Related papers: On the hyperbolic orbital counting problem in conj…

We study an analogue of the conjugacy growth function in finitely generated groups: the automorphic growth function. This counts the number of automorphic orbits that intersect the ball of radius $n$ in the group. We show that this is not a…

Given any countable group $G$, we construct uncountably many quasi-isometry classes of proper geodesic metric spaces with quasi-isometry group isomorphic to $G$. Moreover, if the group $G$ is a hyperbolic group, the spaces we construct are…

We consider mapping class groups \Gamma(M) = pi_0 Diff(M fix \partial M) of smooth compact simply connected oriented 4-manifolds M bounded by a collection of 3-spheres. We show that if M contains CP^2 (with either orientation) as a…

We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms which are uniform as the lattice moves through "congruence"…

Heinz Huber (1956) considered the following problem on the the hyperbolic plane H. Consider a strictly hyperbolic subgroup of automorphisms on H with compact quotient, and choose a conjugacy class in this group. Count the number of vertices…

Let G be the identity component of SO(n,1), acting linearly on a finite dimensional real vector space V. Consider a vector w_0 in V such that the stabilizer of w_0 is a symmetric subgroup of G or the stabilizer of the line Rw_0 is a…

The classical prime geodesic theorem (PGT) gives an asymptotic formula (as $x$ tends to infinity) for the number of closed geodesics with length at most $x$ on a hyperbolic manifold $M$. Closed geodesics correspond to conjugacy classes of…

Let $Y=\Gamma\backslash H^n$ be a quotient of the hyperbolic space by the action of a discrete convex-cocompact group of isometries. We describe certain spaces of $\Gamma$-invariant currents on the sphere at infinity of $H^n$ with support…

We show that the isomorphism problem is solvable in the class of central extensions of word-hyperbolic groups, and that the isomorphism problem for biautomatic groups reduces to that for biautomatic groups with finite centre. We describe an…

For $n\geq 3$ and $\Gamma$ a cocompact lattice acting on the hyperbolic space $\mathbb{H}^n$, we investigate the average behaviour of the error term in the circle problem. First, we explore the local average of the error term over compact…

We establish a precise asymptotic formula for the number of homotopy classes of periodic orbits for the geodesic flow on rank one manifolds of nonpositive curvature. This extends a celebrated result of G. A. Margulis to the nonuniformly…

In this paper we provide a framework for the study of isoperimetric problems in finitely generated group, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions,…

We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a…

We derive geometric formulas for the mass of asymptotically hyperbolic manifolds using coordinate horospheres. As an application, we obtain a new rigidity result of hyperbolic space: if a complete asymptotically hyperbolic manifold has…

Let $\Sigma$ be a hyperbolic surface. We study the set of curves on $\Sigma$ of a given type, i.e. in the mapping class group orbit of some fixed but otherwise arbitrary $\gamma_0$. For example, in the particular case that $\Sigma$ is a…

We study the general theory of asymptotically CAT(0) groups, explaining why such a group has finitely many conjugacy classes of finite subgroups, is $F_\infty$ and has solvable word problem. We provide techniques to combine asymptotically…

In this article, we study the geometry of an infinite dimensional Hyperbolic space. We will consider the group of isometries of the Hilbert ball equipped with the Carath$\acute{e}$odory metric and learn about some special subclasses of this…

For a finite group $G$, we obtain asymptotics for the number of connected components of Hurwitz spaces of marked $G$-covers (of both the affine and projective lines) whose monodromy classes are constrained in a certain way, when the number…

Let G:=SO(n,1)^\circ and \Gamma be a geometrically finite Zariski dense subgroup with critical exponent delta bigger than (n-1)/2. Under a spectral gap hypothesis on L^2(\Gamma \ G), which is always satisfied for delta>(n-1)/2 for n=2,3 and…

Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for…