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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
##### Keywords:
Kleinian group; hyperbolic space; exponent of convergence
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