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Large and small covers of a hyperbolic manifold. (English) Zbl 1272.30062
For a discrete subgroup \(\Gamma\) of the isometry group of the hyperbolic space \(\mathbb H^{n+1}\), we denote by \(\delta(\Gamma)\) the exponent of convergence of its Poincaré series. By work of C. J. Bishop and P. W. Jones [Acta Math. 179, No. 1, 1–39 (1997; Zbl 0921.30032)] it is known that \(\delta(\Gamma)\) coincides with the Hausdorff dimension of the conical limit set of \(\Gamma\).
In the paper under review, the authors focus on the behaviour of \(\delta\) under taking non-trivial normal subgroups. K. Falk and B. O. Stratmann [Tohoku Math. J., II. Ser. 56, No. 4, 571–582 (2004; Zbl 1069.30070)] showed that if \(\hat \Gamma\) is a non-trivial normal subgroup of a non-elementary \(\Gamma\), then \(\delta(\hat \Gamma) \geq \delta(\Gamma)/2\). In the present paper, it is shown that in the case when \(n=1\) and \(\Gamma\) is a Fuchsian group corresponding to a closed hyperbolic surface, this inequality is in a sense best possible: there is a sequence of normal subgroups \(\Gamma_i\) with \(\delta(\Gamma_i)\) tending to \(1/2\). Furthermore, it is also shown that when \(\Gamma\) is non-elementary and convex cocompact for arbitrary \(n\), for any non-trivial normal subgroup \(\hat \Gamma\), the strict inequality \(\delta(\hat \Gamma) > \delta(\Gamma)/2\) holds.
In contrast to these results, the authors also show that in the 3-dimensional cocompact case, there is a normal subgroup with large \(\delta\): when \(\mathbb H^3/\Gamma\) is a closed hyperbolic 3-manifold fibring over a circle, for any \(\epsilon >0\) there is a Schottky subgroup \(G\) of \(\Gamma\) with \(\delta(G)>2-\epsilon\).

MSC:
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
57M50 General geometric structures on low-dimensional manifolds
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