Geometry and Analysis on Groups Research Seminar

"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.