Abstract. We give a precise definition of “generic-case complexity” and show that for a very large class of finitely generated groups the classical decision problems of group theory- the word, conjugacy and membership problems- all have linear-time generic-case complexity. We prove such theorems by using the theory of random walks on regular graphs. Contents 1. Motivation

Abstract. We give a complete proof of Thurston’s celebrated hyperbolic Dehn filling theorem, following the ideal triangulation approach of Thurston and Neumann-Zagier. We avoid to assume that a genuine ideal triangulation always exists, using only a partially flat one, obtained by subdividing an Epstein-Penner decomposition. This forces us to deal with negatively oriented tetrahedra. Our analysis of the set of hyperbolic Dehn filling coefficients is elementary and self-contained. In particular, it does not assume smoothness of the complete point in the variety of deformations. Mathematics Subject Classification (1991): 57M50 (primary), 57Q15 (secondary). Thurston’s hyperbolic Dehn filling theorem is one of the greatest achievements in the geometric theory of 3-dimensional manifolds, and the basis of innumerable results proved over the last twenty years. Despite these facts, we do not think that a completely satisfactory written account of the proof exists in the literature, and the aim of this note is to help filling a gap which could become embarrassing on the long run. We follow the approach through ideal triangulations, sketched by Thurston in his notes [13] and later used by Neumann and Zagier in their beautiful paper [9], to prove volume estimates

Given a closed hyperbolic n-manifold M, we study the flat Lorentzian structures on M × R such that M ×{0} is a Cauchy surface. We show there exist only two maximal structures sharing a fixed holonomy (one future complete and the other one past complete). We study the geometry of those maximal spacetimes in terms of cosmological time. In particular, we study the asymptotic behaviour of the level surfaces of the cosmological time. As a by-product, we get that no affine deformation of the hyperbolic holonomy ρ: π1(M) → SO(n, 1) of M acts freely and properly on the whole Minkowski space. The present work generalizes the case n = 2 treated by Mess, taking from a work of Benedetti and Guadagnini the emphasis on the fundamental rôle played by the cosmological time. In the last sections, we introduce measured geodesic stratifications on M, that in a sense furnish a good generalization of measured geodesic laminations in any dimension and we investigate relationships between measured stratifications on M and Lorentzian structures on M × R. 1.

Abstract not found

Let N be a hyperbolic 3-manifold and B a component of the interior of AH(π1(N)), the space of marked hyperbolic 3-manifolds homotopy equivalent to N. We will give topological conditions on N sufficient to give ρ ∈ B such that for every small neighborhood V of ρ, V ∩ B is disconnected. This implies that B is not manifold with boundary. 1

3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65

simplicial volume

In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles [3, 4]. Given a combinatorial description of a polyhedron, C, Andreev’s Theorem provides five classes of linear inequalities, depending on C, for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing C with the assigned dihedral angles. Andreev’s Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Andreev’s Theorem is both an interesting statement about the geometry of hyperbolic 3-dimensional space, as well as a fundamental tool used in the proof for Thurston’s Hyperbolization Theorem for 3-dimensional Haken manifolds. It is also remarkable to what level the proof of Andreev’s Theorem resembles (in a simpler way) the proof of Thurston. We correct a fundamental error in Andreev’s proof of existence and also provide a readable new proof of the other parts of the proof of Andreev’s Theorem, because

Abstract. We express the colored Jones polynomial as the inverse of the quantum determinant of a matrix with entries in the q-Weyl algebra of q-operators, evaluated at the trivial function (plus simple substitutions). The Kashaev invariant is proved to be equal to another special evaluation of the determinant. We also discuss the similarity between our determinant formula of the Kashaev invariant and the determinant formula of the hyperbolic volume of knot complements, hoping it would lead to a proof of the volume conjecture.

Abstract. We show that the aspherical manifolds produced via the relative strict hyperbolization of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automorphism group, absence of splitting over elementary subgroups, acylindricity, and diffeomorphism finiteness for manifolds with uniformly bounded simplicial volume. In fact, some of these properties hold for any compact aspherical manifold with incompressible aspherical boundary components, provided the fundamental group is hyperbolic relative to fundamental groups of boundary components. 1.