Symmetry in Newcastle

Confirmed speakers are Claudio Bravo, University of Chile, Matt Conder, University of Auckland, and George Willis, University of Newcastle. The meeting will be held in hybrid mode, at room W202 on Callaghan campus (University of Newcastle), and via zoom.

The talks will be recorded and made available on Zero-Dimensional Symmetry YouTube channel https://tinyurl.com/zerodimensionalgroup and website https://zerodimensional.group/. The running times of the talks, titles and abstracts are as follows

10:00 – 11:00 AEST – Claudio Bravo
11:00 – 11:30 AEST – Morning tea
11:30 – 12:30 AEST – Matt Conder
12:30 – 14:00 AEST – Lunch break
14:00 – 15:00 AEST – George Willis

Speaker: Claudio Bravo
Title: Quotients of the Bruhat-Tits tree function filed analogs of the Hecke congruence subgroups
Abstract: Let [latex]C[/latex] be a smooth, projective, and geometrically connected curve defined over a finite field [latex]F[/latex]. For each closed point [latex]P_infty[/latex] of [latex]C[/latex], let [latex]R[/latex] be the ring of functions that are regular outside [latex]P_infty[/latex], and let [latex]K[/latex] be the completion path [latex]P_infty[/latex] of the function field of [latex]C[/latex]. In order to study group of the form [latex]GL_2(R)[/latex], Serre describes the quotient graph [latex]GL_2(R)backslash T[/latex], where [latex]T[/latex] is the Bruhat-Tits tree defined from [latex]SL_2(K)[/latex]. In particular, Serre shows that [latex]GL_2(R)backslash T[/latex] is the union of a finite graph and a finite number of ray shaped subgraphs, which are called cusps. It is not hard to see that finite index subgroups inherit this property.

In this exposition we describe the quotient graph [latex]Hbackslash T[/latex] defined from the action on [latex]T[/latex] of the group [latex]H[/latex] consisting of matrices that are upper triangular modulo [latex]I[/latex], where [latex]I[/latex] is an ideal [latex]R[/latex]. More specifically, we give an explicit formula for the cusp number [latex]Hbackslash T[/latex]. Then By, using Bass-Serre theory, we describe the combinatorial structure of [latex]H[/latex]. These groups play, in the function field context, the same role as the Hecke Congruence subgroups of [latex]SL_2(Z)[/latex]. Moreover, not that the groups studied by Serre correspond to the case where the ideal I coincides with the ring [latex]R[/latex].

Speaker: Matt Conder
Title: Discrete two-generator subgroups of [latex]PSL(2,Q_p)[/latex]
Abstract: Due to work of Gilman, Rosenberger, Purzitsky and many others, discrete two-generator subgroups of [latex]PSL(2,R)[/latex] have been completely classified by studying their action by Möbius transformations on the hyperbolic plane. Here we aim to classify discrete two-generator subgroups of [latex]PSL(2,Q_p)[/latex] by studying their action by isometries on the Bruhat-Tits tree. We first give a general structure theorem for two-generator groups acting by isometries on a tree, which relies on certain Klein-Maskit combination theorems. We will then discuss how this theorem can be applied to determine discreteness of a two-generator subgroup of [latex]PSL(2,Q_p)[/latex]. This is ongoing work in collaboration with Jeroen Schillewaert.

Speaker: George Willis
Title: Groups acting on regular trees and t.d.l.c. groups
Abstract: Groups of automorphisms of regular trees are an important source of examples of and intuition about totally disconnected, locally compact (t.d.l.c.) groups. Indeed, Pierre-Emmanuel Caprice has called them a microcosm the general theory of t.d.l.c. groups. Although much is know about them, many questions remain open.
This talk will survey some of what is known about groups of tree automorphisms and how it relates to the general theory.

The zoom link is https://uonewcastle.zoom.us/j/82596235512.

Contact: Michal Ferov (University of Newcastle)