Geometry and Analysis on Groups Research Seminar

"Geometry and Analysis on Groups" Research Seminar

Time: 21.06.22, 15:00–17:00
Location: Seminarraum 8, Oskar-Morgenstern-Platz 1, 2.Stock
Title: Infinite expanders and heat kernels
Speaker: Mikolaj Fraczyk (Chicago)
Abstract: A family of bounded degree graphs G_n is called an expander family if there is a positive number h (called the Cheeger constant) such that for any subset S of G_n of size less than |G_n|/2, the edge boundary of S has at least h|S| elements. Expander families are widely appreciated for their usefulness in both practical and theoretical applications. In 1998 Itai Benjamini asked the following question: Does there exists an infinite graph G of bounded degree such that the set of all finite balls in G forms an expander family? It is generally believed that the answer is no but the question is still open. In a recent work with Wouter van Limbeek we proved that the answer is no if one replaces finite balls by finite subgraphs weighted by heat kernels on G. Heat kernels can thought of as a smoothening of balls and appear naturally as distributions of random walks on G. I will explain the proof of the theorem and the key role played by the hyperfiniteness of stationary random graphs.