Parabolic Groups Acting on One-dimensional Compact Spaces

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 Z + Z). The Margulis Lemma implies that maximal parabolic subgroups of geometrically finite Kleinian groups are virtually abelian. More generally, the parabolic subgroups of a geometrically finite group on a Hadamard manifold with pinched negative curvature are virtually nilpotent. A natural generalization of the class of geometrically finite groups is the class of relatively hyperbolic groups. They were first introduced by M. Gromov [9], and studied by B. Bowditch [2], and independantly by B. Farb [7]. We will follow Bowditch’s approach (see [2] [13] and [4](appendix) for equivalence of definitions). A finitely generated (f.g.) group Γ is hyperbolic relative to a family of f.g. subgroups G, in the sense of [2], if it acts on a proper hyperbolic length space, such that the action induced on the boundary is a geometrically finite convergence action, whose maximal parabolic subgroups are the elements of G. The definitions are developed in the first section below. The boundary of such a space is shown in [2] to be canonically associated to the pair (Γ,G). We call it the Bowditch boundary of the relatively hyperbolic group. In this paper, we will see that the consequence of the Margulis Lemma mentionned above is, to a large extend, false for relatively hyperbolic groups, but may remains true for certain specific classes of boundaries. In fact, it is easy to construct counterexamples, that is, relatively hyperbolic groups with an arbitrary f.g. parabolic subgroupH: it suffices to consider the free HNN extension H∗{1}. It is hyperbolic relative to the conjugates of H (see Definition 2 in [2], where the graph involved is the Serre tree). However, the Bowditch boundary of this group is not connected. It is a Cantor set ([2], [5]). Our first theorem is a non-trivial generalisation of this example. Theorem 0.1 Let H be an infinite f.g. group which is not a torsion group. Then, there exists a relatively hyperbolic group Γ, containing H as a maximal parabolic subgroup, and whose boundary is connected, locally connected, 1-dimensional compact space without global cut point. To prove this result, we will make use of the Combination Theorem of [5] for amalgamations over infinite cyclic groups. This is done in Section 2. In the beginning of this ∗FIM, ETH Zürich. (2003), E-mail: [email protected]