Let M be a convex cocompact, acylindrical hyperbolic 3-manifold of infinite volume, and let M∗ denote the interior of the convex core of M . In this paper we show that any geodesic plane inM∗ is either closed or dense. We also show that only countably many planes are closed. These are the first rigidity theorems for planes in convex cocompact 3-manifolds of infinite volume that depend only on t...

We describe a construction of cocompact models for the classifying spaces EΓ g,r , where Γ g,r stands for the mapping class group of an oriented surface of genus g with r boundary components and s punctures. Our construction uses a cocompact model for EΓ g,0 as an input, a case which has been dealt with in [3]. We then proceed by induction on r and s.

This article is dedicated to the characterisation of relative hyperbolicity Haglund and Wise’s special groups. More precisely, we introduce a new combinatorial formalism study (virtually) groups, prove that, given cocompact group G finite collection subgroups \({\cal H}\), then hyperbolic H}\) if only (i) each subgroup convex-cocompact, (ii) an almost malnormal collection, (iii) every non-virtu...

We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmüller space. Given a subgroup G of MCG de ning an extension 1 ! 1(S) ! ΓG ! G ! 1, we prove that if ΓG is a word hyperbolic group then G is a convex cocompact subgroup of MCG. When G is free and convex cocompact, called a Schottky ...

We study finitely generated discrete groups of hyperbolic plane isometries (including those which reverse orientation) with non-compact orbit space. A presentation by generators and relations this type group is obtained. To that end, we use the geometrical properties a fundamental region canonical form, apply Macbeath’s classical theorem on presentations simply connected spaces. Our main result...

Let Λ be a minimal Kac–Moody group of rank 2 defined over the finite field Fq , where q = pa with p prime. Let G be the topological Kac–Moody group obtained by completing Λ. An example is G = SL2(K), where K is the field of formal Laurent series over Fq . The group G acts on its Bruhat–Tits building X, a tree, with quotient a single edge. We construct new examples of cocompact lattices in G, ma...