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

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 Poincaré exponent, an invariant of a group which gives more information than the usual Poincaré exponent, which we then use to vastly generalize the Bishop--Jones theorem relating the Hausdorff dimension of the radial limit set to the Poincaré 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.

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 arXiv:1508.06960

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Cited by in corpus (13)

• Extremality and dynamically defined measures, part II: Measures from conformal dynamical systems
• Patterson-Sullivan theory for Anosov subgroups
• Counting loxodromics for hyperbolic actions
• Strong symbolic dynamics for geodesic flow on CAT(-1) spaces and other metric Anosov flows
• Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces
• Random walks, WPD actions, and the Cremona group
• Lectures on complex hyperbolic Kleinian groups
• Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type
• The Polish topology of the isometry group of the infinite dimensional hyperbolic space
• The Martin boundary of an extension by a hyperbolic group
• A central limit theorem for the degree of a random product of Cremona transformations
• Relating boundary and interior solutions of the cohomological equation for cocycles by isometries of negatively curved spaces. The Livsic case
• Regularity of the drift for random walks in groups acting on Gromov hyperbolic spaces