** "Geometry and Analysis on Groups" Research Seminar **

**Time:** 2018.11.20, 15:00–17:00

**Location:** Seminarraum 8, Oskar-Morgenstern-Platz 1, 2.Stock

**Title:** "Growth, ispoperimetry and resistance in finite vertex-transitive graphs."

**Speaker:** Matthew Tointon
(University of Cambridge)

**Abstract:**
A famous result of Varopoulos says that if the simple random walk on an infinite Cayley graph is recurrent then its balls grow sub-cubically. Combined with Gromov's celebrated polynomial-growth theorem, this in turn implies the underlying group has a finite-index subgroup isomorphic to either \(\mathbb{Z}\) or \(\mathbb{Z}^2\). I will discuss an ongoing project, joint with Romain Tessera, in which, amongst other things, we provide analogues of these results for finite vertex-transitive graphs. Highlights will include a technique to reduce various questions about vertex-transitive graphs to questions about Cayley graphs, and a bound on resistance in finite vertex-transitive electric networks conjectured by Benjamini and Kozma in 2002.