M.Sc Thesis

M.Sc StudentSharf Miel
SubjectOn Reduced Horoboundary Action of Groups and Property (EH)
DepartmentDepartment of Mathematics
Supervisor ASSOCIATE PROF. Uri Bader
Full Thesis textFull thesis text - English Version


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.