Tag - Topological groups

Henry Bradford: Quantitative LEF and topological full groups

9th November, 2020

Topological full groups of minimal subshifts are an important source of exotic examples in geometric group theory, as well as being powerful invariants of symbolic dynamical systems. In 2011, Grigorchuk and Medynets proved that TFGs are LEF, that is, every finite subset of the multiplication table occurs in the multiplication table of some finite group. In this talk we explore some ways in which asymptotic properties of the finite groups which occur reflect asymptotic properties of the associated subshift.

Simon Smith: Infinite primitive permutation groups, cartesian decompositions, and topologically simple locally compact groups

21st August, 2020

A non-compact, compactly generated, locally compact group whose proper quotients are all compact is called just-non-compact. Discrete just-non-compact groups are John Wilson’s famous just-infinite groups. In this talk, I'll describe an ongoing project to use permutation groups to better understand the class of just-non-compact groups that are totally disconnected. An important step for this project has recently been completed: there is now a structure theorem for non-compact tdlc groups G that have a compact open subgroup that is maximal. Using this structure theorem, together with Cheryl Praeger and Csaba Schneider’s recent work on homogeneous cartesian decompositions, one can deduce a neat test for whether the monolith of such a group G is a one-ended group in the class 𝒮 of non-discrete, topologically simple, compactly generated, tdlc groups. This class 𝒮 plays a fundamental role in the structure theory of compactly generated tdlc groups, and few types of groups in 𝒮 are known.

Hiroki Matui: Various examples of topological full groups

26th June, 2020

I will begin with the definition of topological full groups and explain various examples of them. The topological full group arising from a minimal homeomorphism on a Cantor set gave the first example of finitely generated simple groups that are amenable and infinite. The topological full groups of one-sided shifts of finite type are viewed as generalization of the Higman-Thompson groups. Based on these two fundamental examples, I will discuss recent development of the study around topological full groups.