M.Sc Student | Sharf Miel |
---|---|

Subject | On Reduced Horoboundary Action of Groups and Property (EH) |

Department | Department of Mathematics |

Supervisor | Professor Uri Bader |

Full Thesis text |

The horocompactification
of a nice enough metric space *X* is a compactification of *X*,
meaning that it is compact and that *X* embeds into it as an open dense
subset. It can be defined as a functional counterpart to the more well-known
Gromov Boundary, which can be thought of as the set of all infinite geodesics,
up to some identification. If a group *G* acts on the metric space *X*
in an isometric manner, then it induces an action on the horocompactification.

Because *G*
preserves *X* itself, it must also preserve the boundary of *X* in
the horocompactification, called the horoboundary. These actions have been studied
extensively in the last couple of decades, mainly in the case of hyperbolic
spaces, in which the orbits of the action are very large. One case was of
specific interest, namely the case in which the space *X* can be
identified with the group *G* equipped with an appropriate metric (e.g., a
word metric, a Riemannian metric, a Carnot-Carathèodory metric, etc.).

We look at a variant of
the horoboundary, called the reduced horoboundary, which is obtained from the
original horoboundary by identifying functions whose difference is bounded. It
is known that if the group *G* is hyperbolic and *X* is taken to be *G*
with some word metric, then the reduced horoboundary can be identified with the
Gromov boundary, and therefore must contain huge *G*-orbits.

We shall study
the other end of the scale - in what cases can we say that the action of *G*
on the reduced horoboundary is trivial, or at least have finite orbits? It is
known to be the case if *G* is abelian and *X* is *G* equipped with
some word metric. In his thesis, Finkelshtein proved that this is also the case
when *G* is the discrete Heisenberg group and *X* is *G*
equipped with some word metric.

We will generalize his methods in
order to prove that this is true when *G* is a discrete generalized
Heisenberg group equipped with a word metric, or one of the smooth generalized
Heisenberg group equipped with some Finsler-Carnot-Carathèodory metric.
The main tool of both proofs will be an identification of certain
characteristics of curves in the Heisenberg groups with properties of curves in
the Euclidean space, such as length and symplectic area, and an isoperimetric
inequality.

We shall also deal with questions
regarding different metrics on the same space *X*, and prove that if the
metrics are "semi-isometric", which is a property moderately stronger
then quasi-isometry, then the reduced horoboundary of *X* does not change
when equipping it with either of the metrics. We use this result to show that
in the special case in which *X* is *G*, if *G* is some group
with a finite index subgroup *H*, if the metrics on *G* and on *H*
are semi-isometric, then *H* has finite reduced horoboundary orbits if and
only if *G* has this property.