Mod p and torsion homology growth in nonpositive curvature

Abstract

We compute the mod p homology growth of residual sequences of finite index normal subgroups of right-angled Artin groups. We find examples where this differs from the rational homology growth, which implies the homology of subgroups in the sequence has lots of torsion. More precisely, the homology torsion grows exponentially in the index of the subgroup. For odd primes p, we construct closed locally CAT(0) manifolds with nonzero mod p homology growth outside the middle dimension. These examples show that Singer’s conjecture on rational homology growth and Lück’s conjecture on torsion homology growth are incompatible with each other, so at least one of them must be wrong.

This paper is about the growth of homology in regular coverings of finite aspherical complexes $X=B\Gamma$. We will content ourselves with the situation when the fundamental group $\Gamma =\pi _1X$ is residually finite. This means there is a nested sequence of finite index normal subgroups $\Gamma _k\vartriangleleft \Gamma$ with $\cap _{k}\Gamma _k=1$. We fix a choice of such a sequence and will be interested in the normalized limits of Betti numbers with coefficients in a field F

$$\begin{aligned} b^{(2)}_{i}(\Gamma ;F):=\limsup _{k} \frac{b_{i}(B\Gamma _k; F)}{[\Gamma :\Gamma _k]}, \end{aligned}$$

where F is either ${\mathbb {Q}}$ or ${\mathbb {F}}_{p}$. When $F={\mathbb {Q}}$ then Lück’s approximation theorem [11] shows this does not depend on the choice of sequence and can be identified with a more analytically defined ith $L^2$-Betti number of the universal cover $E\Gamma$. When $F={\mathbb {F}}_{p}$ we will analogously refer to $b_{i}^{(2)}(\Gamma ;{\mathbb {F}}_{p})$ as the ${\mathbb {F}}_{p}$-$L^2$-Betti number, even though it does not (as far as we know) have an analytic interpretation and it is not even known whether the $\limsup$ depends on the choice of sequence (we abuse notation by omitting the sequence from $b_{i}^{(2)}(\Gamma ;{\mathbb {F}}_{p})$). Note that if the $\limsup$ is independent of the sequence, then it becomes an honest limit.

For a finite aspherical complex $B\Gamma$ it is easy to see that the mod p $L^2$-Betti number is greater or equal to the ordinary $L^2$-Betti number

$$\begin{aligned} b^{(2)}_i(\Gamma ;{\mathbb {F}}_{p})\ge b^{(2)}_i(\Gamma ;{\mathbb {Q}}), \end{aligned}$$

but it might be strictly bigger. We show that this does—in fact—happen for some right-angled Artin groups. This seems to have not been observed previously and contradicts a conjecture of Lück [Conjecture 3.4, [13]] that these numbers are independent of the coefficient field. When $\Gamma$ is a right-angled Artin group we compute $b_{i}^{(2)}(\Gamma ;F)$ completely for any coefficient field, via a residually finite variant of the argument Davis and Leary [9] used to compute the ordinary $L^2$-Betti numbers of such groups.

FormalPara Theorem 1

Let $A_L$ be a right-angled Artin group with defining flag complex L and F any field (e.g. ${\mathbb {Q}}$ or ${\mathbb {F}}_{p}$). Then

$$\begin{aligned} b_{i}^{(2)}(A_L;F)={\bar{b}}_{i-1}(L;F). \end{aligned}$$

Here ${\bar{b}}_{i-1}(L;F)$ denotes the reduced Betti number of L with coefficients in F. In particular, the $\limsup$ is actually a limit, it does not depend on the choice of chain but does depend on the characteristic of the coefficient field.

FormalPara Corollary 2

Suppose that L is a flag triangulation of ${\mathbb {R}}P^2$.

$$\begin{aligned} b_3^{(2)}(A_{L};{\mathbb {Q}})&= 0,\\ b_3^{(2)}(A_{L};{\mathbb {F}}_2)&=1. \end{aligned}$$

The proof of Theorem 1 also shows that F-$L^2$-Betti numbers of finite index subgroups of RAAG’s are multiplicative (see Corollary 10), and we will use this later in Theorem 4. In general, it is not known whether the ${\mathbb {F}}_{p}$-$L^2$-Betti numbers are multiplicative.

By the universal coefficient theorem, $H_n(X,{\mathbb {F}}_p)$ is determined by $H_n(X,{\mathbb {Q}})$ and ${\mathbb {Z}}/p$-summands in $H_n(X,{\mathbb {Z}})$ and $H_{n-1}(X,{\mathbb {Z}})$. In this case, if $A_L$ is as in Corollary 2, then since $A_L$ has a 3-dimensional model for $BA_L$, $H_3(B\Gamma _k; {\mathbb {Z}})$ is torsion-free. Therefore, this discrepancy between ${\mathbb {Q}}$ and ${\mathbb {F}}_2$ homology leads to exponentially growing torsion in homology in degree 2.

FormalPara Corollary 3

The group $A_L$ as in Corollary 2 has exponential $H_2$-torsion growth:

$$\begin{aligned} \limsup _{k} \frac{ \log |H_2(B\Gamma _k;{\mathbb {Z}})_{tors}|}{[A_L:\Gamma _k]} >0. \end{aligned}$$

Furthermore, the rank of the 2-torsion subgroup of $H_2(B\Gamma _k;{\mathbb {Z}})$ grows linearly in $[A_L:\Gamma _k]$.

While it is conjectured that for arithmetic hyperbolic 3-manifold groups the torsion in homology grows exponentially in residual chains of congruence covers, this is the first example of a finitely presented group of any sort where one can prove that homology torsion grows exponentially in a residual chain, answering a query of Bergeron for such a group. By contrast, Abert, Gelander, and Nikolov showed that if L is connected then $H_1$-torsion of $A_L$ grows slower than exponentially [1].

For other groups $\Gamma$ the computation of $L^2$-Betti numbers and homology torsion growth is a difficult problem. A basic vanishing principle which can make computations of $L^2$-Betti numbers simpler is the following conjecture often attributed to Singer.

FormalPara Singer Conjecture

Let $M^n$ be a closed aspherical manifold. Then

$$\begin{aligned} b_{i}^{(2)}(\pi _1(M^n); {\mathbb {Q}}) = 0\quad \text{ for } i \ne \frac{n}{2}. \end{aligned}$$

So in the residually finite setting, the free part of homology should grow sublinearly outside the middle dimension. A more recent vanishing principle regarding torsion growth, motivated by considerations in number theory, is the following conjecture made by Bergeron and Venkatesh in the context of arithmetic locally symmetric spaces [4] (see also [3]).

FormalPara Bergeron–Venkatesh Conjecture

Let G be a semisimple Lie group, $\Gamma$ a cocompact arithmetic lattice in G, and $\Gamma _k$ a sequence of congruence subgroups with $\cap _{k} \Gamma _k = 1$. Then

$$\begin{aligned} \limsup _{k}\frac{ \log |H_i(B\Gamma _k;{\mathbb {Z}})_{tors}|}{[\Gamma :\Gamma _k]}=0 \end{aligned}$$

unless $i = \frac{\dim (G/K)-1}{2}$.

FormalPara Remark

The conjecture is actually more precise and predicts that the limit is positive in some cases, e.g. when G is $SL(3,{\mathbb {R}})$, $SL(4,{\mathbb {R}})$ or SO(m, n) for mn odd.

Partially motivated by this conjecture, in [12] Lück suggested such a vanishing principle could hold quite generally for arbitrary closed aspherical manifolds.

FormalPara Lck Conjecture

(1.12(2), [12]) Let $M^n$ be a closed aspherical n-manifold with residually finite fundamental group. Let $\Gamma _k \vartriangleleft \pi _1(M^n)$ be any normal chain with $\bigcap _{k} \Gamma _k = 1$. If $i\not =(n-1)/2$ then

$$\begin{aligned} \limsup _{k} \frac{\log |H_{i}(B\Gamma _k;{\mathbb {Z}})_{tors}|}{[\pi _1(M^n):\Gamma _k]} = 0. \end{aligned}$$

It is interesting to note that the Singer and Lück Conjectures together imply an ${\mathbb {F}}_{p}$-version of the Singer conjecture.

${\mathbb {F}}_{p}$–Singer Conjecture Let $M^n$ be a closed aspherical n-manifold with residually finite fundamental group. Then

$$\begin{aligned} b_{i}^{(2)}(\pi _1(M^n);{\mathbb {F}}_{p}) = 0\quad \text { for } i \ne \frac{n}{2}. \end{aligned}$$

To see this, suppose we have an n-manifold $M^{n}$ with $b_i^{(2)}(\pi _1(M^{n}); {\mathbb {F}}_{p}) \ne 0$ for $i \ne n/2$. By Poincaré duality, we can assume $i>n/2$. The Künneth formula implies that $M^n \times M^n \times M^n$ has nontrivial ${\mathbb {F}}_{p}$-$L^2$-Betti numbers in dimension 3i. Since the Singer Conjecture predicts that $b_{3i}^{(2)}(\pi _1((M^{n})^{3}); {\mathbb {Q}}) = 0$, the universal coefficient theorem implies exponential homological torsion growth in dimension 3i or $3i - 1$, which lies above the middle dimension, contradicting Lück’s Conjecture.

The ${\mathbb {F}}_{p}$-Singer Conjecture is open even for $n = 3$ (but see [4, 6]). But in high enough dimensions, we show this conjecture is not true for any odd prime p.

FormalPara Theorem 4

For any odd prime p, the ${\mathbb {F}}_{p}$-Singer Conjecture fails in all odd dimensions $\ge 7$ and all even dimensions $\ge 14$.

Our examples are manifolds constructed via right-angled Coxeter groups; in particular they are locally CAT(0), so it follows that the rational homology and torsion homology growth conjectures are incompatible in the CAT(0) setting. On the other hand, our examples are not locally symmetric so even though the Singer conjecture is known for locally symmetric spaces the Bergeron–Venkatesh conjecture remains open.

Here is a brief outline of our construction. In [14], it was shown that if a finite type group $\Gamma$ acts properly on a contractible n-manifold and $b_{i}^{(2)}(\Gamma ;{\mathbb {Q}}) \ne 0$ for $i > \frac{n}{2}$, then there is a counterexample to the Singer Conjecture (in some dimension possibly different from n.) We employ a similar strategy here. Our group is a finite index subgroup of a right-angled Artin group with $b_4^{(2)}(A_L; {\mathbb {F}}_{p}) \ne 0$, and the 7-manifold is going to be the Davis complex corresponding to a right-angled Coxeter group associated to a flag triangulation of a $S^{6}$.

This uses Theorem 1 and the main result of [2]. More precisely, suppose $L=S^2\cup _pD^3$ is a flag triangulation of a complex obtained by gluing a 3-disk to a 2-sphere along a degree p map. Theorem 1 shows that $b_4^{(2)}(A_L; {\mathbb {F}}_{p}) =1$. Since $H_{3}(L; {\mathbb {F}}_{2} )=0$, [2, Theorem 5.1] shows that a related flag complex OL (the link of a vertex in the Salvetti complex of $A_{L}$) embeds into a flag triangulation T of $S^6$. This is where we need $p \ne 2$; interestingly this goes back to the fact that van Kampen’s obstruction to embedding d-dimensional simplicial complexes into ${\mathbb {R}}^{2d}$ is an order two invariant.

Now, $A_L$ is commensurable to the right-angled Coxeter group $W_{OL}$ by [8], and $W_{OL}$ is a subgroup of $W_{T}$. This acts properly on the associated Davis complex, a contractible 7-manifold, so we obtain the desired proper action for a finite index subgroup of $A_L$.

We then show that the ${\mathbb {F}}_{p}$-Singer Conjecture fails for either the right-angled Coxeter group associated to T or to a link of an odd-dimensional simplex in T. In other words, there must be a right-angled Coxeter group counterexample in one of the dimensions 3, 5, or 7.

Taking cartesian products of counterexamples and surface groups, we get counterexamples in all the dimensions stated in the theorem. In this way, we also get a single closed aspherical manifold contradicting ${\mathbb {F}}_{p}$-Singer for a finite collection of primes. This suggests the following.

FormalPara Question 5

Given a closed aspherical manifold $M^n$, is there a number N so that for all primes $p > N$, the ${\mathbb {F}}_{p}$-Singer Conjecture holds for $M^n$?

Of course, this is at least as difficult as the ordinary Singer conjecture, but it seems interesting (and open) in many cases where the ordinary Singer conjecture is known. Along the same lines, one can modify the Lück conjecture by ignoring the contributions to torsion coming from a finite collection of exceptional primes which should be determined by the geometry of the manifold $M^n$ (akin to how the exceptional primes for right-angled Artin groups are determined by the complex L.) This seems particularly interesting for locally symmetric spaces and might be easier than the Bergeron–Venkatesh conjecture.

1 Right-angled Artin and Coxeter groups

We collect some facts about right-angled Artin groups (RAAG’s), right-angled Coxeter groups (RACG’s) and relations between one and the other which we will need later. The philosophy to keep in mind is that RAAGs are the things we can compute, RACGs are the things related to closed aspherical manifolds, and translating from the former to the later involves a bit of (classical) embedding theory.

Let L be a flag complex with vertex set V. The one-skeleton of L determines two group presentations. A presentation for the RAAG $A_L$ has generators $\{g_v\}_{v\in V}$; there are relations $[g_v, g_{v'}] = 1$ (i.e., $g_v$ and $g_{v'}$ commute) whenever $\{v,v'\}\in L^{(1)}$. The RACG $W_L$ is the quotient of $A_L$ formed by adding the relations $(g_v)^2=1$, for all $v\in V$.

We now describe a standard classifying space for a RAAG $A_L$. More precisely, let $T^V$ denote the product $(S^1)^V$. Each copy of $S^1$ is given a cell structure with one vertex $e_0$ and one edge. For each simplex $\sigma \in L$, $T(\sigma )$ denotes the subset of $T^V$ consisting of points $(x_v)_{v\in V}$ such that $x_v=e_0$ whenever v is not a vertex of $\sigma$. So, $T(\sigma )$ is a $(\dim \sigma +1)$-dimensional standard subtorus of $T^V$. The Salvetti complex for $A_L$ is the subcomplex $BA_L$ of $T^V$ defined as the union of the subtori $T(\sigma )$ over all simplices $\sigma$ in L:

$$\begin{aligned} BA_L:= \bigcup _{\sigma \subset L} T(\sigma ). \end{aligned}$$

The link of the unique vertex is a flag complex of the same dimension as L, and is usually denoted OL (and called the octahedralization of L.)

We now give a similar construction of a classifying space for the commutator subgroup $C_L$ of $W_L$ (which is torsion-free and finite index in $W_L$.) Let $I^V$ denote the product $([-1,1])^V$. Each copy of $[-1,1]$ is given a cell structure with two vertices and one edge. For each simplex $\sigma \subset L$, $I(\sigma )$ denotes the subset of $I^V$ consisting of those points $(x_v)_{v\in V}$ such that $x_v \in \{\pm 1\}$ whenever v is not a vertex of $\sigma$. So, $I(\sigma )$ is a disjoint union of parallel faces of $I^V$ of dimension $\dim \sigma +1$. The standard classifying space for $C_L$ is the subcomplex $BC_L$ of $I^V$ defined as the union of the $I(\sigma )$ over all simplices $\sigma$ in L:

$$\begin{aligned} BC_L:= \bigcup _{\sigma \in L} I(\sigma ). \end{aligned}$$

The link of each vertex of $BC_L$ is a copy of L. The universal cover of $BC_L$ is denoted $\Sigma _L$ and called the Davis complex of $W_L$. $({\mathbb {Z}}/2)^V$ acts on $I^V$ and preserves the subcomplex $BC_L$. The lifts of this induced action to $\Sigma _L$ are precisely $W_L$ and we have the exact sequence

$$\begin{aligned} 1 \rightarrow C_L \rightarrow W_L \rightarrow ({\mathbb {Z}}/2)^V \rightarrow 1. \end{aligned}$$

Lemma 6

Let L be a flag complex.

• (1) $A_L$ is commensurable to $W_{OL}$ [8].

• (2) $W_L$ is linear, and hence residually finite. Therefore, so is $A_L$.

• (3) If L is a triangulation of $S^{n-1}$, then $\Sigma _L$ is a contractible n-manifold [7].

With Davis in [2], we studied the minimal dimension of aspherical manifolds with right-angled Artin fundamental groups. Constructing such manifolds involves embedding right-angled Artin groups $A_L$ into manifold Coxeter groups $W_{S^{n-1}}$. This boils down to finding PL-embeddings of OL into spheres. The complexes OL have “join-like” properties which make them difficult to embed directly but one can compute when the van Kampen embedding obstruction vanishes for these complexes. It is a complete obstruction to PL-embedding d-complexes in $S^{2d}$, except when $d=2$, and gives the following embedding criterion.

Theorem 7

(2.2, 5.1 and 5.4, [2]) Suppose L is a d-dimensional flag complex, $d \ne 2$. Then OL embeds as a full subcomplex into a flag PL-triangulation of $S^{2d}$ if and only if $H_d(L; {\mathbb {F}}_2) = 0$.

The prime 2 plays a special role in this theorem because van Kampen’s obstruction looks at what happens to pairs of distinct points under a generic map $OL\rightarrow S^{2d}$, which leads to an order two (co)-homological invariant. Therefore, if $H_d(L;{\mathbb {F}}_2)=0$ we get an embedding $OL\hookrightarrow S^{2d}$ irrespective of the ${\mathbb {F}}_{p}$-homology of L for odd primes p. This observation is key to the proof of Theorem 4.

2 Proof of Theorem 1

Let L be a flag complex, $\Gamma =A_L$ the right-angled Artin group defined by this complex, $B\Gamma$ its Salvetti complex, and F any field. By Lemma 6, we can choose a chain $\Gamma _k\vartriangleleft \Gamma$ of normal, finite index subgroups with $\bigcap _{k}\Gamma _k=1$.

Consider the cover of the Salvetti complex $B\Gamma$ by the standard maximal tori $T_\alpha$. Its nerve is a simplex $\Delta$ since all the tori intersect at the base-point. For a simplex $\sigma$ in $\Delta$, we denote the intersection of the corresponding tori by $T_{\sigma }=\bigcap _{\alpha \in \sigma } T_{\alpha }$. We look at the finite cover $B\Gamma _k$ or equivalently, look at the coefficient module $V=F[\Gamma /\Gamma _k]$. The Mayer–Vietoris spectral sequence (see VII.4 [5]) corresponding to the cover $\{T_\alpha \}$ has $E^1$ term

$$\begin{aligned} E^1_{i,j}:=C_i(\Delta ;H_j(T_{\sigma };V))\implies H_{i+j}(B\Gamma ;V)=H_{i+j}(B\Gamma _k;F). \end{aligned}$$

The following lemma is crucial.

Lemma 8

$$\begin{aligned} \lim _{k \rightarrow \infty } \frac{\dim _{F} H_j(T_\sigma ; F[\Gamma /\Gamma _k])}{[\Gamma :\Gamma _k]}= {\left\{ \begin{array}{ll} 1 &{} \text {if } T_{\sigma }=pt, \text { and } j=0,\\ 0 &{} \text { otherwise.} \end{array}\right. } \end{aligned}$$

Proof

Since covers of tori are tori, the only way the homology of covers of $T_{\sigma }$ can grow linearly is if the number of components of the preimage of $T_{\sigma }$ in $B\Gamma _k$ grows linearly in the index. Since $\Gamma _k$ is a residual sequence of normal covers, the number of components grows linearly if and only if $T_{\sigma }$ is a point. In more detail, since the cover is normal, the number of components is the ratio of indices $\frac{[\Gamma :\Gamma _k]}{|\pi _1T_{\sigma }:\pi _1T_{\sigma }\cap \Gamma _k|}$, and since the sequence $\Gamma _k$ is residual, the denominator grows with k as long as $\pi _1T_{\sigma }$ is infinite. $\square$

Therefore, up to an error whose dimension is sublinear in the index $[\Gamma :\Gamma _k]$, the spectral sequence is concentrated on the $E^1_{i,0}$ line. This implies

$$\begin{aligned} \limsup _{k} \frac{\dim _{F} E^2_{i,0}}{[\Gamma :\Gamma _k]}=\limsup _{k} \frac{b_i(B\Gamma _k;F)}{[\Gamma :\Gamma _k]}. \end{aligned}$$

(1)

Next, we approximate the chain complex $E^1_{i,0}$ by something that we will be able to compute exactly. For this, set

$$\begin{aligned} V_{\sigma }:= {\left\{ \begin{array}{ll} F[\Gamma /\Gamma _k] &{} \text {if } T_{\sigma }=pt,\\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Then the projection $E^1_{i,0}\rightarrow C_i(\Delta ;V_{\sigma })$ is a chain map and its kernel has dimension that is sublinear in the index $[\Gamma :\Gamma _k]$. Therefore

$$\begin{aligned} \limsup _{k} \frac{ \dim _{F} E^2_{i,0}}{[\Gamma :\Gamma _k]}= \limsup _{k} \frac{ \dim _{F} H_i(\Delta ;V_{\sigma })}{[\Gamma :\Gamma _k]}. \end{aligned}$$

(2)

The quantity in the limit on the right can be computed exactly, just in terms of the topology of L.

Lemma 9

$$\begin{aligned} \frac{\dim _{F} H_i(\Delta ;V_{\sigma })}{[\Gamma :\Gamma _k]}={\bar{b}}_{i-1}(L;F). \end{aligned}$$

Proof

The complex $\Delta$ has a subcomplex ${\mathcal {L}}\subset \Delta$ whose simplices are those intersections of tori that consist of more than one point. In other words,

$$\begin{aligned} {\mathcal {L}}:=\{\sigma \subset \Delta \mid V_{\sigma }=0\}. \end{aligned}$$

This complex ${\mathcal {L}}$ is precisely the nerve of the cover of L by maximal simplices, so it is homotopy equivalent to L. From the definition of ${\mathcal {L}}$ and $V_{\sigma }$ we get the exact sequence of chain complexes

$$\begin{aligned} 0\rightarrow C_*({\mathcal {L}};F)\otimes V\rightarrow C_*(\Delta ;F)\otimes V\rightarrow C_*(\Delta ;V_{\sigma })\rightarrow 0. \end{aligned}$$

Since $\Delta$ is a simplex, this implies

$$\begin{aligned} H_i(\Delta ;V_{\sigma })\cong \bar{H}_{i-1}(L;F)\otimes V. \end{aligned}$$

This finishes the proof since V is a $[\Gamma :\Gamma _k]$-dimensional F-vector space. $\square$

The theorem follows from this lemma, together with (1) and (2). Note that we only used normality of the $\Gamma _k$ in Lemma 8. The proof goes through for chains $\Gamma _k$ where the number of lifts in $B\Gamma _k$ of standard n-tori in $B\Gamma$ for $n > 0$ grows sublinearly. For example, this occurs for normal chains in a finite index subgroup of $\Gamma$.

Corollary 10

For a finite index subgroup H of a RAAG $\Gamma$ we have

$$\begin{aligned} b_i^{(2)}(H;F) = [\Gamma :H]b_i^{(2)}(\Gamma ;F). \end{aligned}$$

Proof

Let $\Gamma _k \vartriangleleft H$ be a normal chain. The cover BH of $B\Gamma$ has $\le [\Gamma :H]$ lifts of each torus. For each lifted torus ${\tilde{T}}_{\sigma }$ in BH, the number of lifts in $B\Gamma _k$ is given by

$$\begin{aligned} \frac{[H:\Gamma _k]}{|\pi _1{\tilde{T}}_{\sigma }:\pi _1{\tilde{T}}_{\sigma }\cap \Gamma _k|}, \end{aligned}$$

which is sublinear. Hence, $b_i^{(2)}(\Gamma ;F) = \lim \frac{b_{i}(B\Gamma _k; F)}{[\Gamma :\Gamma _k]}$, which implies the multiplicativity formula. $\square$

3 Mayer–Vietoris sequences for F-$L^2$-Betti numbers of Coxeter groups

Let L be a flag complex and $W_L$ the corresponding RACG. Look at a decomposition $L = A \cup _{C} B$ where A, B and hence C are full subcomplexes of L. The Coxeter group $W_L$ splits as an amalgamated product $W_L = W_{A} *_{W_{C}} W_{B}$, and our goal in this section is to describe relations between F-$L^2$-Betti numbers that arise from such splittings. Everywhere in this section coefficients are in an arbitrary field F and will be omitted to improve readability.

Let $\Gamma _k$ be a chain of finite index torsion-free normal subgroups with $\bigcap _{k} \Gamma _k = 1$ (note that any residual normal chain in $W_L$ is eventually torsion-free.) Let $\Sigma _L$ be the associated Davis complex for $W_L$, and let $Y_L^{k} = \Sigma _L/\Gamma _k$.

Given any full subcomplex A of L, the RACG $W_A$ is a subgroup of $W_L$, and the corresponding Davis complex $\Sigma _{A}$ is naturally a subcomplex of $\Sigma _{L}$. The stabilizer of $\Sigma _A$ in $W_{L}$ is precisely $W_{A}$. The $W_L$-orbit of $\Sigma _{A}$ in $\Sigma _L$ is a disjoint union of copies of $\Sigma _{A}$. The intersections of $\Gamma _k$ with $W_{A}$ give a corresponding chain of finite index subgroups $W_{A} \cap \Gamma _k \vartriangleleft W_A$.

We let $Y_A^{k}$ denote the image of this orbit in $Y_{L}^{k}$, so that $Y_A^{k}$ is a disjoint union of $\frac{[W_L:\Gamma _k]}{[W_{A}:\Gamma _k \cap W_{A}]}$ copies of $\Sigma _{A}/ W_{A} \cap \Gamma _k$. It follows that we can compute $b^{(2)}_{i}(W_{A})$ (with respect to the chain $W_{A} \cap \Gamma _k$) using $Y_{A}^{k}$:

$$\begin{aligned} b^{(2)}_{i}( W_{A}) =\limsup _{k} \frac{b_{i}(Y_{A}^{k})}{[W_{L}: \Gamma _k ]}. \end{aligned}$$

Suppose that $L=A\cup _CB$ where A, B and hence C are full subcomplexes of L. We then have a decomposition of spaces:

$$\begin{aligned} Y_L^{k} = Y_{A}^{k} \cup _{Y_{C}^{k}} Y_{B}^{k}, \end{aligned}$$

and hence a Mayer–Vietoris sequence

$$\begin{aligned} \dots \rightarrow H_{i}(Y_{C}^{k}) \rightarrow H_{i}(Y_{A}^{k}) \oplus H_{i}(Y_{B}^{k}) \rightarrow H_{i}(Y_{L}^{k}) \rightarrow \dots \end{aligned}$$

By the above discussion taking $\limsup$ of dimensions of the homology groups in this sequence divided by $[W_{L}: \Gamma _k]$ gives F-$L^{2}$-Betti numbers of the corresponding Coxeter groups. Since $\limsup$ is subadditive, it follows that having $b^{(2)}_{i}=0$ for one of the terms gives the usual inequalities between the nearby terms.

The decomposition we will use is when $A = {{\,\mathrm{St}\,}}(v)$ is the star of a vertex v, $B = L-v$ is its complement and $C = {{\,\mathrm{Lk}\,}}(v)$ is the link of v. In this case, the Mayer–Vietoris sequence leads to the following inequalities.

Lemma 11

• (1) $b^{(2)}_i(W_{L})\le b_i^{(2)}(W_{L-v})$      if $b^{(2)}_{i-1}(W_{{{\,\mathrm{Lk}\,}}(v)})=0$,

• (2) $b^{(2)}_i(W_{L})\ge b_i^{(2)}(W_{L-v})$      if $b^{(2)}_{i}(W_{{{\,\mathrm{Lk}\,}}(v)})=0$.

Proof

Removing the vertex v from L gives a Mayer–Vietoris sequence

$$\begin{aligned}&\dots \rightarrow H_{i}(Y_{{{\,\mathrm{Lk}\,}}(v)}^{k}) \xrightarrow {i_{1*}\oplus i_{2*}} H_{i}(Y_{{{\,\mathrm{St}\,}}(v))}^{k}) \oplus H_{i}(Y_{L - v}^{k}) \rightarrow H_{i}(Y_L^{k}) \rightarrow H_{i-1}(Y_{{{\,\mathrm{Lk}\,}}(v)}^{k}) \rightarrow \dots \end{aligned}$$

The map $i_1:Y^k_{{{\,\mathrm{Lk}\,}}(v)}\rightarrow Y^k_{{{\,\mathrm{St}\,}}(v)}$ is an inclusion of the form $Y\times \{\pm 1\}\hookrightarrow Y\times [-1,1]$, so $i_{1*}$ maps $H_{i}(Y_{{{\,\mathrm{Lk}\,}}(v)}^{k})$ onto $H_{i}(Y_{{{\,\mathrm{St}\,}}(v)}^{k})$. The stated inequalities follow from this. $\square$

Iteratively removing vertices leads to the following lemma. It lets us reduce dimension by passing from complexes to their links.

Lemma 12

• (1) If A is a flag complex with $b^{(2)}_{i}(W_A) \ne 0$, then there exists a vertex $v \in A$ and a full subcomplex B of ${{\,\mathrm{Lk}\,}}_{A}(v)$ with $b_{i-1}^{(2)}(W_B) \ne 0$.

• (2) If L is a flag complex with $b^{(2)}_{i}(W_{L}) = 0$ and if A is a full subcomplex of L with $b^{(2)}_{i}(W_A) \ne 0$, then there exists a vertex $v \in L$ and a full subcomplex B of ${{\,\mathrm{Lk}\,}}_{L}(v)$ with $b_{i}^{(2)}(W_B) \ne 0$.

Proof

Assume that all the link terms have $b^{(2)}_{i-1}=0$. Then removing vertices from A one at a time until we are left with a single vertex leads, by the first part of Lemma 11, to

$$\begin{aligned} b^{(2)}_i(W_A)\le b^{(2)}_i(W_{A-v})\le \dots \le b^{(2)}_i(W_{pt})=0 \end{aligned}$$

which contradicts the assumption that $b_i^{(2)}(W_A)>0$. This proves the first part.

Now, assume that all the link terms have $b^{(2)}_i=0$. Then removing vertices from L one at a time until we are left with A leads, by the second part of Lemma 11, to

$$\begin{aligned} b^{(2)}_i(W_L)\ge b^{(2)}_i(W_{L-v})\ge \dots \ge b^{(2)}_i(W_{A}) \end{aligned}$$

which contradicts the assumption $b^{(2)}_i(W_L)=0<b^{(2)}_i(W_A)$. This proves the second part. $\square$

4 Proof of Theorem 4

We are now ready for the proof of Theorem 4. Fix an odd prime p. Let $L=S^2\cup _pD^3$ be a flag triangulation of a complex obtained by gluing a 3-disk to a 2-sphere via a degree p map. Since $H_{3} (L; {\mathbb {F}}_{2}) = 0$, by Theorem 7 the octahedralization OL embeds as a full subcomplex of a flag PL-triangulation T of $S^6$.

Using commensurability we choose a common finite index subgroup N of $A_L$ and $W_{OL}$, which is normal in $W_{OL}$. We fix a torsion-free normal residual chain in $W_{T}$ which intersects $W_{OL}$ inside N. These are abundant as there is an obvious retraction $r:W_T \rightarrow W_{OL}$ so we can intersect any residual chain with $r^{-1}(N)$.

Since $\Sigma _{T}$ is a 7-manifold, the ${\mathbb {F}}_{p}$-Singer Conjecture predicts vanishing of $b^{(2)}_{*}(W_{T};{\mathbb {F}}_{p})$, and similarly, vanishing for the links of odd-dimensional simplices.

Proposition 13

The ${\mathbb {F}}_{p}$-Singer Conjecture fails either for T, or for one of the links of 1 or 3-dimensional simplices.

Proof

Suppose the ${\mathbb {F}}_{p}$-Singer Conjecture holds for T, and in particular $b^{(2)}_{4}(W_{T};{\mathbb {F}}_{p})= 0$. Since the chain in $W_{OL}$ is contained in N and N has finite index in $A_{L}$, Corollary 10 implies $b^{(2)}_{4}(W_{OL};{\mathbb {F}}_{p}) \ne 0$. By the second part of Lemma 12 applied to OL and T, there is a full subcomplex B of ${{\,\mathrm{Lk}\,}}_{T}(v)$ with $b_4^{(2)}(W_B; {\mathbb {F}}_{p}) \ne 0$. Now we apply the first part of Lemma 12 to B, to get a full subcomplex C of ${{\,\mathrm{Lk}\,}}_{B}(u)$ with $b_3^{(2)}(W_{C}; {\mathbb {F}}_{p}) \ne 0$. Note that ${{\,\mathrm{Lk}\,}}_{B}(u)$, and therefore C, is a full subcomplex of ${{\,\mathrm{Lk}\,}}_{{{\,\mathrm{Lk}\,}}_{T}(v)}(u)={{\,\mathrm{Lk}\,}}_{T}(uv)\approx S^{4}$.

If the ${\mathbb {F}}_{p}$-Singer Conjecture still holds for ${{\,\mathrm{Lk}\,}}_{T}(uv)$, we can repeat this argument to get get a full subcomplex D of ${{\,\mathrm{Lk}\,}}_{T}(\sigma ^{3})\approx S^{2}$ with $b_2^{(2)}(W_{D}; {\mathbb {F}}_{p}) \ne 0$. Now, the ${\mathbb {F}}_{p}$-Singer Conjecture must fail for ${{\,\mathrm{Lk}\,}}_{T}(\sigma ^{3})$, because repeating this argument once more produces a subcomplex of $S^{0}$ with $b^{(2)}_{1} \ne 0$, which is clearly impossible. $\square$

It follows that the ${\mathbb {F}}_{p}$-Singer Conjecture must fail in at least one of the dimensions 3, 5, or 7. Since a closed surface $S_g$ with $g \ge 2$ has $b_1^{(2)}(\pi _1(S_g); {\mathbb {F}}_{p}) \ne 0$, taking cartesian products between surface groups and our counterexamples gives, via the Künneth formula, counterexamples in all odd dimensions $\ge 7$ and all even dimensions $\ge 14$.

Remark

The reason why we used a 3-dimensional complex $S^2\cup _p D^3$ instead of a 2-dimensional complex $S^1\cup _p D^2$ above is twofold. First, for 2-dimensional complexes there are other obstructions to embedding in $S^{4}$ besides the classical van Kampen obstruction [10]. Second, in codimension 2 there is a problem of extending a given triangulation on the complex to a triangulation of $S^{4}$ (the embedding might be locally knotted.) If for a flag triangulation L of $S^1\cup _pD^2$ one can exhibit its octahedralization OL as a subcomplex of $S^4$, then our method would yield a 5-dimensional counterexample to the ${\mathbb {F}}_{p}$-Singer Conjecture.