We prove that the relative homological dimension of a Kleinian group G does not exceed 1 + the critical exponent of G. As an application of this result we show that for a geometrically finite Kleinian group G, if the topological dimension of the limit set of G equals its Hausdorff dimension, then the limit set is a round sphere.

Given a closed oriented surface S of genus ≥ 3 we describe those cohomology classes χ ∈ H1(S,C) which appear as the period characters of abelian differentials for some choice of complex structure τ = τ(χ) on S consistent with the orientation. In other words, we describe the union Uτ∈T(S) H1,0(S,ℂ), where T(S) is the Teichmüller space of S. The proof is based upon Ratners solution of Raghunathans conjecture.

We prove that an irreducible lattice in a semisimple algebraic group is virtually isomorphic to an arithmetic lattice if and only if it admits a faithful self-similar action on a rooted tree of finite valency.

We prove that for every finitely-presented group G there exists a 2-dimensional irreducible complex-projective variety W with the fundamental group G, so that all singularities of W are normal crossings and Whitney umbrellas.

We prove that the discreteness problem for two-generated nonelementary subgroups of SL(2, C) is undecidable in the Blum-Shub-Smale (BSS) computability model.

We compare various notions of proper discontinuity for group actions. We also discuss fundamental domains and criteria for cocompactness.

We prove, under the assumption of the virtual fibration conjecture for arithmetic hyperbolic 3-manifolds, that all arithmetic lattices in O(n,1), n> 4, and different from 7, are non-coherent. We also establish noncoherence of uniform arithmetic lattices of the simplest type in SU(n,1), n> 1, and of uniform lattices in SU(2,1) which have infinite abelianization.

We construct discrete and faithful representations into the isometry group of a hyperbolic space of the fundamental groups of acute negatively curved even-sided polygons of finite groups.

These are lectures on discrete groups of isometries of complex hyperbolic spaces, aimed to discuss interactions between the function theory on complex hyperbolic manifolds and the theory of discrete groups.

We study side-lengths of triangles in path metric spaces. We prove that unless such a space X is bounded, or quasi-isometric to line or half-line, every triple of real numbers satisfying the strict triangle inequalities, is realized by the side-lengths of a triangle in X. We construct an example of a complete path metric space quasi-isometric to the Euclidean plane, for which every degenerate triangle has one side which is shorter than a certain uniform constant.