Skip to content

Events for July 23, 2018 from General and Seminar calendars

Workshop in Analysis and Probability Seminar

Time: 3:00PM - 3:50PM

Location: BLOC 220

Speaker: Beata Randrianantoanina, Miami University

Title: Bilipschitz embeddings of Cayley graphs of lamplighter groups

Abstract: We prove that the sequence of Cayley graphs of finite lamplighter groups $\{\mathbb{Z}_2\wr\mathbb{Z}_n\}_{n\ge 2}$ with a standard set of generators embeds bilipschitzly with uniformly bounded distortions into any non-superreflexive Banach space, and that this sequence is a is a set of test-spaces for superreflexivity in the sense of Ostrovskii. Our proof is inspired by the well known identification of Cayley graphs of infinite lamplighter groups with the horocyclic product of trees. We cover $\mathbb{Z}_2\wr\mathbb{Z}_n$ by three sets with a structure similar to a horocyclic product of trees, which enables us to construct well-controlled embeddings. Using known results, we also observe that, for any $q$, the Cayley graph of the infinite lamplighter group $\mathbb{Z}_q\wr\mathbb{Z}$ with respect to any finite generating set is a test space for superreflexivity. Joint work with Mikhail Ostrovskii.