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A finiteness result for groups which quasi-act on hyperbolic spaces. (English) Zbl 1262.20048

Summary: Let \((X,d)\) be a Gromov-hyperbolic metric space endowed with a measure having finite entropy \(H\) and such that the measure of every ball of radius \(R>0\) is finite and bounded from below by a positive function of \(R\). In this paper we look at the set \(Q(X;L,C,D)\) of the isomorphism classes of torsion-free groups \(\Gamma\) which admit a discrete, \(D\)-co-bounded \((L,C)\)-quasi-action on \(X\) (\(D>0\), \(L\geq 1\), \(C\geq 0\)) and we describe some algebraic conditions which, imposed on the groups \(\Gamma\), define finite subsets of \(Q(X;L,C,D)\), provided \(C<\varepsilon\) for some \(\varepsilon>0\). As an example, these conditions are satisfied when \(\Gamma\) is assumed to admit a faithful, discrete, \(m\)-dimensional representation over some local field (in this case \(\varepsilon=\varepsilon(m,H,L)\)). In particular (set \(C=0\), \(L=1\)), our results apply when the groups are assumed to act by isometries.

MSC:

20F67 Hyperbolic groups and nonpositively curved groups
20F05 Generators, relations, and presentations of groups
20F65 Geometric group theory
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57M60 Group actions on manifolds and cell complexes in low dimensions
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