Boundaries of relatively hyperbolic groups

Abstract

In this thesis we will investigate two issues on relatively hyperbolic groups which will be treated independently in two parts.

In the first part of this thesis we characterise relatively hyperbolic groups as geometrically finite convergence groups.  More precisely, we show the following. Suppose M is a non-empty perfect compact metrisable space, and suppose that a group, Γ, acts as a convergence group on M such that M consists only conical limit points and bounded parabolic points.  Suppose also that the stabiliser of each bounded parabolic point is finitely generated.  Then Γ is relatively hyperbolic, and M is equivariantly homeomorphic to the boundary of Γ.  We also give another aspect of this characterisation by showing under the above assumptions that Γ acts also as cusp uniform group on the space of triples of M.

In the second part, we describe a condition on the minimal compactifications of maximal parabolic subgroups of a relatively hyperbolic group, Γ.  We prove that the dynamical system of Γ on its boundary is finitely presented if we assume that this conditions is satisfied by maximal parabolic subgroups of Γ.  We also give  examples of groups where this condition is satisfied.

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