Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Geometry and dynamics in Gromov hyperbolic metric spaces: With an
  emphasis on non-proper settings   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.

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The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
6
The categories of this paper are: astro-ph
17
The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: quant-ph
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The categories of this paper are: q-bio