In this article we consider the Picard group SL(2, Z[;']), viewed as a discrete subgroup of the isometries of hyperbolic space. We fix a canonical choice of generators and then construct a Markov partition for the action of the group on the sphere at infinity. Our main application is to the study of the zeta function associated to the associated three-dimensional hyperbolic manifold.

We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature (n−1, 1) is “thin”, namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg’s theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many ...

We construct an example of a torsion free freely indecomposable finitely presented nonquasiconvex subgroup H of a word hyperbolic group G such that the limit set of G is not the limit set of a quasiconvex subgroup of G. In particular, this gives a counterexample to the conjecture of G.Swarup that a finitely presented one-ended subgroup of a word hyperbolic group is quasiconvex if and only if it...

We prove that all hierarchically hyperbolic groups have nite asymptotic dimension. One application of this result is to obtain the sharpest known bound on the asymptotic dimension of the mapping class group of a nite type surface: improving the bound from exponential to at most quadratic in the complexity of the surface. We also apply the main result to various other hierarchically hyperbolic g...

Our main goal in this paper is to construct the first explicit fundamental domain of the Picard modular group acting on the complex hyperbolic space CH . The complex hyperbolic space is a Hermitian symmetric space, its bounded realization is the unit ball in C equipped with the Bergman metric. The Picard modular group is a discontinuous holomorphic automorphism subgroup of SU(2, 1) with Gaussia...

We prove that the mapping torus group Fn ⋊α Z of any automorphism α of a free group Fn of finite rank n ≥ 2 is weakly hyperbolic relative to the canonical (up to conjugation) family H(α) of subgroups of Fn which consists of (and contains representatives of all) conjugacy classes that grow polynomially under iteration of α. Furthermore, we show that Fn ⋊α Z is strongly hyperbolic relative to the...

In this paper we consider type-preserving representations of the fundamental group three-holed projective plane into Projective Linear Group $\operatorname{PGL}(2,\mathbb{R})$ and study connected components with non-maximal euler class. We show that in class zero for all such there is a simple closed curve which non-hyperbolic, while $\pm 1$ are $6$ where curves sent to hyperbolic elements $2$ ...

We present some constraints on the intersection pairing in hyperbolic manifolds. We also show that, given a closed negatively curved manifold M of dimension at least three, only nitely many rank k oriented vector bundles over M can admit complete hyperbolic metrics. x0. Introduction Hyperbolic manifolds are locally symmetric Riemannian manifolds of negative sectional curvature. Any complete hyp...

The following discourse is inspired by the works on hyperbolic groups of Epstein, and Neumann/Reeves. In [E], it is shown that geometrically finite hyperbolic groups are biautomatic. In [NR1], it is shown that virtually central extensions of word hyperbolic groups are biautomatic. We prove the following generalisation: Theorem 1. Let H be a geometrically finite hyperbolic group. Let σ ∈ H(H) an...

An infinite group G is called indicable (resp. virtually indicable) if G (resp. a subgroup of finite index in G) admits a homomorphism onto Z. This is a powerful property for a group to have; for example in the context of infinite fundamental groups of aspherical 3-manifolds it remains one of the outstanding open questions to prove such groups are virtually indicable. To continue on the 3-manif...