Locally compact groups acting on discrete structures

01 July 2020 to 31 December 2020


Alejandra Garrido, The University of Newcastle, Australia
Colin Reid, The University of Newcastle, Australia
Stephan Tornier, The University of Newcastle, Australia
George Willis, The University of Newcastle, Australia

Locally compact groups are ubiquitous in mathematics and are of fundamental importance to describing our universe. Those that act on discrete structures (the totally disconnected locally compact groups) arise in combinatorial geometry, number theory and algebra. Their study is an essential part of understanding the structure of general locally compact groups with significant advances being made in the last 25 years through varied approaches: an emerging local structure theory, decomposition theory and refinement of scale methods (analogous to eigenvalues) are some highlights. Their relation to actions on discrete structures allows ideas from profinite and geometric group theory to be brought to bear, and to transfer techniques and questions between t.d.l.c. groups and other fields, including symbolic dynamics and number theory.

This semester aims to open and strengthen collaborations between diverse areas where totally disconnected locally compact groups can be used to make new discoveries or motivate research. The programme is focused on three major themes: scale methods, self-similar groups and dynamics; local-to-global phenomena for actions and for subgroups; links with geometric group theory. Connections and cross-fertilisation with other areas will also be emphasised.

Starting with a summer school, several workshops in these fields will gather experts in order to improve our understanding of t.d.l.c. groups through their actions on discrete structures.