,t_i}) \le \Phi (t_0)$. Hence, $\delta (p)\le 136K + 5\Phi (t_0)$ and $G_3$ is $(136K + 5\Phi (t_0))$-hyperbolic by Lemma A. $\square$

Let G be a periodic graph with quasi-exponential decay. Fix $a\le b$ in $\{-\infty \} \cup \mathbb {Z}\cup \{\infty \}$. Define $G_3^{a,b}\subseteq G_3$ as the geodesic metric space given by $\big ( \cup _{a\le n\le b+1} T^n \eta _0([t_0,\infty )) \big )\cup \big ( \cup _{a\le n\le b, t\ge t_0} V_{n,t} \big )$. Lemmas B and 5.7 have the following consequence.

Corollary 5.8

Let G be any periodic graph with quasi-exponential decay. Then there exists a constant $\delta$ such that $G_3^{a,b}$ is $\delta$-hyperbolic for every $a\le b$ in $\{-\infty \} \cup \mathbb {Z}\cup \{\infty \}$.

Next, some results on curves which are shown to be quasi-geodesic are given. The aim will be to construct a quasi-geodesic quadrilateral with large $\delta$. Recall the definition of $D_G(z,w)$ given before Corollary 3.5.

Let G be a periodic graph. In the next lemma, for $t\in \mathbb {R}$ and fixed $s_1<s_2$, define $\phi _t$ as a geodesic in G joining $\eta _0(s_2+t)$ with $T\eta _0(s_2+t)$, $\psi _t$ as a geodesic joining $\eta _0(s_1-t)$ with $T\eta _0(s_1-t)$, and the curves $\xi _{n,t}:=\eta _0\big ([s_2,s_2+t]\big )\cup \phi _t \cup T\phi _t\cup \cdots \cup T^{n-1}\phi _t \cup T^{n}\eta _0\big ([s_2,s_2+t]\big )$, $\zeta _{n,t}:=\eta _0\big ([s_1,s_1-t]\big )\cup \psi _t \cup T\psi _t\cup \cdots \cup T^{n-1}\psi _t \cup T^{n}\eta _0\big ([s_1,s_1-t]\big )$ parameterized by arc-length.

Lemma 5.9

Let G be a periodic graph with $\inf _{z\in \gamma _0} d_{G}(z,Tz)=0$. Let $s_1<s_2$ and define the constants $c_1:=d_G\big (\eta _0(s_1), T\eta _0(s_1)\big )$, $c_2:=d_G\big (\eta _0(s_2), T\eta _0(s_2)\big )$ and $c^*:=\max \{c_1,c_2\}$. Let $n\in \mathbb {N}$ and $c\in \mathbb {R}^+$ be so that $c^*n\le 2 (s_2-s_1)$ and $d_G\big (\eta _0(s), T\eta _0(s)\big )\ge c$ for all $s\in [s_1,s_2]$. If $r,u \ge 0$ satisfy $L(\xi _{n,r})=\min _{t\ge 0} L(\xi _{n,t})$ and $L(\zeta _{n,u})=\min _{t\ge 0} L(\zeta _{n,t})$, then the quadrilateral $Q:=\{\eta _0\big ([s_1,s_2]\big ),\xi _{n,r},T^n \eta _0\big ([s_1,s_2]\big ),\zeta _{n,u}\}$ is a $(3c^*/c,2c^*)$-quasigeodesic quadrilateral and $\delta (Q) \ge c(n-2)/12$. In particular, if n is the integer part of $2(s_2-s_1)/c^*$, then $\delta (Q) \ge c(s_2-s_1)/(6c^*)-c/4$.

Proof

To show that Q is a quasi-geodesic quadrilateral, it suffices to show that $\xi _{n,r}$ and $\zeta _{n,t}$ are quasi-geodesics. In fact, by symmetry, it is enough to show it just for, e.g., $\xi _{n,r}$.

Let $\xi _{n,r} (s)$ and $\xi _{n,r} (t)$ be any two points on $\xi _{n,r}$. Without loss of generality, $t\ge s$. Since $\xi _{n,r}$ is parameterized by arc-length, $d_G\big (\xi _{n,r}(s),\xi _{n,r}(t))\le L_G(\xi _{n,r}|_{[s,t]}\big ) = t-s$.

For the lower bound, suppose $\xi _{n,r}(s) \in T^{j_1} G^*$, $\xi _{n,r}(t) \in T^{j_2-1} G^*$ with $0\le j_1 < j_2 \le n$. Assume that $\xi _{n,r}(s), \xi _{n,r}(t) \notin \eta _0\big ([s_2,s_2+r]\big ) \cup T^n\eta _0\big ([s_2,s_2+r]\big )$ (the other cases are similar). Let $t_1\le s\le t \le t_2$ be so that $\xi _{n,r}(t_1) \in T^{j_1} \gamma _0$ and $\xi _{n,r}(t_2) \in T^{j_2} \gamma _0$.

Recall the definition of $D_G$. By Corollary 3.5, it will be enough to bound $D_G$ below.

Note that $D_G\big (\xi _{n,r}(t_1),\xi _{n,r}(t_2)\big ) = \sum _{j=j_1}^{j_2-1} \big (d_G(x_j, T^{-1}x_{j+1}) + d_G(T^{-1}x_{j+1}, x_{j+1}) \big ) + d_G\big (x_{j_2},\xi _{n,r}(t_2)\big )$ for appropriate $\{x_j\}$. Choose i so that $j_1 \le i < j_2$ and $d_G\big (T^{-1}x_{i+1}, x_{i+1}\big ) = \min _{j_1 \le j < j_2} d_G\big (T^{-1}x_{j+1}, x_{j+1}\big )$. Consider $\eta _k:=T^k \eta _0$ as a parametrization of $T^k \gamma _0$ for any $k\in \mathbb {Z}$. Then

$$\begin{aligned}&d_G\big (\xi _{n,r}(t_1), T^{j_1-i-1}x_{i+1}\big ) \nonumber \\&\quad \quad + (j_2-j_1) \, d_G\big (T^{-1}x_{i+1}, x_{i+1}\big ) + d_G\big (T^{j_2-i-1}x_{i+1}, \xi _{n,r}(t_2)\big ) \nonumber \\&\quad \le \sum _{j=j_1}^{j_2-1} \big (d_G(x_j, T^{-1}x_{j+1}) + d_G(T^{-1}x_{j+1}, x_{j+1}) \big ) + d_G(x_{j_2},\xi _{n,r}(t_2)) \nonumber \\&\quad \le (j_2-j_1) \, d_G\big (\eta _0(s_2+r),T\eta _0(s_2+r)\big ) \,. \end{aligned}$$

(5.3)

If the second inequality in (5.3) is an equality, then $D_G(\xi _{n,r}(t_1),\xi _{n,r}(t_2))=t_2-t_1$ and $d_G(\xi _{n,r}(t_1),\xi _{n,r}(t_2))\ge (t_2-t_1)/3$. Otherwise, the second inequality in (5.3) is strict.

Define $a:=\eta _{i+1}^{-1} (x_{i+1})$. Then (5.3) gives that $L(\xi _{n,a-s_2}) < L(\xi _{n,r})$. Therefore $a<s_2$ by the definition of $\xi _{n,r}$. Also, $a>s_1$, since otherwise $L(\xi _{n,r}) > L(\xi _{n,a-s_2}) > 2(s_2-s_1) \ge c_2n = L(\xi _{n,0}) \ge L(\xi _{n,r})$.

Hence $s_1 < a < s_2$ and then $d_G(T^{-1}x_{i+1}, x_{i+1}) \ge c = d_G\big (\eta _0(s_2),T\eta _0(s_2)\big ) c/c_2$ and (5.3) gives

$$\begin{aligned} \begin{aligned} D_G\big (\xi _{n,r}(t_1),\xi _{n,r}(t_2)\big )&\ge d_G\big (\xi _{n,r}(t_1), T^{j_1-i-1}x_{i+1}\big ) + (j_2-j_1) \, d_G\big (T^{-1}x_{i+1}, x_{i+1}\big ) \\&\quad +d_G\big (T^{j_2-i-1}x_{i+1}, \xi _{n,r}(t_2)\big )\\&\ge \frac{c}{c_2} (j_2-j_1) \, d_G\big (\eta _0(s_2),T\eta _0(s_2)\big ) \\&\ge \frac{c}{c_2}\, (j_2-j_1) \, d_G\big (\eta _0(s_2+r),T\eta _0(s_2+r)\big ) \\&= \frac{c}{c_2}\, (t_2-t_1) \,. \end{aligned} \end{aligned}$$

By Corollary 3.5, $(t_2-t_1) c/(3c_2) \le d_G\big (\xi _{n,r}(t_1),\xi _{n,r}(t_2)\big )$, and, by the triangle inequality,

$$\begin{aligned} d_G\big (\xi _{n,r}(s),\xi _{n,r}(t)\big )\ge & {} d_G\big (\xi _{n,r}(t_1),\xi _{n,r}(t_2)\big ) - 2c_2 \ge \frac{c}{3c_2} (t_2-t_1)- 2c_2 \\\ge & {} \frac{c}{3c_2} (t-s)- 2c_2\, . \end{aligned}$$

Any other case gives the same inequality. Thus, $\xi _{n,r}$ is a $(3c_2/c,2c_2)$-quasigeodesic.

Finally, let us estimate $\delta (Q)$.

Let p be the midpoint in $\eta _0([s_1,s_2])$. By Corollary 3.5,

$$\begin{aligned} d_G\big (p,\xi _{n,r}\cap (\cup _k T^k \gamma _0)\big ) \ge d_G(p,\eta _0(s_2)) = \frac{s_2-s_1}{2} \ge \frac{ c^* n}{4}\, . \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned} d_G(p,\xi _{n,r})&\ge d_G\big (p,\xi _{n,r}\cap (\cup _k T^k \gamma _0)\big ) - (1/2) d_G\big (\eta _0(s_2+r),T\eta _0(s_2+r)\big ) \\&\ge d_G\big (p,\xi _{n,r}\cap (\cup _k T^k \gamma _0)\big ) - (1/2) d_G\big (\eta _0(s_2),T\eta _0(s_2)\big ) \ge \frac{c^* n}{4} - \frac{c^*}{2}\\&= \frac{ c^* (n-2)}{4}\,. \end{aligned} \end{aligned}$$

Similarly, $d_G (p,\zeta _{n,u}) \ge c^*(n-2)/4$.

As above, $D_G \big (p,T^n \eta _0([s_1,s_2])\big ) \ge \min \{ cn, \,(s_2-s_1)/2\} \ge \min \{ cn, \,c^* n/4\} \ge cn/4$ and then, by Corollary 3.5, $d_G \big (p,T^n \eta _0([s_1,s_2])\big ) \ge cn/12$ and, since $c\le c^*$, $\delta (Q) \ge c(n-2)/12$. $\square$

For Lemma 5.10 below, it will be useful to keep in mind the definition of fine triangles. Given a geodesic triangle $T=\{x,y,z\}$ in a geodesic metric space X, let $T_E$ be a Euclidean triangle with sides of the same length than T. Since there is no possible confusion, denote the corresponding points in T and $T_E$ by the same letters. The maximum inscribed circle in $T_E$ meets the side [xy] (respectively [yz], [zx]) in a point $z^{\prime }$ (respectively $x^{\prime }$, $y^{\prime }$) such that $d(x,z^{\prime })=d(x,y^{\prime })$, $d(y,x^{\prime })=d(y,z^{\prime })$ and $d(z,x^{\prime })=d(z,y^{\prime })$. We call the points $x^{\prime },y^{\prime },z^{\prime },$ the internal points of $\{x,y,z\}$. There is a unique isometry f of the triangle $\{x,y,z\}$ onto a tripod (a star graph with one vertex w of degree 3, and three vertices $x_0,y_0,z_0$ of degree one, such that $d(x_0,w)=d(x,z^{\prime })=d(x,y^{\prime })$, $d(y_0,w)=d(y,x^{\prime })=d(y,z^{\prime })$, and $d(z_0,w)=d(z,x^{\prime })=d(z,y^{\prime })$). The triangle $\{x,y,z\}$ is $\delta$-fine if $f(p)=f(q)$ implies that $d(p,q)\le \delta$. The space X is $\delta$-fine if every geodesic triangle in X is $\delta$-fine.

There are two definitions of Gromov hyperbolicity (the second one is the definition of fine space) whose equivalence will be useful to quantify (see, e.g, [17, Proposition 2.21, p. 41]):

Theorem A

Let us consider a geodesic metric space X.

(1) If X is $\delta$-hyperbolic, then it is $4\delta$-fine.

(2) If X is $\delta$-fine, then it is $\delta$-hyperbolic.

Finally, for Lemma 5.10 below, some notation needs to be introduced. Let G be a periodic graph. Fix a parametrization $\eta _0$ of $\gamma _0$ and $t_0\in \mathbb {R}$. Consider points $x \in T^{n} G^*$, $y \in T^{n+k}G^*$, with $n\in \mathbb {N}$, $k\ge 4$, so that if $\gamma$ is a straight geodesic in G from x to y, then there exists $x_j\in \gamma \cap T^{n+j} \gamma _0$ with $s_j:=\eta _{n+j}^{-1}(x_j) \ge t_0$ for $2\le j \le k-1$.

In $G_1$, consider the curves $g_{j} := U_{n+j,s_j} \cup [x_{j+1}Tx_j]$ joining $x_{j}$ and $x_{j+1}$ for $2\le j \le k-2$, and the curve $g := [xx_{1}] \cup [x_{1}x_{2}] \cup (\cup _{(2\le j \le k-2)} g_j) \cup [x_{k-1}x_{k}] \cup [x_{k}y]$ joining x and y in $G_1$.

Lemma 5.10

With the above notation, if G satisfies (5.1) and $G^*$ is hyperbolic, then g with its arc-length parametrization is an $(\alpha ,\beta )$-quasi-geodesic in $G_1$ and $\mathcal {H}_{G_1}(g,\gamma ) \le H$, where $\alpha ,\beta$ and H are constants depending just on $\delta (G_1^*)$ and $M:= \sup _{t \ge t_0} d_G\big (\eta _0(t), T\eta _0(t)\big )$. In fact, $(\alpha ,\beta ) = \big (3,8\delta (G_1^*)+6M\big )$.

Proof

Let $\gamma :[0,l_0] \rightarrow G$ be an arc-length parametrization of $\gamma$ and let $g:[0,l] \rightarrow G_1$ be an arc-length parametrization of g; then $d_{G_1}\big (g(t_1),g(t_2)\big )\le |t_1-t_2|$ for every $t_1,t_2 \in [0,l]$.

To obtain a lower bound, note that $M<\infty$ by (5.1); then every segment $U_{n,t}$ with $t \ge t_0$ has length at most M. Fix $t_1,t_2 \in [0,l]$ with $t_1< t_2$. Assume first that $g(t_1),g(t_2)\in T^{n+j} G_1^*$ for some j with $2\le j \le k-2$. Consider the geodesic triangle $\mathcal {T}_j=\{[x_jx_{j+1}],U_{n+j,s_j}, [x_{j+1}Tx_j]\}$ in $T^{n+j} G_1^*$. Since $G^*$ is hyperbolic, $G_1^*$ is hyperbolic by Lemma 5.2 and the triangle $\mathcal {T}_j$ is $4\delta (G_1^*)$-fine by Theorem A.

Let $[a_0,b_0]:=\gamma ^{-1}\big ([x_jx_{j+1}]\big )$, $[a,b]:=g^{-1}(g_j)$ and $c:=g^{-1}(Tx_j)$. By the triangle inequality, $b_0-a_0 \le b-a$, thus one can choose $c_1, c_2 \in [a,b]$ such that $c-c_1 = c_2-c >0$ satisfying $(c_1-a)+(b-c_2) = b_0-a_0$. Finally, pick $c_0\in [a_0, b_0]$ with $c_1-a = c_0-a_0$ and $b-c_2 = b_0-c_0$.

Define $u:[a,b] \rightarrow [a_0, b_0]$ as the piecewise linear continuous function

$$\begin{aligned} u(t):= {\left\{ \begin{array}{ll} t-a+a_0, \qquad &{} \text {if } t\in [a,c_1], \\ c_0, \qquad &{} \text {if } t\in (c_1,c_2), \\ t-b+b_0, \qquad &{} \text {if } t\in [c_2,b]. \end{array}\right. } \end{aligned}$$

Since $\mathcal {T}_j$ is $4\delta (G_1^*)$-fine, $d_{G_1}\big (g(t),\gamma (u(t))\big ) \le 4\delta (G_1^*) + c-c_1 \le 4\delta (G_1^*) + M$.

Therefore, by the triangle inequality,

$$\begin{aligned} \begin{aligned} d_{G_1}\big (g(t_1),g(t_2)\big )&\ge d_{G_1}\big (\gamma (u(t_1)),\gamma (u(t_2))\big ) - 8\delta (G_1^*) - 2M\\&= u(t_2) - u(t_1) - 8\delta (G_1^*) - 2M \\&\ge t_2-t_1 -(c_2-c_1) - 8\delta (G_1^*) - 2M \ge t_2-t_1 - 8\delta (G_1^*) - 4M . \end{aligned} \end{aligned}$$

Since $[x x_{1}] \cup [x_{1}x_{2}]$ and $[x_{k-1}x_{k}] \cup [x_{k}y]$ are geodesics in $G_1$, the above inequality also holds if $g(t_1),g(t_2)\in T^{n+j} G_1^*$ for some $j\in \{0,1, k-1,k \}$.

Assume now that $g(t_1)\in T^{n+j_1} G_1^*$ and $g(t_2)\in T^{n+j_2} G_1^*$ with $j_1 < j_2$. Let $r_1,r_2 \in [t_1,t_2]$ such that $g(r_1)=x_{j_1+1}$ and $g(r_2)=x_{j_2}$. The previous argument with the function u provides $t_1^*,t_2^*$ satisfying $\gamma (t_1^*) \in T^{n+j_1} G_1^*$, $\gamma (t_2^*) \in T^{n+j_2} G_1^*$, $d_{G_1}\big (g(t_1),\gamma (t_1^*)\big ) \le 4\delta (G_1^*) + M$, $d_{G_1}\big (g(t_2),\gamma (t_2^*)\big ) \le 4\delta (G_1^*) + M$, $d_{G_1}\big (\gamma (t_1^*),x_{j_1+1}\big ) \ge r_1-t_1 - 2M$ and $d_{G_1}\big (\gamma (t_2^*),x_{j_2}\big ) \ge t_2-r_2 - 2M$. Now, using Corollary 3.5,

$$\begin{aligned} \begin{aligned} d_{G_1}\big (g(t_1),g(t_2)\big )&\ge d_{G_1}\big (\gamma (t_1^*),\gamma (t_2^*)\big ) - 8\delta (G_1^*) - 2M \\&= d_{G_1}\big (\gamma (t_1^*),x_{j_1+1}\big ) + d_{G_1}\big (x_{j_1+1},x_{j_2}\big ) + d_{G_1}\big (\gamma (t_2^*),x_{j_2}\big ) \\&\quad - 8\delta (G_1^*) - 2M \\&\ge r_1-t_1 - 2M + \frac{1}{3}\,(r_2-r_1) + t_2-r_2 - 2M - 8\delta (G_1^*) - 2M \\&\ge \frac{1}{3}\,(t_2-t_1) - 8\delta (G_1^*) - 6M , \end{aligned} \end{aligned}$$

and we conclude that g is a $\big (3,8\delta (G_1^*)+6M\big )$-quasi-geodesic in $G_1$. Since $G_1^*$ is hyperbolic, the geodesic stability gives that $\mathcal {H}_{G_1}\big ( g_j,[x_{j}x_{j+1}]\big ) = \mathcal {H}_{T^{n+j} G_1^*}\big ( g_j,[x_{j}x_{j+1}]\big ) \le H$ for $2\le j \le k-2$, where H is a constant depending just on $\delta (G_1^*)$ and M. Hence, $\mathcal {H}_{G_1}( g,\gamma ) \le H$. $\square$

Remark 5.11

The argument in the proof of Lemma 5.10 proves, in fact, a more general result. On the one hand, the conclusion holds (with the same constants) if one replaces $g_j$ by $[x_{j}x_{j+1}]$ for any subset of $\{2\le j \le k-2\}$. On the other hand, the conclusion also holds (with the same constants) for non-straight geodesics: it suffices to consider each connected subcurve of $\gamma \cap T^{n+j}G^*$ joining $T^{n+j}\gamma _0$ with $T^{n+j+1}\gamma _0$ instead of $[x_{j}x_{j+1}]$ (if a connected subcurve of $\gamma \cap T^{n+j}G^*$ joins two points in $T^{n+j}\gamma _0$ one can replace it, in order to obtain g, by the geodesic contained in $T^{n+j}\gamma _0$ with the same endpoints; in a similar way, if it joins two points in $T^{n+j+1}\gamma _0$ one can replace it by the geodesic contained in $T^{n+j+1}\gamma _0$ with the same endpoints).

Lemma 5.12

Consider a periodic graph G and a parametrization $\eta _0$ of $\gamma _0$ satisfying both (5.1) and $\lim _{t \rightarrow -\infty } d_G\big (\eta _0(t), T\eta _0(t)\big )= \infty$. If $G^*$ is hyperbolic, then there exists a constant $t_0$ with the following properties:

(1) If $x \in T^{n} \gamma _0$, $y \in T^{n+1} \gamma _0$ and [xy] is a geodesic in $T^nG^*$ joining them, then there exist $p, s_x, s_y$ so that $p\in [xy]$ and $s_x,s_y\ge t_0 + 6 \delta (G^*)$ with $d_{G}\big (p,T^{n} \eta _0(s_x)\big ) \le 2 \delta (G^*)$ and $d_{G}\big (p, T^{n+1} \eta _0(s_y)\big ) \le 2 \delta (G^*)$.

(2) Let $\gamma =[xy]$ be a geodesic in G, with $x\in T^{n}(G^*)$, $y\in T^{n+k}(G^*)$ and $k\ge 3$. Let $x_{j} \in T^{n+j}\gamma _0\cap \gamma$, $2\le j\le k-1$. Then $x_{j} = T^{n+j} \eta _0(s_j)$ with $s_j \ge t_0$ for $2\le j \le k-1$.

Proof

(1) Given $x \in T^{n} \gamma _0$ and $y \in T^{n+1} \gamma _0$, since $\liminf _{t \rightarrow +\infty } d_G\big (\eta _0(t), T\eta _0(t)\big )= 0$, there exists t large enough such that the geodesic $[T^{n} \eta _0(t) T^{n+1} \eta _0(t)]$ in $T^{n} G^*$ satisfies $d_G\big ([xy],[T^{n} \eta _0(t) T^{n+1} \eta _0(t)]\big )> 2 \delta (G^*)$. Consider the geodesic quadrilateral $Q:=\{x,y , T^{n+1} \eta _0(t), T^{n} \eta _0(t)\}$ in $T^{n} G^*$, that is $2 \delta (G^*)$-thin. Then for every $q\in [xy]$ one has $d_G\big (q,[x T^{n} \eta _0(t)]\cup [y T^{n+1} \eta _0(t)]\big )\le 2 \delta (G^*)$. Hence, there exist a point $p\in [xy]$ such that $d_G\big (p,[x T^{n} \eta _0(t)]\big )\le 2 \delta (G^*)$ and $d_G\big (p,[y T^{n+1} \eta _0(t)]\big )\le 2 \delta (G^*)$. Choose $s_x,s_y$ such that $d_{G}\big (p,T^{n} \eta _0(s_x)\big ) \le 2 \delta (G^*)$ and $d_{G}\big (p, T^{n+1} \eta _0(s_y)\big ) \le 2 \delta (G^*)$. Then $d_{G}\big (T^{n} \eta _0(s_x), T^{n+1} \eta _0(s_y)\big ) \le 4 \delta (G^*)$ and by Corollary 3.4, $d_{G}\big (T^{n} \eta _0(s_x), T^{n+1} \eta _0(s_x)\big ) \le 2d_{G}\big (T^{n} \eta _0(s_x), T^{n+1} \gamma _0\big ) \le 2d_{G}\big (T^{n} \eta _0(s_x), T^{n+1} \eta _0(s_y)\big ) \le 8 \delta (G^*)$.

A symmetric argument gives $d_{G}\big (T^{n} \eta _0(s_y), T^{n+1} \eta _0(s_y)\big ) \le 8 \delta (G^*)$. Since $\lim _{t \rightarrow -\infty } d_G\big (\eta _0(t), T\eta _0(t)\big )= \infty$, there exists a constant $t_0$ such that $d_G\big (\eta _0(t), T\eta _0(t)\big ) > 8 \delta (G^*)$ for every $t < t_0 + 6 \delta (G^*)$; hence, $s_x,s_y \ge t_0 + 6 \delta (G^*)$.

(2) Fix $x_{j} = T^{n+j} \eta _0(s_j)$ with $2\le j \le k-1$. By (1), there exist $p \in [x_{j-1}x_{j}]\cap T^{n+j-1} G^*$, $p^{\prime } \in [x_{j}x_{j+1}]\cap T^{n+j}G^*$ and $s,s^{\prime } \ge t_0 + 6 \delta (G^*)$ such that $d_{G}\big (p,T^{n+j} \eta _0(s)\big ) \le 2 \delta (G^*)$ and $d_{G}(p^{\prime }, T^{n+j} \eta _0(s^{\prime })) \le 2 \delta (G^*)$.

By symmetry, assume that $s\ge s^{\prime }$. Assume also that $s_j < s^{\prime }$, since otherwise $s_j\ge s^{\prime }\ge t_0+6\delta (G^*)$. Thus

$$\begin{aligned} \begin{aligned} d_{G}(p,p^{\prime })&\le d_{G}\big (p,T^{n+j} \eta _0(s)\big ) + d_{G}\big (T^{n+j} \eta _0(s), T^{n+j} \eta _0(s^{\prime })\big ) \\&\quad +d_{G}\big (T^{n+j} \eta _0(s^{\prime }), p^{\prime }\big ) \\&\le 4 \delta (G^*) + d_{G}\big (T^{n+j} \eta _0(s), T^{n+j} \eta _0(s^{\prime })\big ) ,\\ \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned}&d_{G}\big (x_j,T^{n+j} \eta _0(s^{\prime })\big ) + d_{G}\big (T^{n+j} \eta _0(s^{\prime }), T^{n+j} \eta _0(s)\big ) \\&\quad = d_{G}\big (x_j, T^{n+j} \eta _0(s)\big )\\&\quad \le d_{G}\big (x_j, p\big ) + d_{G}\big (p, T^{n+j} \eta _0(s)\big ) \\&\quad \le d_{G}\big (x_j, p\big ) + 2 \delta (G^*) \le d_{G}\big (p^{\prime }, p\big ) + 2 \delta (G^*)\\&\quad \le 6 \delta (G^*) + d_{G}\big (T^{n+j} \eta _0(s), T^{n+j} \eta _0(s^{\prime })\big ) , \end{aligned} \end{aligned}$$

and thus $d_{G}\big (x_j,T^{n+j} \eta _0(s^{\prime })\big ) \le 6 \delta (G^*)$. Since $6\delta (G^*) \ge d_{G}\big (x_j, T^{n+j} \eta _0(s^{\prime })\big ) = s^{\prime }- s_j \ge t_0 + 6 \delta (G^*) -s_j$, one gets $s_j \ge t_0$. $\square$

Lemma 5.13

Let G be a periodic graph with quasi-exponential decay and $G^*$ hyperbolic. Then there exists a constant N such that $\mathcal {H}_G(g_1, g_2) \le N$ for every geodesics $g_1, g_2$ in G with the same endpoints and $g_1 \subset \gamma _0$.

Proof

Consider first the case $g_2 \subset \cup _{j\ge 0} T^{j} G^*$. Define $n_2:=\max \{j\in \mathbb {Z}: \, g_2 \cap T^{j} G^* \ne \emptyset \}$. Let $\{g_j^{1}, \dots , g_j^{r_j} \}$ be the connected components of $g_2 \cap T^{j} G^*$ and $\mathcal {G}:=\{ g_j^{i}|\, 1\le i\le r_j, 0 \le j \le n_2 \}$.

If $n_2=0$, then $\mathcal {H}_{G}( g_1, g_2) \le H(\delta (G^*),1,0)$, where H is the function of the geodesic stability (see the paragraph after Lemma A).

If $n_2 >0$, for each $g_{n_2}^{i}$, define $\gamma _{n_2}^{i}$ as follows: if $g_{n_2}^{i}$ joins $T^{n_2} \eta _0(s^{i})$ and $T^{n_2} \eta _0(t^{i})$ with $s^{i} \le t^{i}$, then $\gamma _{n_2}^{i}:= T^{n_2} \eta _0([s^{i},t^{i}])$. Let $g_2^{\prime }$ be the geodesic in $\cup _{0\le j\le n_2-1} T^{j} G^*$ obtained from $g_2$ by replacing each $g_{n_2}^{i}$ by $\gamma _{n_2}^{i}$; then $\mathcal {H}_{G}( g_2, g_2^{\prime }) \le H(\delta (G^*),1,0)$. In a similar way one can find a geodesic $g_2^{\prime \prime }$ contained in $\cup _{0\le j\le n_2-2} T^{j} G^*$ with $\mathcal {H}_{G}( g_2, g_2^{\prime \prime }) \le 2 H(\delta (G^*),1,0)$ (if $n_2 \ge 2$). Hence, if $n_2 \le 2$, then $\mathcal {H}_G(g_1, g_2) \le 3 H(\delta (G^*),1,0)$. Assume now that $n_2 \ge 3$.

For each $g_j^{i} \in \mathcal {G}$ with $1\le j \le n_2-2$, define $\gamma _j^{i}$ as follows: if $g_j^{i}$ joins $T^j \eta _0(s_j^{i})$ and $T^{j+1} \eta _0(t_j^{i})$ with $s_j^{i} \le t_j^{i}$, then $\gamma _j^{i}:= T^j \eta _0([s_j^{i},t_j^{i}]) \cup U_{j,t_j^{i}}$; if $s_j^{i} > t_j^{i}$, then $\gamma _j^{i}:= T^j \eta _0([t_j^{i},s_j^{i}]) \cup U_{j,s_j^{i}}$; if $g_j^{i}$ joins $T^j \eta _0(s_j^{i})$ and $T^{j} \eta _0(t_j^{i})$ with $s_j^{i} \le t_j^{i}$, then $\gamma _j^{i}:= T^j \eta _0([s_j^{i},t_j^{i}])$; if $g_j^{i}$ joins $T^{j+1} \eta _0(s_j^{i})$ and $T^{j+1} \eta _0(t_j^{i})$ with $s_j^{i} \le t_j^{i}$, then $\gamma _j^{i}:= T^{j+1} \eta _0([s_j^{i},t_j^{i}])$. Define I as the set of indices $1\le i \le r_0$ such that $g_0^{i}$ joins $T \eta _0(s_0^{i})$ and $T \eta _0(t_0^{i})$ with $s_0^{i} \le t_0^{i}$; define $\gamma _0^{i}:= T \eta _0([s_0^{i},t_0^{i}])$ for every $i\in I$. By Lemma 5.5, the relation (5.1) holds and then, by Lemma 5.12, $s_j^{i},t_j^{i} \ge t_0$, where $t_0$ is the constant in Lemma 5.12, and therefore $\gamma _j^{i} \subset G_1$. By Remark 5.11, $\mathcal {H}_{G_1}( g_j^{i}, \gamma _j^{i}) \le H_0$, where $H_0$ is a constant depending just on $\delta (G_1^*)$ and on $\sup _{t \ge t_0} d_G(\eta _0(t), T\eta _0(t))$.

Define $\gamma _2:= \big (g_2^{\prime \prime } {\setminus } \big (\big (\cup _{j=1}^{n_2-2} \cup _{i=1}^{j_r} g_j^{i}\big ) \cup \big ( \cup _{i\in I} g_0^{i}\big ) \big )\big ) \cup \big (\cup _{j=1}^{n_2-2} \cup _{i=1}^{j_r} \gamma _j^{i}\big )\cup \big ( \cup _{i\in I} \gamma _0^{i}\big )$. Therefore, $\mathcal {H}_{G_1}( g_2, \gamma _2) \le H_1:= H_0 + 2H(\delta (G^*_1),1,0)$.

By Remark 5.11, $\gamma _2$ is an $(\alpha ,\beta )$-quasigeodesic in $G_1$ (with its arc-length parametrization), where $\alpha ,\beta$ are the constants in Lemma 5.10. Let $\gamma _{2}^{\prime }:=\gamma _2 \cap \big ( \cup _{j=1}^{n_2-2} T^j G^{*}\big ) \subset G_2$. Note that $\gamma _{2}^{\prime }$ is connected and joins two points in $T\gamma _0$. Since $d_{G_1} \le d_{G_2}$ on $G_2$, $\gamma _{2}^{\prime }$ is also an $(\alpha ,\beta )$-quasigeodesic in $G_2$.

By Lemma 5.5, $\sup \big \{t_2-t_1: \, \Phi (t_1)=\Phi (t_2),\, t_2\ge t_1 \ge t_0 \big \} < \infty$ and (5.1) holds. Hence, by Lemma 5.3, there exists a quasi-isometry $f^{-1}:G_2 \rightarrow G_3$ and there also exist constants $\alpha ^{\prime },\beta ^{\prime },$ which just depend on G, such that $f^{-1}(\gamma _{2}^{\prime })$ is an $(\alpha ^{\prime },\beta ^{\prime })$-quasigeodesic in $G_3$. Note that $G_3$ is hyperbolic by Lemma 5.7; therefore, if $\gamma _{3}^{\prime } \subset T\gamma _0$ is the geodesic joining the endpoints of $f^{-1}(\gamma _{2}^{\prime })$ in $G_3$, then $\mathcal {H}_{G_3}\big ( \gamma _{3}^{\prime }, f^{-1}(\gamma _{2}^{\prime })\big ) \le H_3:= H\big (\delta (G_3),\alpha ^{\prime },\beta ^{\prime }\big )$. Since f is the identity map on $\cup _{n\in \mathbb {Z}} T^n \eta _0([t_0,\infty ))$, $f(\gamma _{3}^{\prime }) \subset T\gamma _0$ is a geodesic in $G_2$ joining the endpoints of $\gamma _{2}^{\prime }$; since f is a quasi-isometry, there exists a constant $H_4,$ which just depend on G, such that $\mathcal {H}_{G_2}( f(\gamma _{3}^{\prime }), \gamma _{2}^{\prime }) \le H_4$. Since $d_{G_1} \le d_{G_2}$ on $G_2$, $\mathcal {H}_{G_1}( f(\gamma _{3}^{\prime }), \gamma _{2}^{\prime }) \le H_4$. Define $\gamma _3:= (\gamma _2 {\setminus } \gamma _2^{\prime }) \cup f(\gamma _3^{\prime }) \subset G$; then $\mathcal {H}_{G_1}( \gamma _{3}, \gamma _{2})= \mathcal {H}_{G_1}( f(\gamma ^{\prime }_{3}), \gamma ^{\prime }_{2}) \le H_4$ and $\mathcal {H}_{G}( g_2, \gamma _{3}) = \mathcal {H}_{G_1}( g_2, \gamma _{3}) \le H_1+ H_4$. Since $\gamma _3$ is a geodesic in $G^*$ with the same endpoints that $g_1$, one gets $\mathcal {H}_{G}( \gamma _{3}, g_{1}) \le H(\delta (G^*),1,0)$ and $\mathcal {H}_{G}( g_1, g_{2}) \le H_1+ H_4 + H(\delta (G^*),1,0)$.

Hence, if $g_2 \subset \cup _{j\ge 0} T^{j} G^*$ the lemma holds with $N=H_1+ H_4 + H(\delta (G^*),1,0)$. If $g_2 \subset \cup _{j< 0} T^{j} G^*$, the same result holds by symmetry. The general case follows by applying these two cases to the connected components $g_{2,1}, \dots , g_{2,m}$ of $g_2 \cap \cup _{j\ge 0} T^{j} G^*$ and to the closure of the connected components of $g_{2} {\setminus } \cup _{j=1}^m g_{2,j}$. $\square$

Corollary 5.14

Let G be a periodic graph with quasi-exponential decay and $G^*$ hyperbolic. Then for each geodesic $\gamma$ in G there exists a straight geodesic $\gamma ^{\prime }$ with the same endpoints and $\mathcal {H}_G(\gamma , \gamma ^{\prime }) \le N$, where N is the constant in Lemma 5.13.

Proof

Fix a geodesic $\gamma : [a,b] \rightarrow G$ with $\gamma (a) \in T^{n_1}G^*$, $\gamma (b) \in T^{n_2}G^*$ and $n_1 \le n_2$. Assume that $\gamma \cap T^{n_1}\gamma _0 \ne \emptyset$ (otherwise, we consider $T^{n_1+1}\gamma _0$ instead of $T^{n_1}\gamma _0$) and that $\gamma \cap T^{n_2+1}\gamma _0 \ne \emptyset$ (otherwise, we consider $T^{n_2}\gamma _0$ instead of $T^{n_2+1}\gamma _0$). Define inductively $s_j,t_j$ ($n_1 \le j \le n_2+1$) as follows: $s_{n_1}:=\min \{t\in [a,b] : \, \gamma (t) \in T^{n_1} \gamma _0 \}$, $t_{n_1}:=\max \{t\in [a,b] : \, \gamma (t) \in T^{n_1} \gamma _0 \}$, $s_j:=\min \{t\in (t_{j-1},b] : \, \gamma (t) \in T^j \gamma _0 \}$, $t_j:=\max \{t\in (t_{j-1},b] : \, \gamma (t) \in T^j \gamma _0 \}$. We define also $\gamma ^j:=[\gamma (s_j)\gamma (t_j)] \subset T^j \gamma _0$ for $n_1 \le j \le n_2+1$.

By Lemma 5.13, $\mathcal {H}_G(\gamma ([s_j,t_j]), \gamma ^j) \le N$. Then $\gamma ^{\prime }:=\big (\gamma {\setminus } \cup _{j= n_1}^{n_2+1} \gamma ([s_j,t_j])\big ) \cup \big (\cup _{j= n_1}^{n_2+1} \gamma ^j\big )$ is a straight geodesic in G and that $\mathcal {H}_G(\gamma , \gamma ^{\prime }) \le N$. $\square$

Finally, let us show the proof of the second part of Theorem 1.1.

Proof

(Second part of Theorem 1.1 ). Assume that G is hyperbolic. Lemma B implies that $G^*$ is also hyperbolic.

Since $\inf _{z\in \gamma _0} d_{G}(z,Tz)=0$, without loss of generality one can consider only arc-length parametrizations $\eta _0$ of $\gamma _0$ for which $\liminf _{t\rightarrow +\infty } d_G\big (\eta _0(t),T\eta _0(t)\big ) =0$. Fix one of these. It will be shown that $\lim _{t\rightarrow -\infty } F(t)=\infty$, where $F(t) := d_G(\eta _0(t), T\eta _0(t))$. Indeed,

(a) Assume that $\liminf _{t\rightarrow -\infty } F(t)=0$. Then there exists a sequence of positive numbers $\{c_k\}$ converging to 0 and two sequences $\{s_{1,k}\},\{s_{2,k}\} \subset \mathbb {R}$ such that $\lim _{k\rightarrow \infty } s_{2,k} = \infty$, $\lim _{k\rightarrow \infty } s_{1,k}= -\infty$, $F(s_{1,k})= F(s_{2,k})= c_k$, and $F(t)\ge c_k$ for every $t \in [s_{1,k},s_{2,k}]$ and every k. Therefore, Lemmas 2.1 and 5.9 imply that G is not hyperbolic.

(b) If $0< \liminf _{t\rightarrow -\infty } F(t)$ and $\limsup _{t\rightarrow -\infty } F(t) < \infty$, one can also easily construct quasi-geodesic quadrilaterals Q with $\delta (Q)$ arbitrarily large, and thus G is not hyperbolic (by lemmas 2.1 and 5.9). (The Cayley graph of $\mathbb {Z}^2$, for which $1\le F(t) \le \frac{3}{2}$, is a basic example of this situation.)

(c) Assume that $\liminf _{t\rightarrow -\infty } F(t)<\infty$ and $\limsup _{t\rightarrow -\infty } F(t) = \infty$. Note that F is a Lipschitz function; in fact, $|F(t_1)-F(t_2)| \le 2 |t_1-t_2|$. Fix a constant $c>\liminf _{t\rightarrow -\infty } F(t)$. There exist two sequences $\{s_{1,k}\},\{s_{2,k}\} \subset \mathbb {R}^-$ such that $F(s_{1,k})= F(s_{2,k})= c$, $F(t)\ge c$ for every $t \in [s_{1,k},s_{2,k}]$ and $F(t_k) \ge k$ for some $t_k \in [s_{1,k},s_{2,k}]$, for every k. Since F is 2-Lipschitz, $s_{2,k} - s_{1,k} \ge k-c$ for every k, and then $\lim _{k\rightarrow \infty } (s_{2,k} - s_{1,k})= \infty$. Therefore, Lemmas 2.1 and 5.9 give that G is not hyperbolic.

Thus, $\lim _{t\rightarrow -\infty } F(t) = \infty$.

The argument in (c) also gives $\limsup _{t\rightarrow +\infty } F(t)<\infty$ since $\liminf _{t\rightarrow +\infty } F(t)=0$; then (5.1) holds.

Assume that G has not quasi-exponential decay, so $\sup _{s_2 \ge s_1 \ge 0} (s_2-s_1)\Phi (s_2)/\Phi (s_1) = \infty$. By Lemma 5.6, $\mathfrak {L}(G_3)=\infty$ and $G_3$ is not hyperbolic and, by Lemma 5.4, since G is hyperbolic, $\sup \big \{t_2-t_1: \, \Phi (t_1)=\Phi (t_2),\, t_2\ge t_1 \ge 0 \big \} = \infty$. Consider $t_2> t_1 > 0$ with $\Phi (t_1)=\Phi (t_2)<\Phi (0)$ which are maximal in the following sense: $\Phi (t_1-\varepsilon )>\Phi (t_1)$ and $\Phi (t_2)>\Phi (t_2+\varepsilon )$ for every $\varepsilon >0$. Therefore, $\Phi (t_1)=F(t_1)=\Phi (t_2)=F(t_2)$ and $F(t) \ge F(t_1)=F(t_2)$ for every $t \in [t_1, t_2]$. Lemma 5.9 (taking $c_1=c_2=c^*=c=F(t_1)< \Phi (0)$) provides a $(3,2\Phi (0))$-quasigeodesic quadrilateral Q with $\delta (Q) \ge (t_2-t_1)/6-\Phi (0)/4$. Hence, Lemma 2.1 shows that G is not hyperbolic. This is a contradiction. Therefore, G has quasi-exponential decay.

Let us show the other direction by assuming that $G^*$ is hyperbolic and G has quasi-exponential decay. By Lemma 5.5, $\sup \{t_2-t_1:\, \Phi (t_1)=\Phi (t_2),\, t_2\ge t_1\ge t_0 \}<\infty$ for any fixed $t_0$, and (5.1) holds.

Fix any geodesic triangle $\mathcal {T}_0:=\{z_1,z_2,z_3\}$ in G, with $z_{i} \in T^{n_i} G^*$ for $1\le i \le 3$ and $n_1 \le n_2 \le n_3$. One just needs to deal with the case $n_1 + 4 \le n_2 \le n_3 - 4$; the other cases are similar and simpler.

By Corollary 5.14, without loss of generality, assume that the geodesics of $\mathcal {T}_0$ are straight.

By Lemma 5.12 there exists a constant $t_0$ such that if $x \in \mathcal {T}_0\cap T^{n} \gamma _0$ with either $n_1+2 \le n \le n_2-1$ or $n_2+2 \le n \le n_3-1$, then $(T^{n} \eta _0)^{-1}(x) \ge t_0$. Consider the geodesic metric spaces $G_1$ and $G_2$ defined after (5.1) (with this constant $t_0$) and recall $G_1=G\cup G_2$; since G is an isometric subspace of $G_1$, $\mathcal {T}_0$ is also a geodesic triangle in $G_1$.

Since $(T^{n} \eta _0)^{-1}(x) \ge t_0$ if $x \in \mathcal {T}_0\cap T^{n} \gamma _0$ with either $n_1+2 \le n \le n_2-1$ or $n_2+2 \le n \le n_3-1$, and the geodesics of $\mathcal {T}_0$ are straight, by Lemma 5.10, there exist $(\alpha , \beta )$-quasigeodesics $g_{12},g_{13}$ and $g_{23}$ in $G_1$ such that $g_{ij}$ joins $z_i$ and $z_j$, and $\mathcal {H}_{G_1}(g_{ij},[z_iz_j]) \le H$, where H only depends on $\delta (G_1^*)$ and $\nu := \sup _{t\ge t_0} d_G(\eta _0(t), T\eta _0(t))$, $\alpha = 3$ and $\beta = 8\delta (G_1^*)+6\nu$ (recall that $G_1^*$ is hyperbolic by Lemma 5.2). Furthermore, $g_{12}=[z_1z_2]$ in $T^{n_1} G_1^*\cup T^{n_1+1} G_1^*\cup T^{n_2-1} G_1^*\cup T^{n_2} G_1^*$, $g_{23}=[z_2z_3]$ in $T^{n_2} G_1^*\cup T^{n_2+1} G_1^*\cup T^{n_3-1} G_1^*\cup T^{n_3} G_1^*$, $g_{13}=[z_1z_3]$ in $T^{n_1} G_1^*\cup T^{n_1+1} G_1^*\cup T^{n_2-1} G_1^*\cup T^{n_2} G_1^*\cup T^{n_2+1} G_1^*\cup T^{n_3-1} G_1^*\cup T^{n_3} G_1^*$, $g_{12}\cap (\cup _{n_1+1<n<n_2-1} T^{n} G_1^*)\subset G_2$, $g_{23}\cap (\cup _{n_2+1<n<n_3-1} T^{n} G_1^*)\subset G_2$, $g_{13}\cap \{(\cup _{n_1+1<n<n_2-1} T^{n} G_1^*)\cup (\cup _{n_2+1<n<n_3-1} T^{n} G_1^*)\}\subset G_2$. Then $\mathcal {T}_1:=\{g_{12},g_{13},g_{23}\}$ is an $(\alpha ,\beta )$-quasi-geodesic triangle in $G_1$.

Define $G_2(\mathcal {T}_1)$ and $G_3(\mathcal {T}_1)$ as the geodesic metric spaces given by

$$\begin{aligned} \begin{aligned} G_2(\mathcal {T}_1)&:= T^{n_1} G_1^*\cup T^{n_1+1} G_1^* \cup \big ( \cup _{n_1+1<n<n_2-1, t\ge t_0} U_{n,t} \big )\\&\quad \cup T^{n_2-1} G_1^*\cup T^{n_2} G_1^*\cup T^{n_2+1} G_1^* \\&\quad \cup \big ( \cup _{n_2+1<n<n_3-1, t\ge t_0} U_{n,t} \big ) \cup T^{n_3-1} G_1^*\cup T^{n_3} G_1^*, \\ G_3(\mathcal {T}_1)&:= T^{n_1} G_1^*\cup T^{n_1+1} G_1^* \cup \big ( \cup _{n_1+1<n<n_2-1, t\ge t_0} V_{n,t} \big )\\&\quad \cup T^{n_2-1} G_1^*\cup T^{n_2} G_1^*\cup T^{n_2+1} G_1^* \\&\quad \cup \big ( \cup _{n_2+1<n<n_3-1, t\ge t_0} V_{n,t} \big ) \cup T^{n_3-1} G_1^*\cup T^{n_3} G_1^*. \end{aligned} \end{aligned}$$

Note that $G_2(\mathcal {T}_1)$ is contained in $G_1$.

By Corollary 5.8 there exists a constant $\delta$, which does not depend on $n_1,n_2,n_3,\mathcal {T}_0,$ such that the subspaces $\cup _{n_1+1<n<n_2-1, t\ge t_0} V_{n,t}$ and $\cup _{n_2+1<n<n_3-1, t\ge t_0} V_{n,t}$ are $\delta$-hyperbolic.

Since $G^*$ is hyperbolic, by Lemma 5.2 there exists a constant $\delta ^*$, which does not depend on $n_1,n_2,n_3,\mathcal {T}_0,$ such that $G_1^*$ is $\delta ^*$-hyperbolic. By Lemma B, $T^{n_1} G_1^*\cup T^{n_1+1} G_1^*$, $T^{n_2-1} G_1^*\cup T^{n_2} G_1^*\cup T^{n_2+1} G_1^*$ and $T^{n_3-1} G_1^*\cup T^{n_3} G_1^*$ are $(120)^2\delta ^*$-hyperbolic. Hence, by Lemma B, $G_3(\mathcal {T}_1)$ is $(120)^4\max \{\delta ,(120)^2\delta ^*\}$-hyperbolic.

As in the proof of Lemma 5.3, one can check that $G_3(\mathcal {T}_1)$ and $G_2(\mathcal {T}_1)$ are quasi-isometric (with constants which just depend on $G^*$); thus, by invariance of hyperbolicity, there exists a constant $\delta _2$ which does not depend on $n_1,n_2,n_3,\mathcal {T}_0,$ such that $G_2(\mathcal {T}_1)$ is $\delta _2$-hyperbolic. Since $\mathcal {T}_1$ is also an $(\alpha ,\beta )$-quasi-geodesic triangle in $G_2(\mathcal {T}_1)\subset G_1$, $\mathcal {T}_1$ is $\delta _2^{\prime }$-thin, where $\delta _2^{\prime }$ is a constant that does not depend on $n_1,n_2,n_3,\mathcal {T}_0$. Since $d_{G_1} \le d_{G_2(\mathcal {T}_1)}$, we have that $\mathcal {T}_1$ is also $\delta _2^{\prime }$-thin in $G_1$. Since $\mathcal {H}_{G_1}(g_{ij},[z_iz_j]) \le H$, the triangle $\mathcal {T}_0$ is $(\delta _2^{\prime }+2H)$-thin in $G_1$. Since $\mathcal {T}_0 \subset G$ and G is an isometric subspace of $G_1$, the geodesic triangle $\mathcal {T}_0$ is also $(\delta _2^{\prime }+2H)$-thin in G. $\square$