On the horofunction boundary of discrete Heisenberg group
- PDF / 444,264 Bytes
- 15 Pages / 439.37 x 666.142 pts Page_size
- 61 Downloads / 223 Views
On the horofunction boundary of discrete Heisenberg group Uri Bader1 · Vladimir Finkelshtein2 Received: 19 June 2019 / Accepted: 24 January 2020 © The Author(s) 2020
Abstract We consider a finitely generated group endowed with a word metric. The group acts on itself by isometries, which induces an action on its horofunction boundary. The conjecture is that nilpotent groups act trivially on their reduced boundary. We will show this for the Heisenberg group. The main tool will be a discrete version of the isoperimetric inequality. Keywords Heisenberg group · Horofunction compactification · Cayley graph 2010 Mathematics Subject Classification 20F65 · 20F18
1 Introduction Every metric space embeds in the space of continuous functions on it, and its image there, modulo the constant functions is precompact. The functions in the closure are denoted horofunctions, the closure of the image is denoted the horofunction compactification and the boundary is denoted the horofunction boundary or the horoboundary. This notion is due to Gromov [3]. The horoboundary carries a natural equivalence relation. The corresponding quotient space is called the reduced horoboundary. Given a group with a specified set of generators, one obtains a metric space by considering the corresponding word metric on the group, and thus one gets corresponding horoboundary and reduced horoboundary. The group acts naturally on those spaces. Both of those spaces might depend on the choice of generators, but in some cases topological and dynamical properties of the action do not. A well-known example is given by hyperbolic groups, for which the reduced horoboundary coincides with the Gromov boundary and, in particular, does not depend on choice of
B
Vladimir Finkelshtein [email protected] Uri Bader [email protected]
1
Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, 234 Herzl Street, 7610001 Rehovot, Israel
2
Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße 3-5, 37073 Göttingen, Germany
123
Geometriae Dedicata
generators (while the horoboundary does). Hyperbolic groups indeed provide a rich class of examples for groups with non-trivial actions on their reduced horoboundaries. On the other extreme, the reduced horoboundary of a finitely generated abelian group depends on a choice of generators, but the boundary behavior is rather simple, as seen in the following theorem. Theorem A Given a finitely generated abelian group endowed with any finite set of generators, the corresponding reduced horoboundary is finite and the group action on it is trivial. More generally, we conjecture the following. Conjecture Given a finitely generated nilpotent group endowed with any finite set of generators, the action of the group on its reduced horoboundary is trivial. The purpose of this paper is to establish this conjecture for the first non-trivial case. Theorem B Given any finite set of generators of the discrete Heisenberg group, the action of the group on the corresponding reduced horoboundary is trivial
Data Loading...
