1 Background
Ideal boundaries of hyperbolic spaces Suppose that
dvipng error! exitcode was 2 (signal 0), transscript follows:is a hyperbolic metric space. Pick a base-point
dvipng error! exitcode was 2 (signal 0), transscript follows:. This defines the Gromov product
dvipng error! exitcode was 2 (signal 0), transscript follows:for points
dvipng error! exitcode was 2 (signal 0), transscript follows:. The ideal boundary
dvipng error! exitcode was 2 (signal 0), transscript follows:of
dvipng error! exitcode was 2 (signal 0), transscript follows:is the collection of equivalence classes
dvipng error! exitcode was 2 (signal 0), transscript follows:of sequences
dvipng error! exitcode was 2 (signal 0), transscript follows:in
dvipng error! exitcode was 2 (signal 0), transscript follows:where
dvipng error! exitcode was 2 (signal 0), transscript follows:if and only if
dvipng error! exitcode was 2 (signal 0), transscript follows:
The topology on
dvipng error! exitcode was 2 (signal 0), transscript follows:is defined as follows. Let
dvipng error! exitcode was 2 (signal 0), transscript follows:. Define
dvipng error! exitcode was 2 (signal 0), transscript follows:-neighborhood of
dvipng error! exitcode was 2 (signal 0), transscript follows:to be
dvipng error! exitcode was 2 (signal 0), transscript follows:
Then the basis of topology at
dvipng error! exitcode was 2 (signal 0), transscript follows:consists of
dvipng error! exitcode was 2 (signal 0), transscript follows:. We will refer to the resulting ideal boundary
dvipng error! exitcode was 2 (signal 0), transscript follows:as the Gromov--boundary of
dvipng error! exitcode was 2 (signal 0), transscript follows:. One can check that the topology on
dvipng error! exitcode was 2 (signal 0), transscript follows:is independent of the choice of the base-point. Moreover, if
dvipng error! exitcode was 2 (signal 0), transscript follows:is a quasi-isometry then it induces a homeomorphism
dvipng error! exitcode was 2 (signal 0), transscript follows:. The Gromov product extends to a continuous function
dvipng error! exitcode was 2 (signal 0), transscript follows:
The geodesic boundary of
dvipng error! exitcode was 2 (signal 0), transscript follows:admits a family of visual metrics
dvipng error! exitcode was 2 (signal 0), transscript follows:defined as follows. Pick a positive parameter
dvipng error! exitcode was 2 (signal 0), transscript follows:. Given points
dvipng error! exitcode was 2 (signal 0), transscript follows:consider various chains
dvipng error! exitcode was 2 (signal 0), transscript follows:(where
dvipng error! exitcode was 2 (signal 0), transscript follows:varies) so that
dvipng error! exitcode was 2 (signal 0), transscript follows:. Given such a chain, define
dvipng error! exitcode was 2 (signal 0), transscript follows:
where
dvipng error! exitcode was 2 (signal 0), transscript follows:. Finally,
dvipng error! exitcode was 2 (signal 0), transscript follows:
where the infimum is taken over all chains connecting
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:. Taking different values of
dvipng error! exitcode was 2 (signal 0), transscript follows:results in H\"older--equivalent metrics. Each quasi-isometry
dvipng error! exitcode was 2 (signal 0), transscript follows:yields a quasi-symmetric homeomorphism (see section \ref{analytical} for the definition)
dvipng error! exitcode was 2 (signal 0), transscript follows:
Conversely, each quasi-symmetric homeomorphism as above extends to a quasi-iso\-metry
dvipng error! exitcode was 2 (signal 0), transscript follows:, see MR1395067.
Then the ideal boundary of a Gromov--hyperbolic group
dvipng error! exitcode was 2 (signal 0), transscript follows:is defined as
dvipng error! exitcode was 2 (signal 0), transscript follows:
where
dvipng error! exitcode was 2 (signal 0), transscript follows:is a Cayley graph of
dvipng error! exitcode was 2 (signal 0), transscript follows:. Hence
dvipng error! exitcode was 2 (signal 0), transscript follows:is well-defined up to a quasi-symmetric homeomorphism.
Ideal boundaries of dvipng error! exitcode was 2 (signal 0), transscript follows:
spaces Consider a
dvipng error! exitcode was 2 (signal 0), transscript follows:space
dvipng error! exitcode was 2 (signal 0), transscript follows:. Two geodesic rays
dvipng error! exitcode was 2 (signal 0), transscript follows:are said to be equivalent if there exists a constant
dvipng error! exitcode was 2 (signal 0), transscript follows:such that
dvipng error! exitcode was 2 (signal 0), transscript follows:
The geodesic boundary
dvipng error! exitcode was 2 (signal 0), transscript follows:of
dvipng error! exitcode was 2 (signal 0), transscript follows:is defined to be the set of equivalence classes
dvipng error! exitcode was 2 (signal 0), transscript follows:of geodesic rays
dvipng error! exitcode was 2 (signal 0), transscript follows:in
dvipng error! exitcode was 2 (signal 0), transscript follows:. Fix a base-point
dvipng error! exitcode was 2 (signal 0), transscript follows:. If
dvipng error! exitcode was 2 (signal 0), transscript follows:is locally compact (which we will assume from now on), then there exists a unique a representative
dvipng error! exitcode was 2 (signal 0), transscript follows:in each equivalence class
dvipng error! exitcode was 2 (signal 0), transscript follows:so that
dvipng error! exitcode was 2 (signal 0), transscript follows:. With this convention the visual topology on
dvipng error! exitcode was 2 (signal 0), transscript follows:is defined as the compact-open topology on the space of maps
dvipng error! exitcode was 2 (signal 0), transscript follows:. One can check that this topology is independent of the choice of the base-point and that isometries
dvipng error! exitcode was 2 (signal 0), transscript follows:induce homeomorphisms
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Example. If
dvipng error! exitcode was 2 (signal 0), transscript follows:then
dvipng error! exitcode was 2 (signal 0), transscript follows:is homeomorphic to
dvipng error! exitcode was 2 (signal 0), transscript follows:.
If
dvipng error! exitcode was 2 (signal 0), transscript follows:is a
dvipng error! exitcode was 2 (signal 0), transscript follows:space then it is also Gromov-hyperbolic. Then the two ideal boundaries of
dvipng error! exitcode was 2 (signal 0), transscript follows:(one defined via sequences and the other defined via geodesic rays) are canonically homeomorphic to each other. More specifically, each geodesic ray
dvipng error! exitcode was 2 (signal 0), transscript follows:defines sequences
dvipng error! exitcode was 2 (signal 0), transscript follows:, for
dvipng error! exitcode was 2 (signal 0), transscript follows:diverging to infinity. The equivalence class of such
dvipng error! exitcode was 2 (signal 0), transscript follows:is independent of
dvipng error! exitcode was 2 (signal 0), transscript follows:and one gets a homeomorphism from the
dvipng error! exitcode was 2 (signal 0), transscript follows:-boundary to the Gromov-boundary.
In general, quasi-isometries of
dvipng error! exitcode was 2 (signal 0), transscript follows:spaces to not extend to the ideal boundaries in any sense. Moreover, Bruce Kleiner and Chris Croke constructed examples MR1746908 of pairs of
dvipng error! exitcode was 2 (signal 0), transscript follows:spaces
dvipng error! exitcode was 2 (signal 0), transscript follows:which admit geometric (i.e. isometric, discrete, cocompact) actions by the same group
dvipng error! exitcode was 2 (signal 0), transscript follows:so that
dvipng error! exitcode was 2 (signal 0), transscript follows:are not homeomorphic.
Therefore, given a
dvipng error! exitcode was 2 (signal 0), transscript follows:--group
dvipng error! exitcode was 2 (signal 0), transscript follows:one can talk only of the collection of
dvipng error! exitcode was 2 (signal 0), transscript follows:boundaries of
dvipng error! exitcode was 2 (signal 0), transscript follows:, i.e. the set
dvipng error! exitcode was 2 (signal 0), transscript follows:
where the actions
dvipng error! exitcode was 2 (signal 0), transscript follows:are geometric.