Metric-functional boundary of Cayley graphs

Ariel Yadin (Ben-Gurion U, Beer-Sheva)

Jul 18. 2023, 14:00 — 14:40

The horofunction boundary is a compactification of metric spaces with roots in works of Busemann and Gromov.  It has been widely used in the context of hyperbolic spaces and hyperbolic groups.  

Our recent work focuses on studying the structure of these boundaries for "small" finitely generated groups.  These have not been studied as extensively, and many basic questions are still open.  An important motivation is the so-called "Gap Conjecture" by Grigorchuk.  

As a starting example, we prove the following:

A finitely generated group is of linear growth (equivalently, virtually Z) if and only if every Cayley graph has only finitely many Busemann points in its horofunction boundary (i.e. a finite Busemann boundary).  

This is based on joint works with L. Ron-George and M. Tointon.

All notions will be explained, no prior knowledge is assumed.

Further Information
ESI Schrödinger and Boltzmann Lecture Hall
Associated Event:
Geometric and Asymptotic Group Theory with Applications 2023 - Groups and Dynamics (Workshop)
Christopher Cashen (U of Vienna)
Javier de la Nuez González (KIAS, Seoul)
Alexandra Edletzberger (U of Vienna)
Yash Lodha (U of Hawaii, Honolulu)