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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 infinitedimensional rankone 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 finitedimensional 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 BishopJones 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 nonproper settings. We construct PattersonSullivan measures for groups of divergence type without any compactness assumption. This is carried out by first constructing such measures on the SamuelSmirnov 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 PattersonSullivan 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 

Publisher  American Mathematical Society 
Number of pages  369 
Publication status  Accepted/In press  15 Apr 2016 
Publication series
Name  Mathematical Surveys and Monographs 

Publisher  American Mathematical Society 
ISSN (Print)  00765376 
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/1508.06960Keywords
 math.DS
 math.GR
 math.GT
 math.MG
Projects
 1 Finished

Programme GrantNew Frameworks in metric Number Theory
1/06/12 → 30/11/18
Project: Research project (funded) › Research