Abstract We exhibit some finite-volume cusped hyperbolic 5-manifolds that fiber over the circle. These include smallest 5-manifold known, discovered by Ratcliffe and Tschantz. As a consequence, we build finite type subgroup of group is not hyperbolic.

We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorp...

In this paper, we study some group-theoretic constructions associated to arithmetic fundamental groups of hyperbolic curves over finite fields. One of the main results of this paper asserts that any Frobeniuspreserving isomorphism between the geometrically pro-l fundamental groups of hyperbolic curves with one given point removed induces an isomorphism between the geometrically pro-l fundamenta...

Let Γ be a torsion-free Kleinian group, so that M = H/Γ is an orientable hyperbolic 3-manifold. The non-trivial elements of Γ are classified as either parabolic or hyperbolic. If γ ∈ Γ is hyperbolic, then γ has an axis in H which projects to a closed geodesic gγ in M (which depends only on the conjugacy class of γ in Γ). The element γ acts on its axis by translating and possibly rotating around...

We survey some of our recent results on length series identities for hyperbolic (cone) surfaces, possibly with cusps and/or boundary geodesics; classical Schottky groups; representations/characters of the one-holed torus group to SL(2,C); and hyperbolic 3 manifolds obtained by hyperbolic Dehn surgery on punctured torus bundles over the circle. These can be regarded as generalizations and variat...

There are three main observations which make up the proof. The first observation, discussed in Section 2, is that convex co-compact subgroups of a fundamental group of a hyperbolic 3-manifold N persist in the approximates given by hyperbolic Dehn surgery. This result is stated formally as Proposition 2.1. The second observation, discussed in Section 4, is that a collection of distinct hyperboli...

We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group A(K) has such a subgroup if its defining graph K contains an n-hole (i.e. an induced cycle of length n) with n ≥ 5. We construct another eight “forbidden” graphs and show that every graph K on ≤ 8 vertices either contains one of our examples, or co...

We study random walks on the three-strand braid group B3, and in particular compute the drift, or average topological complexity of a random braid, as well as the probability of trivial entanglement. These results involve the study of magnetic random walks on hyperbolic graphs (hyperbolic Harper-Hofstadter problem), what enables to build a faithful representation of B3 as generalized magnetic t...

We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n−1)-submanifold S . The main result is that the fundamental group of M \S is relatively hyperbolic, relative to fundamental groups of the ends of M \S , and M \S admits a complete finite volume A -regular Riemannian metric of negative sectional curvature. It fo...

Following Bowditch [5], we study representations of the free group on two generators into SL(2, C), and the connection with generalized Markoff maps. We show that Bowditch’s Q-conditions for generalized Markoff maps are sufficient for the generalized McShane identity to hold for the corresponding representations. These conditions are very close to being necessary as well, and a large class of r...