Hausdorff Dimension

Friday 21st September 2018, Room 104, Senate House, University of London Local organiser: Brita Nucinkis

This is the fifth meeting of the Research Group Functor Categories for Groups (FCG). We shall focus on the study of Hausdorff dimension for profinite groups, initiated by Abercrombie in the 90s.

The timetable is as follows:

  • 12-1pm: Welcome (opportunity to eat packed lunch)
  • 1-2pm: Yiftach Barnea (Royal Holloway, University of London), Introduction to Hausdorff Dimension in Pro-p Groups
  • 2:15-3:15pm: Anitha Thillaisundaram (University of Lincoln), The normal Hausdorff spectrum of pro-p groups
  • 3:15-4pm: Coffee
  • 4-5pm: Alejandra Garrido (Heinrich Heine Universität Düsseldorf), Hausdorff dimension of free-like pro-p groups

Abstracts

Yiftach Barnea (Royal Holloway, University of London), Introduction to Hausdorff Dimension in Pro-p Groups

Given an infinite metric profinite group G, Hausdorff dimension can be used to distinguish the size of closed subgroups of G of infinite index. Thus, finite subgroups have Hausdorff dimension zero and closed subgroups of finite index have Hausdorff dimension one, all other closed subgroups have Hausdorff dimension between zero and one. The Hausdorff Dimension Spectrum of G is the set of all possible Hausdorff dimensions of closed subgroups of G.

In this talk I will introduce the above notions. I will explain that these depend on the metric rather than just the topology. I will present some of my previous work on the spectrum and as time permit some important applications of Hausdorff dimension for profinite groups.

This will require no background in Hausdorff dimension (actually I am not even going to formally define it), but will require some elementary background in profinite groups.

Anitha Thillaisundaram (University of Lincoln), The normal Hausdorff spectrum of pro-p groups

The normal Hausdorff spectrum of a pro-p group G is the set of Hausdorff dimensions of all closed normal subgroups of G. For finitely generated pro-p groups, previously known normal Hausdorff spectra were finite,
often just the trivial set {0,1}. Shalev asked in 2000 whether there exists a finitely generated pro-p group with infinite normal Hausdorff spectrum, and whether this spectrum can contain an interval. We give a positive answer to both questions, using wreath products, and we state some open problems. This is joint work with Benjamin Klopsch and Amaia Zugadi-Reizabal.

Alejandra Garrido (Heinrich Heine Universität Düsseldorf), Hausdorff dimension of free-like pro-p groups

I will report on ongoing joint work with O. Garaialde and B. Klopsch on the Hausdorff spectrum of free pro-p groups and their relatives (eg pro-p groups of positive rank gradient), with respect to metrics induced by different natural filtrations of the groups (eg Frattini series, Zassenhaus series). Among other things, we show that all possible Hausdorff dimensions are realised by closed subgroups of these groups.