Abstract. We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.

Abstract. We describe a family of 4-dimensional hyperbolic orbifolds, constructed by deforming an infinite volume orbifold obtained from the ideal, hyperbolic 24-cell by removing two walls. This family provides an infinite number of infinitesimally rigid, infinite covolume, geometrically finite discrete subgroups of Isom(H 4 ). It also leads to finite covolume Coxeter groups which are the homomorphic image of the group of reflections in the hyperbolic 24-cell. The examples are constructed very explicitly, both from an algebraic and a geometric point of view. The method used can be viewed as a 4-dimensional, but infinite volume, analog of 3-dimensional hyperbolic Dehn filling.

We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dense faithfully embedded surface group. Similarly, we show that any connected semisimple Lie group contains a dense copy of any fully residually free group.Comment: This is the version published by Geometry & Topology on 4 October 200

The space AH(M ) of marked hyperbolic 3-manifold homotopy equivalent to a compact 3-manifold with boundary M sits inside the PSL2(C)-character variety X(M ) of π1(M ). We study the dynamics of the action of Out(π1(M )) on both AH(M ) and X(M ). The nature of the dynamics reflects the topology of M .The quotient AI(M ) = AH(M )/Out(π1(M )) may naturally be thought of as the moduli space of unmarked hyperbolic 3-manifolds homotopy equivalent to M and its topology reflects the dynamics of the action.One may combine Theorems 1.2 and 1.3 to completely characterize when Out(π 1 (M )) acts properly discontinuously on AH(M ) in the case that M has no toroidal boundary components. Corollary 1.4. Let M be a compact hyperbolizable 3-manifold with no toroidal boundary components and non-abelian fundamental group. The group Out(π 1 (M )) acts properly discontinuously on AH(M ) if and only if M contains no primitive essential annuli. Moreover, AI(M ) is Hausdorff if and only if M contains no primitive essential annuli.It is a consequence of the classical deformation theory of Kleinian groups (see Bers [5] or Canary-McCullough [17, Chapter 7] for a survey of this theory) that Out(π 1 (M )) acts properly discontinuously on the interior int(AH(M )) of AH(M ). If H n is the handlebody of genus n ≥ 2, Minsky [43] exhibited an explicit Out(π 1 (H n ))-invariant open subset P S(H n ) of X(H n ) such that int(AH(H n )) is a proper subset of P S(H n ) and Out(π 1 (H n )) acts properly discontinuously on AH(H n ).If M is a compact hyperbolizable 3-manifold with incompressible boundary and no toroidal boundary components, which is not an interval bundle, then we find an open set W (M ) strictly bigger than int(AH(M )) which Out(π 1 (M )) acts properly discontinuosly on. See Theorem 9.1 and its proof for a more precise description of W (M ). We further observe, see Lemma 8.1, that W (M ) ∩ ∂AH(M ) is a dense open subset of ∂AH(M ) in this setting.Theorem 1.5. Let M be a compact hyperbolizable 3-manifold with nonempty incompressible boundary and no toroidal boundary components, which is not an interval bundle. Then there exists an open Out(π 1 (M ))-invariant subset W (M ) of X(M ) such that Out(π 1 (M )) acts properly discontinuously on W (M ) and int(AH(M )) is a proper subset of W (M ).It is conjectured that if M is an untwisted interval bundle over a closed surface S, then int(AH(M )) is the maximal open Out(π 1 (M ))-invariant subset of X(M ) on which Out(π 1 (M )) acts properly discontinuously. One may show that no open domain of discontinuity can intersect ∂AH(S × I) (see [34]). Further evidence for this conjecture is provided by results of Bowditch [8], Goldman [21], Souto-Storm [49], Tan-Wong-Zhang [54] and Cantat [19].Michelle Lee [34] has recently shown that if M is an twisted interval bundle over a closed surface, then there exists an open Out(π 1 (M ))-invariant subset W of X(M ) such that Out(π 1 (M )) acts properly discontinuously on

This paper investigates the relationship between the topology of hyperbolizable 3-manifolds M with incompressible boundary and the volume of hyperbolic convex cores homotopy equivalent to M . Specifically, it proves a conjecture of Bonahon stating that the volume of a convex core is at least half the simplicial volume of the doubled manifold DM , and this inequality is sharp. This paper proves that the inequality is in fact sharp in every pleating variety of AH(M ). t ).Convex core volume is continuous in the strong topology [31]. Therefore Vol(C Nε ) ≤ Vol(C Nt ).Applying an identical argument to ε ′ shows that Vol(C Nε ) ≤ Vol(C N ε ′ ).