CARMA Research Group
Symmetry
About us
Leader: George Willis
Symmetry is accounted for mathematically through the algebraic concept of a 'group'. Our research focusses on '0-dimensional', aka 'totally disconnected, locally compact', groups, which arise as symmetries of graphs (in the sense of networks). We aim to analyse the structure of these groups and thus to understand the types of symmetries that graphs may possess. Whereas connected locally compact groups are well understood, much remains to be done in the totally disconnected case and we are filling significant gaps in knowledge. The research also has significant links with harmonic analysis, number theory and geometry — ideas from these fields are used in our research and our results feed back to influence these other fields. Another part of our research aims to develop computational tools for working with the groups and for visualising the corresponding symmetries.
We collaborate with researchers in Europe, Asia and North and South America in this work, which is being supported by Australian Research Council funds of $2.8 million in the period 2018-22.
More information about us and our research can be at https://zerodimensional.group. Feel free to contact us at contact@zerodimensional.group.
People
Members of this research group:
- George Willis
- Michal Ferov
- Alejandra Garrido
- Colin Reid
- David Robertson
- Stephan Tornier
Activities
-
Symmetry in Newcastle
10:00 am, Friday, 6th Mar 2020
(Location to be decided)Schedule:
10.00-11.00: Waltraud Lederle
11.00-11.30: Morning Tea
11.30-12.30: Mark Pengitore
12.30-14.00: Lunch
14.00-15.00: Jeroen Schillewaert
15.00-15.30: Afternoon Tea
15.30-16.00: François Thilmany"Fixed points for group actions on $2$-dimensional affine buildings"
— Senior Lecturer Jeroen Schillewaert
Abstract:
We prove a local-to-global result for fixed points of groups acting on $2$-dimensional affine buildings (possibly non-discrete, and not of type $\tilde{G}_{2}$). In the discrete case, our theorem establishes two conjectures by Marquis. (joint work with Koen Struyve and Anne Thomas)"Lattices of minimal covolume in $\mathrm{SL}_n$"
— Dr François Thilmany
Abstract:
A classical result of Siegel asserts that the (2,3,7)-triangle group attains the smallest covolume among lattices of $\mathrm{SL}_2(\mathbb{R})$. In general, given a semisimple Lie group $G$ over some local field $F$, one may ask which lattices in $G$ attain the smallest covolume. A complete answer to this question seems out of reach at the moment; nevertheless, many steps have been made in the last decades. Inspired by Siegel's result, Lubotzky determined that a lattice of minimal covolume in $\mathrm{SL}_2(F)$ with $F=\mathbb{F}_q((t))$ is given by the so-called characteristic $p$ modular group $\mathrm{SL}_2(\mathbb{F}_q[1/t])$. He noted that, in contrast with Siegel’s lattice, the quotient by $\mathrm{SL}_2(\mathbb{F}_q[1/t])$ was not compact, and asked what the typical situation should be: "for a semisimple Lie group over a local field, is a lattice of minimal covolume a cocompact or nonuniform lattice?".
In the talk, we will review some of the known results, and then discuss the case of $\mathrm{SL}_n(\mathbb{R})$ for $n > 2$. It turns out that, up to automorphism, the unique lattice of minimal covolume in $\mathrm{SL}_n(\mathbb{R})$ ($n > 2$) is $\mathrm{SL}_n(\mathbb{Z})$. In particular, it is not uniform, giving a partial answer to Lubotzky’s question in this case."Conjugacy and dynamics in the almost automorphism group of a tree"
— Dr Waltraud Lederle
Abstract:
We define the almost automorphism group of a regular tree, also known as Neretin's group, and determine when two elements are conjugate. (joint work with Gil Goffer)"Translation-like actions on nilpotent groups"
— Assistant Prof Mark Pengitore
Abstract:
Whyte introduced translation-like actions of groups as a geometric generalization of subgroup containment. He then proved a geometric reformulation of the von Neumann conjecture by demonstrating that a finitely generated group is non amenable if and only it admits a translation-like action by a non-abelian free group. This provides motivation for the study of what groups can act translation-like on other groups. As a consequence of Gromov’s polynomial growth theorem, virtually nilpotent groups can act translation-like on other nilpotent groups. We demonstrate that if two nilpotent groups have the same growth, but non-isomorphic Carnot completions, then they can't act translation-like on each other. (joint work with David Cohen) -
Symmetry in Newcastle
12:00 pm, Friday, 3rd Apr 2020
(Location to be decided)