# Geometry and dynamics in Gromov hyperbolic metric spaces: With an emphasis on non-proper settings

Tushar Das, David Simmons, Mariusz Urbański

Research output: Book/ReportBook

## Abstract

Our monograph presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Our work unifies and extends a long list of results by many authors. We make it a point to avoid any assumption of properness/compactness, keeping in mind the motivating example of $\mathbb H^\infty$, the infinite-dimensional rank-one symmetric space of noncompact type over the reals. The monograph provides a number of examples of groups acting on $\mathbb H^\infty$ which exhibit a wide range of phenomena not to be found in the finite-dimensional theory. Such examples often demonstrate the optimality of our theorems. We introduce a modification of the Poincar\'e exponent, an invariant of a group which gives more information than the usual Poincar\'e exponent, which we then use to vastly generalize the Bishop--Jones theorem relating the Hausdorff dimension of the radial limit set to the Poincar\'e exponent of the underlying semigroup. We give some examples based on our results which illustrate the connection between Hausdorff dimension and various notions of discreteness which show up in non-proper settings. We construct Patterson--Sullivan measures for groups of divergence type without any compactness assumption. This is carried out by first constructing such measures on the Samuel--Smirnov compactification of the bordification of the underlying hyperbolic space, and then showing that the measures are supported on the bordification. We study quasiconformal measures of geometrically finite groups in terms of (a) doubling and (b) exact dimensionality. Our analysis characterizes exact dimensionality in terms of Diophantine approximation on the boundary. We demonstrate that some Patterson--Sullivan measures are neither doubling nor exact dimensional, and some are exact dimensional but not doubling, but all doubling measures are exact dimensional.
Original language English American Mathematical Society 369 Accepted/In press - 15 Apr 2016

### Publication series

Name Mathematical Surveys and Monographs American Mathematical Society 0076-5376

### Bibliographical note

A previous version of this document included Section 12.5 (Tukia's isomorphism theorem). The results of that subsection have been split off into a new document which is available at http://arxiv.org/abs/1508.06960

## Keywords

• math.DS
• math.GR
• math.GT
• math.MG
• ### Programme Grant-New Frameworks in metric Number Theory

EPSRC

1/06/1230/11/18

Project: Research project (funded)Research