A finiteness result for groups which quasi-act on hyperbolic spaces

作者：Zuddas Fabio^*

来源：Geometriae Dedicata, 2011, 150(1): 35-47.

DOI：10.1007/s10711-010-9492-9

• Gromov-hyperbolic space
• Word-hyperbolic group
• Quasi-action
• Quasi-isometry
• Entropy
• Finiteness result

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 I" which admit a discrete, D-co-bounded (L, C)-quasi-action on X (D > 0, L a parts per thousand yen 1, C a parts per thousand yen 0) and we describe some algebraic conditions which, imposed on the groups I", define finite subsets of Q(X; L, C, D), provided C < epsilon for some epsilon > 0. As an example, these conditions are satisfied when I" is assumed to admit a faithful, discrete, m-dimensional representation over some local field (in this case epsilon = epsilon(m, H, L)). In particular (set C = 0, L = 1), our results apply when the groups are assumed to act by isometries.

• 出版日期2011-2