Parabolic groups acting on one-dimensional compact spaces

Details Parabolic groups acting on one-dimensional compact spaces Parabolic groups acting on one-dimensional compact spaces

2004-01-07

arxiv.org/abs/math/0401059v2

Int. J. of Alg. and Comp., Vol. 15, Nos. 5-6 (2005) 893-906

Mathematics

Group Theory

10 pages. Added a precision on local connectedness for Lemma 2.3, thanks to B. Bowditch

Scientific paper

10.1142/S0218196705002530

Given a class of compact spaces, we ask which groups can be maximal parabolic subgroups of a relatively hyperbolic group whose boundary is in the class. We investigate the class of 1-dimensional connected boundaries. We get that any non-torsion infinite f.g. group is a maximal parabolic subgroup of some relatively hyperbolic group with connected one-dimensional boundary without global cut point. For boundaries homeomorphic to a Sierpinski carpet or a 2-sphere, the only maximal parabolic subgroups allowed are virtual surface groups (hyperbolic, or virtually $\mathbb{Z} + \mathbb{Z}$).

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