A Note on Complex-Hyperbolic Kleinian Groups

Abstract

Let $\varGamma$ be a discrete group of isometries acting on the complex hyperbolic n-space $\mathbb {H}^n_\mathbb {C}$. In this note, we prove that if $\varGamma$ is convex-cocompact, torsion-free, and the critical exponent $\delta (\varGamma )$ is strictly lesser than 2, then the complex manifold $\mathbb {H}^n_\mathbb {C}/\varGamma$ is Stein. We also discuss several related conjectures.

The theory of complex hyperbolic manifolds and complex-hyperbolic Kleinian groups (i.e., discrete holomorphic isometry groups of complex hyperbolic spaces $\mathbb {H}^n_\mathbb {C}$) is a rich mixture of Riemannian and complex geometry, topology, dynamics, symplectic geometry and complex analysis. The purpose of this note is to discuss interactions of the theory of complex-hyperbolic Kleinian groups and the function theory of complex-hyperbolic manifolds. Let $\varGamma$ be a discrete group of isometries acting on the complex-hyperbolic n-space, $\mathbb {H}^n_\mathbb {C}$, the unit ball $\mathbf {B}^n\subset \mathbb {C}^n$ equipped with the Bergman metric. A fundamental numerical invariant associated with $\varGamma$ is the critical exponent $\delta (\varGamma )$ of $\varGamma$, defined by

$$\begin{aligned} \delta (\varGamma ) = \inf \left\{ s : \sum _{\gamma \in \varGamma } e^{-s\cdot d(x,\gamma x)} <\infty \right\} , \end{aligned}$$

where $x\in \mathbb {H}^n_\mathbb {C}$ is any¹ point. The critical exponent measures the rate of exponential growth the $\varGamma$-orbit $\varGamma x\subset \mathbb {H}^n_\mathbb {C}$; it also equals the Hausdorff dimension of the conical limit set of $\varGamma$, see [7, 8].

Our main result is:

FormalPara Theorem 1

Suppose that $\varGamma < {\text {Aut}}(\mathbf {B}^n)$ is a convex-cocompact, torsion-free discrete subgroup satisfying $\delta (\varGamma )<2$. Then, $M_\varGamma = \mathbf {B}^n/\varGamma$ is Stein.

The condition on the critical exponent in the above theorem is sharp, since for a complex Fuchsian subgroup $\varGamma < {\text {Aut}}(\mathbf {B}^n)$, $\delta (\varGamma ) = 2$, but the quotient $M_\varGamma = \mathbf {B}^n/\varGamma$ is non-Stein, because the convex core of $M_\varGamma$ is a complex curve, see Example 4. On the other hand, if $\varGamma$ is a torsion-free real Fuchsian subgroup or a small deformation of such (see Example 3), then $\varGamma$ satisfies the condition of the above theorem.

The main ingredients in the proof of Theorem 1 are Proposition 11 and Theorem 15. The condition “convex-cocompact” is only used in Proposition 11, whereas Theorem 15 holds for any torsion-free discrete subgroup $\varGamma <{\text {Aut}}(\mathbf {B}^n)$ satisfying $\delta (\varGamma )<2$.

FormalPara Conjecture 2

Theorem 1 holds if we omit the “convex-cocompact” assumption on $\varGamma$.

In Sect. 4, we discuss other conjectural generalizations of Theorem 1 and supporting results.

1 Preliminaries

In this section, we recall some definitions and basic facts about the n-dimensional complex hyperbolic space, we refer to [9, 11] for details.

Consider the n-dimensional complex vector space $\mathbb {C}^{n+1}$ equipped with the pseudo-hermitian bilinear form:

$$\begin{aligned} \langle z, w\rangle =-z_0 \bar{w}_0 + \sum _{k=1}^n z_k \bar{w}_k \end{aligned}$$

(1)

and define the quadratic form q(z) of signature (n, 1) by $q(z):= \langle z, z\rangle$. Then, q defines the negative light cone $V_-:= \{z: q(z)<0\} \subset \mathbb {C}^{n+1}$. The projection of $V_-$ in the projectivization of $\mathbb {C}^{n+1}$, $\mathbb {P}^{n}$, is an open ball which we denote by $\mathbf {B}^n$.

The tangent space $T_{[z]} \mathbb {P}^n$ is naturally identified with $z^\perp$, the orthogonal complement of $\mathbb {C}z$ in V, taken with respect to $\langle \cdot , \cdot \rangle$. If $z\in V_-$, then the restriction of q to $z^\perp$ is positive-definite, hence, $\langle \cdot , \cdot \rangle$ project to a hermitian metric h (also denoted $\langle \cdot , \cdot \rangle _h$) on $\mathbf {B}^n$. The complex hyperbolic n-space $\mathbb {H}^n_{\mathbb {C}}$ is $\mathbf {B}^n$ equipped with the hermitian metric h. The boundary $\partial \mathbf {B}^n$ of $\mathbf {B}^n$ in $\mathbb {P}^n$ gives a natural compactification of $\mathbf {B}^n$.

In this note, we usually denote the complex hyperbolic n-space by $\mathbf {B}^n$. The real part of the hermitian metric h defines a Riemannian metric g on $\mathbf {B}^n$. The sectional curvature of g varies between $-4$ and $-1$. We denote the distance function on $\mathbf {B}^n$ by d. The distance function satisfies

$$\begin{aligned} \cosh ^2(d(0, z))= (1-|z|^2)^{-1}. \end{aligned}$$

(2)

A real linear subspace $W\subset \mathbb {C}^{n+1}$ is said to be totally real with respect to the form (1) if for any two vectors $z, w\in W$, $\langle z, w\rangle \in \mathbb {R}$. Such a subspace is automatically totally real in the usual sense: $J W\cap W=\{0\}$, where J is the almost complex structure on V. (Real) geodesics in $\mathbf {B}^n$ are projections of totally real indefinite (with respect to q) 2-planes in $\mathbb {C}^{n+1}$ (intersected with $V_-$). For instance, geodesics through the origin $0\in \mathbf {B}^n$ are Euclidean line segments in $\mathbf {B}^n$. More generally, totally-geodesic real subspaces in $\mathbf {B}^n$ are projections of totally real indefinite subspaces in $\mathbb {C}^{n+1}$ (intersected with $V_-$). They are isometric to the real hyperbolic space $\mathbb {H}^n_\mathbb {R}$ of constant sectional curvature $-1$.

Complex geodesics in $\mathbf {B}^n$ are projections of indefinite complex 2-planes. Complex geodesics are isometric to the unit disk with the hermitian metric:

$$\begin{aligned} \frac{dz d{\bar{z}}}{(1-|z|^2)^2}, \end{aligned}$$

which has constant sectional curvature $-4$. More generally, k-dimensional complex hyperbolic subspaces $\mathbb {H}^k_\mathbb {C}$ in $\mathbf {B}^n$ are projections of indefinite complex $(k+1)$-dimensional subspaces (intersected with $V_-$).

All complete totally-geodesic submanifolds in $\mathbb {H}^n_\mathbb {C}$ are either real or complex hyperbolic subspaces.

The group ${\mathrm {U}}(n,1)\cong {\mathrm {U}}(q)$ of (complex) automorphisms of the form q projects to the group ${\text {Aut}}(\mathbf {B}^n) \cong {\mathrm {PU}}(n,1)$ of complex (biholomorphic, isometric) automorphisms of $\mathbf {B}^n$. The group ${\text {Aut}}(\mathbf {B}^n)$ is linear, its matrix representation is given, for instance, by the adjoint representation, which is faithful since ${\text {Aut}}(\mathbf {B}^n)$ has trivial center.

A discrete subgroup $\varGamma$ of ${\text {Aut}}(\mathbf {B}^n)$ is called a complex-hyperbolic Kleinian group. The accumulation set of an(y) orbit $\varGamma x$ in $\partial \mathbf {B}^n$ is called the limit set of $\varGamma$ and denoted by $\varLambda (\varGamma )$. The complement of $\varLambda (\varGamma )$ in $\partial \mathbf {B}^n$ is called the domain of discontinuity of $\varGamma$ and denoted by $\varOmega (\varGamma )$. The group $\varGamma$ acts properly discontinuously on $\mathbf {B}^n \cup \varOmega (\varGamma )$.

For a (torsion-free) complex-hyperbolic Kleinian group $\varGamma$, the quotient $\mathbf {B}^n/\varGamma$ is a Riemannian orbifold (manifold) equipped with push-forward of the Riemannian metric of $\mathbf {B}^n$. We reserve the notation $M_\varGamma$ to denote this quotient. The convex core of $M_\varGamma$ is the the projection of the closed convex hull of $\varLambda (\varGamma )$ in $\mathbf {B}^n$. The subgroup $\varGamma$ is called convex-cocompact if the convex core of $M_\varGamma$ is a nonempty compact subset. Equivalently (see [3]), $\overline{M}_\varGamma =(\mathbf {B}^n \cup \varOmega (\varGamma ))/\varGamma$ is compact.

Below are two interesting examples of convex-cocompact complex-hyperbolic Kleinian groups which will also serve as illustrations our results.

Example 3

(Real Fuchsian subgroups) Let $\mathbb {H}^2_\mathbb {R}\subset \mathbf {B}^n$ be a totally real hyperbolic plane. This inclusion is induced by an embedding $\rho :{{\,\mathrm{Isom}\,}}(\mathbb {H}^2_\mathbb {R}) = \mathrm {PSL}(2,\mathbb {R}) \rightarrow {\text {Aut}}(\mathbf {B}^n)$, whose image preserves $\mathbb {H}^2_\mathbb {R}$. Let $\varGamma '< {{\,\mathrm{Isom}\,}}(\mathbb {H}^2_\mathbb {R})$ be a uniform lattice. Then $\varGamma = \rho (\varGamma ')$ preserves $\mathbb {H}^2_\mathbb {R}$ and acts on it cocompactly. Such subgroups $\varGamma <{\text {Aut}}(\mathbf {B}^n)$ will be called real Fuchsian subgroups. The compact surface-orbifold $\varSigma = \mathbb {H}^2_\mathbb {R}/{\varGamma }$ is the convex core of $M_{\varGamma }$. The critical exponent $\delta (\varGamma )$ is 1.

Let $\varGamma _t$, $t\ge 0$, be a continuous family of deformations of $\varGamma _0 = \varGamma$ in ${\text {Aut}}(\mathbf {B}^n)$ such that $\varGamma _t$’s, for $t>0$, are convex-cocompact but not real Fuchsian. Such deformation exist as long as $\varGamma _t$ is, say, torsion-free, see, e.g., [15]. The groups $\varGamma _t$, $t>0$, are called real quasi-Fuchsian subgroups. The critical exponents of such subgroups are strictly greater than 1.

Example 4

(Complex Fuchsian subgroups) Let $\varGamma '$ be a cocompact subgroup of ${\mathrm {SU}}(1,1)$, the identity component isometry group of the real-hyperbolic plane (modulo ${\mathbb {Z}}_2$) and let ${\mathrm {SU}}(1,1) \rightarrow {\mathrm {SU}}(n,1)$ be any embedding. Note that ${\mathrm {SU}}(n,1)$ modulo center (isomorphic to ${\mathbb {Z}}_{n+1}$) is isomorphic to ${\mathrm {PU}}(n,1)$. By taking compositions, we get a representation $\rho :\varGamma '\rightarrow {\mathrm {PU}}(n,1)$. Then $\varGamma := \rho (\varGamma ')$ leaves a complex geodesic invariant in $\mathbf {B}^n$. Such subgroups $\varGamma$ will be called complex Fuchsian subgroups. In this case, $\mathrm {core}(M_{\varGamma }) = \mathbb {H}^1_\mathbb {C}/\varGamma$ is a compact complex curve in $M_{\varGamma }$ where $\mathbb {H}^1_\mathbb {C}$ is the $\varGamma$-invariant complex geodesic. The critical exponent $\delta (\varGamma )$ is 2.

2 Generalities on Complex Manifolds

By a complex manifold with boundary M, we mean a smooth manifold with (possibly empty) boundary $\partial M$ such that ${\text {int}}(M)$ is equipped with a complex structure and that there exists a smooth embedding $f: M\rightarrow X$ to an equidimensional complex manifold X, biholomorphic on ${\text {int}}(M)$. A holomorphic function on M is a smooth function which admits a holomorphic extension to a neighborhood of M in X.

Let X be a complex manifold and $Y\subset X$ is a codimension 0 smooth submanifold with boundary in X. The submanifold Y is said to be strictly Levi-convex if every boundary point of Y admits a neighborhood U in X such that the submanifold with boundary $Y\cap U$ can be written as

$$\begin{aligned} \{\phi \le 0\}, \end{aligned}$$

for some smooth submersion $\phi : U\rightarrow \mathbb {R}$ satisfying ${{\,\mathrm{Hess}\,}}(\phi ) >0,$ where ${{\,\mathrm{Hess}\,}}(\phi )$ is the holomorphic Hessian:

$$\begin{aligned} \left( \frac{\partial ^2 \phi }{\partial \bar{z}_i\partial z_j}\right) . \end{aligned}$$

Definition 5

A strongly pseudoconvex manifold M is a complex manifold with boundary which admits a strictly Levi-convex holomorphic embedding in an equidimensional complex manifold.

Definition 6

An open complex manifold Z is called holomorphically convex if for every discrete closed subset $A\subset Z$ there exists a holomorphic function $Z\rightarrow \mathbb {C}$ which is proper on A.

Alternatively,² one can define holomorphically convex manifolds as follows: For a compact K in a complex manifold M, the holomorphic convex hull $\hat{K}_M$ of K in M is

$$\begin{aligned} \hat{K}_M= \{z\in M: |f(z)|\le \sup _{w\in K} |f(w)|, \forall f\in {\mathcal {O}}_M\}. \end{aligned}$$

In the above, ${\mathcal {O}}_M$ denotes the ring of holomorphic functions on M. Then M is holomorphically convex iff for every compact $K\subset M$, the hull $\hat{K}_M$ is also compact.

Theorem 7

(Grauert [10]) The interior of every compact strongly pseudoconvex manifold M is holomorphically convex.

Definition 8

A complex manifold M is called Stein if it admits a proper holomorphic embedding in $\mathbb {C}^n$ for some n.

Equivalently, M is Stein iff it is holomorphically convex and holomorphically separable: That is, for every distinct points $x,y\in M$, there exists a holomorphic function $f: M\rightarrow \mathbb {C}$ such that $f(x) \ne f(y)$. We will use:

Theorem 9

(Rossi [13], Corollary on page 20) If a compact complex manifold M is strongly pseudoconvex and contains no compact complex subvarieties of positive dimension, then ${\text {int}}(M)$ is Stein.

We now discuss strong quasiconvexity and Stein property in the context of complex-hyperbolic manifolds. A classical example of a complex submanifold with Levi-convex boundary is a closed round ball $\overline{\mathbf {B}}^n$ in $\mathbb {C}^n$. Suppose that $\varGamma < {\text {Aut}}(\mathbf {B}^n)$ is a discrete torsion-free subgroup of the group of holomorphic automorphisms of $\mathbf {B}^n$ with (nonempty) domain of discontinuity $\varOmega = \varOmega (\varGamma )\subset \partial \mathbf {B}^n$. The quotient

$$\begin{aligned} \overline{M}_\varGamma = (\mathbf {B}^n \cup \varOmega )/\varGamma \end{aligned}$$

is a smooth manifold with boundary.

Lemma 10

$\overline{M}_\varGamma$ is strongly pseudoconvex.

Proof

We let $T_\varLambda$ denote the union of all projective hyperplanes in $P_\mathbb {C}^n$ tangent to $\partial \mathbf {B}^n$ at points of $\varLambda$, the limit set of $\varGamma$. Let $\widehat{\varOmega }$ denote the connected component of $P_\mathbb {C}^n{\setminus }T_\varLambda$ containing $\mathbf {B}^n$. It is clear that $\mathbf {B}^n\cup \varOmega \subset {\widehat{\varOmega }}$ is strictly Levi-convex. By the construction, $\varGamma$ preserves $\widehat{\varOmega }$. It is proven in [5, Thm. 7.5.3] that the action of $\varGamma$ on ${\widehat{\varOmega }}$ is properly discontinuous. Hence, $X:={\widehat{\varOmega }}/\varGamma$ is a complex manifold containing $\overline{M}_\varGamma$ as a strictly Levi-convex submanifold with boundary. $\square$

Specializing to the case when $\overline{M}_\varGamma$ is compact, i.e. $\varGamma$ is convex-cocompact, we obtain:

Proposition 11

Suppose that $\varGamma$ is torsion-free, convex-cocompact and $n>1$. Then:

• 1. $\partial \overline{M}_\varGamma$ is connected.

• 2. If ${\text {int}}(\overline{M}_\varGamma ) = M_\varGamma$ contains no compact complex subvarieties of positive dimension, then $M_\varGamma$ is Stein.

For example, as it was observed in [4], the quotient-manifold $\mathbf {B}^2/\varGamma$ of a real-Fuchsian subgroup $\varGamma < {\text {Aut}}(\mathbf {B}^2)$ is Stein while the quotient-manifold of a complex-Fuchsian subgroup $\varGamma < {\text {Aut}}(\mathbf {B}^2)$ is non-Stein.

3 Proof of Theorem 1

In this section, we construct certain plurisubharmonic functions on $M_\varGamma$, for each finitely generated, discrete subgroup $\varGamma < {\text {Aut}}(\mathbf {B}^n)$ satisfying $\delta (\varGamma ) < 2$. We use these functions to show that $M_\varGamma$ has no compact subvarieties of positive dimension. At the end of this section, we prove the main result of this paper.

Let X be a complex manifold. Recall that a continuous function $f:X\rightarrow \mathbb {R}$ is called plurisubharmonic³ if for any homomorphic map $\phi : V( \subset \mathbb {C}) \rightarrow X$, the composition $f\circ \phi$ is subharmonic. Plurisubharmonic functions f satisfy the maximum principle; in particular, if f restricts to a nonconstant function on a connected complex subvariety $Y\subset X$, then Y is noncompact.

Now, we turn to our construction of plurisubharmonic functions. Let $\varGamma < {\text {Aut}}(\mathbf {B}^n)$ be a discrete subgroup. Consider the Poincaré series

$$\begin{aligned} \sum _{\gamma \in \varGamma } ({1 - |\gamma (z)|^2}), \quad z\in \mathbf {B}^n. \end{aligned}$$

(3)

Lemma 12

Suppose that $\delta (\varGamma )<2$. Then (3) uniformly converges on compact sets.

Proof

Since $\delta (\varGamma ) < 2$, the Poincaré series

$$\begin{aligned} \sum _{\gamma \in \varGamma } e^{ -2 d(0,\gamma (z))} \end{aligned}$$

uniformly converges on compact subsets in $\mathbf {B}^n$. By (2), we get

$$\begin{aligned} e^{-2d(0,\gamma (z))} \le (1 - |\gamma (z)|^2) \le 4 e^{-2d(0,\gamma (z))}. \end{aligned}$$

(4)

Then, the result follows from the upper inequality. $\square$

Remark 13

Note that when $\delta (\varGamma )>2$, or when $\varGamma$ is of divergent type (e.g., convex-cocompact) and $\delta (\varGamma ) = 2$, then (3) does not converge. This follows from the lower inequality of (4).

Assume that $\delta (\varGamma )<2$. Define $F: \mathbf {B}^n\rightarrow {\mathbb {R}}$,

$$\begin{aligned} F(z) = \sum _{\gamma \in \varGamma } ({ |\gamma (z)|^2 - 1}). \end{aligned}$$

Since F is $\varGamma$-invariant, i.e., $F(\gamma z) = F(z)$, for all $\gamma \in \varGamma$ and all $z\in \mathbf {B}^n$, F descends to a function

$$\begin{aligned} f : M_\varGamma \rightarrow {\mathbb {R}}. \end{aligned}$$

Lemma 14

The function $f : M_\varGamma \rightarrow {\mathbb {R}}$ is plurisubharmonic.

Proof

Enumerate $\varGamma$ as $\varGamma = \{\gamma _1,\gamma _2,\ldots \}$. Consider the sequence of partial sums of the series F:

$$\begin{aligned} S_k(z) = \sum _{j\le k} ({ |\gamma _j(z)|^2 - 1}). \end{aligned}$$

Since each summand in the above is plurisubharmonic⁴, $S_k$ is plurisubharmonic for each $k\ge 1$. Moreover, the sequence of functions $S_k$ is monotonically decreasing. Thus, the limit $F = \lim _{k\rightarrow \infty } S_k$ is also plurisubharmonic, and hence so is f. $\square$

Note, however, that at this point, we do not yet know that the function f is nonconstant.

Now, we prove the main result of this section.

Theorem 15

Let $\varGamma$ be a torsion-free discrete subgroup of ${\text {Aut}}(\mathbf {B}^n)$. If $\delta (\varGamma ) < 2$, then $M_\varGamma$ contains no compact complex subvarieties of positive dimension.

Proof

Suppose that Y is a compact connected subvariety of positive dimension in $M_\varGamma$. Since $\pi _1(Y)$ is finitely generated, so is its image $\varGamma '$ in $\varGamma =\pi _1(M_\varGamma )$. Since $\delta (\varGamma ')\le \delta (\varGamma )$, by passing to the subgroup $\varGamma '$ we can (and will) assume that the group $\varGamma$ is finitely generated.

We construct a sequence of functions $F_k: \mathbf {B}^n\rightarrow \mathbb {R}$ as follows. For $k \in \mathbb {N}$, let $\varSigma _k\subset \varGamma -\{1\}$ denote the subset consisting of $\gamma \in \varGamma$ satisfying $d(0,\gamma (0)) \le k$. Since $\varGamma$ is a finitely generated linear group, it is residually finite and, hence, there exists a finite index subgroup $\varGamma _k<\varGamma$ disjoint from $\varSigma _k$. For each $k\in \mathbb {N}$, define $F_k: \mathbf {B}^n\rightarrow \mathbb {R}$ as the sum:

$$\begin{aligned} F_k(z) = \sum _{\gamma \in \varGamma _k} ({ |\gamma (z)|^2 - 1}). \end{aligned}$$

Since

$$\begin{aligned} \bigcap _{k\in \mathbb {N}} \varGamma _k=\{1\}, \end{aligned}$$

the sequence of functions $F_k$ converges to $(|z|^2-1)$ uniformly on compact subsets of $\mathbf {B}^n$. As before, each $F_k$ is plurisubharmonic (cf. Lemmata 12, 14).

Let ${\widetilde{Y}}$ be a connected component of the preimage of Y under the projection map $\mathbf {B}^n \rightarrow M_\varGamma$. Since ${\widetilde{Y}}$ is a closed, noncompact subset of $\mathbf {B}^n$, the function $(|z|^2-1)$ is nonconstant on ${\widetilde{Y}}$. As the sequence $(F_k)$ converges to $(|z|^2-1)$ uniformly on compacts, there exists $k\in \mathbb {N}$, such that $F_k$ is nonconstant on ${\widetilde{Y}}$. Let $f_k : M_k = M_{\varGamma _k} \rightarrow \mathbb {R}$ denote the function obtained by projecting $F_k$ to $M_k$, and $Y_k$ be the image of ${\widetilde{Y}}$ under the projection map $\mathbf {B}^n\rightarrow M_k$. Since $M_k$ is a finite covering of $M_\varGamma$, the subvariety $Y_k\subset M_k$ is compact. Moreover, $f_k$ is a nonconstant plurisubharmonic function on $Y_k$ since $F_k$ is such a function on ${\widetilde{Y}}$. This contradicts the maximum principle. $\square$

Remark 16

Regarding Remark 13: The failure of convergence of the series (3) as pointed out in Remark 13 is not so surprising. In fact, if $\varGamma$ is a complex Fuchsian group, then $\delta (\varGamma ) = 2$ and the convex core of $M_\varGamma$ is a compact Riemann surface, see Example 4. Thus, our construction of F must fail in this case.

We conclude this section with a proof of the main result of this paper.

Proof of Theorem 1

By Theorem 15, $M_\varGamma$ does not have compact complex subvarieties of positive dimensions. Then, by the second part of Proposition 11, $M_\varGamma$ is Stein. $\square$

4 Further Remarks

In relation to Theorem 1, it is also interesting to understand the case when $\delta (\varGamma ) = 2$, that is: For which convex-cocompact, torsion-free subgroups $\varGamma$ of ${\text {Aut}}(\mathbf {B}^n)$ satisfying $\delta (\varGamma )=2$, is the manifold $M_\varGamma$ Stein? It has been pointed out before that a complex Fuchsian subgroup $\varGamma < {\text {Aut}}(\mathbf {B}^n)$ satisfies $\delta (\varGamma ) = 2$, but the manifold $M_\varGamma$ is not Stein. In fact, the convex core of $M_\varGamma$ is a complex curve, see Remark 16. We conjecture that complex Fuchsian subgroups are the only such non-Stein examples.

Conjecture 17

Let $\varGamma <{\text {Aut}}(\mathbf {B}^n)$ be a convex-cocompact, torsion-free subgroup such that $\delta (\varGamma ) = 2$. Then, $M_\varGamma$ is non-Stein if and only if $\varGamma$ is a complex Fuchsian subgroup.

We illustrate this conjecture in the following very special case: Let $\phi : \pi _1(\varSigma ) \rightarrow {\text {Aut}}(\mathbf {B}^n)$ be a faithful convex-cocompact representation where $\varSigma$ is a compact Riemann surface of genus $g \ge 2$. Then $\phi$ induces a (unique) equivariant harmonic map:

$$\begin{aligned} F: {\widetilde{\varSigma }} \rightarrow \mathbf {B}^n. \end{aligned}$$

which descends to a harmonic map $f: \varSigma \rightarrow M_\varGamma$.

Proposition 18

Suppose that F is a holomorphic immersion. Then $\varGamma = \phi (\pi _1(\varSigma ))$ satisfies $\delta (\varGamma )\ge 2$. Moreover, if $\delta (\varGamma )= 2$, then $\varGamma$ preserves a complex line. In particular, $\varGamma$ is a complex Fuchsian subgroup of ${\text {Aut}}(\mathbf {B}^n)$.

Proof

Noting that $M_\varGamma$ contains a compact complex curve, namely, $f(\varSigma )$, the first part follows directly from Theorem 1.

For the second part, we let Y denote the surface ${{\tilde{\varSigma }}}$ equipped with the Riemannian metric obtained via pull-back of the Riemannian metric g on $\mathbf {B}^n$. The entropy⁵h(Y) of Y is bounded above by $\delta (\varGamma )$, that is

$$\begin{aligned} h(Y) \le 2. \end{aligned}$$

(5)

This can be seen as follows: The distance function $d_Y$ on Y satisfies

$$\begin{aligned} d_Y(y_1,y_2) \ge d(F(y_1), F(y_2)). \end{aligned}$$

Therefore, the exponential growth-rate $\delta _Y$ of $\pi _1(\varSigma )$-orbits in Y satisfies $\delta _Y\le \delta (\varGamma )$. On the other hand, the quantity $\delta _Y=h(Y)$ since $\pi _1(\varSigma )$ acts cocompactly on Y.

Assume that ${\widetilde{\varSigma }}$ is endowed with a conformal Riemannian metric of constant $-4$ sectional curvature. Since ${{\tilde{\varSigma }}}$ is a symmetric space, we have

$$\begin{aligned} h^2(Y)\mathrm {Area}(Y/\varGamma ) \ge h^2({\widetilde{\varSigma }})\mathrm {Area}(\varSigma ), \end{aligned}$$

see [1, p. 624]. The inequality (5) together with the above implies that

$$\begin{aligned} \mathrm {Area}(Y/\varGamma ) \ge \mathrm {Area}(\varSigma ). \end{aligned}$$

On the other hand, since $f: Y/\varGamma \rightarrow M_\varGamma$ is holomorphic, $4\cdot \mathrm {Area}(Y/\varGamma )$ equals to the Toledo invariant $c(\phi )$ (see [14]) of the representation $\phi$. Since $c(\phi ) \le 4\pi (g-1)$, the inequality $\mathrm {Area}(Y/\varGamma ) \ge \mathrm {Area}(\varSigma ) = \pi (g-1)$ shows that $\mathrm {Area}(Y/\varGamma ) = \pi (g-1)$ or, equivalently, $c(\phi ) = 4\pi (g-1)$. By the main result of [14], $\varGamma$ preserves a complex-hyperbolic line in $\mathbf {B}^n$. $\square$

Remark 19

The assumption that F is an immersion can be eliminated: Instead of working with a Riemannian metric, one can work with a Riemannian metric with finitely many singularities.

Motivated by Theorem 15, we also make the following conjecture.

Conjecture 20

If $\varGamma < {\text {Aut}}(\mathbf {B}^n)$ is discrete, torsion-free, and $\delta (\varGamma )<2k$, then $M_\varGamma$ does not contain compact complex subvarieties of dimension $\ge k$.

We conclude this section with a verification of this conjecture under a stronger hypothesis.

Proposition 21

If $\varGamma < {\text {Aut}}(\mathbf {B}^n)$ is discrete, torsion-free, and $\delta (\varGamma )<2k-1$, then $M_\varGamma$ does not contain compact complex subvarieties of dimension $\ge k$.

Proof

Note that if $\varGamma$ is elementary (i.e., virtually abelian), then $\delta (\varGamma ) = 0$. In this case, the result follows from Theorem 15. For the rest, we assume that $\varGamma$ is nonelementary.

By [2, Sec. 4], there is a natural map $f: M_\varGamma \rightarrow M_\varGamma$ homotopic to the identity map ${\mathrm {id}}_{M_\varGamma }: M_\varGamma \rightarrow M_\varGamma$ and satisfying

$$\begin{aligned} |\mathrm {Jac}_p(f)| \le \left( \frac{\delta (\varGamma )+1}{p}\right) ^p,\quad 2\le p\le 2n, \end{aligned}$$

where $\mathrm {Jac}_p(f)$ denotes the p-Jacobian of f. When $\delta (\varGamma )<2k-1$, we have $|\mathrm {Jac}_p(f)|<1$, for $p\in [2k,2n]$. This means that f strictly contracts the volume form on each p-dimensional tangent space at every point $x\in M_\varGamma$, for $p\in [2k,2n]$.

Let $Y\subset M_\varGamma$ be a compact complex subvariety of dimension $\ge k$ (real dimension $\ge 2k$). Then, Y is also a volume minimizer in its homology class. Since f strictly contracts volume on Y, f(Y) has volume strictly lesser than that of Y. However, f being homotopic to ${\mathrm {id}}_{M_\varGamma }$, f(Y) belongs to the homology class of Y. This is a contradiction to the fact that Y minimizes volume its homology class. $\square$

Remark 22

Note that Proposition 21 gives an alternative proof of Theorem 15 (hence Theorem 1) under a stronger hypothesis, namely $\delta (\varGamma )\in (0,1)$. However, this method fails to verify Theorem 15 in the case when $\delta (\varGamma )\in [1,2)$.

Finally, we note that the papers [6, 16] contain other interesting results and conjectures on Stein properties of complex-hyperbolic manifolds.