Groups of dynamical origin

Supervisor: Dr Christopher Bruce

School: Mathematics and Statistics


Interactions between group theory and topological dynamics has a long history of symbiosis: Many important properties of groups can be characterised in terms of actions on certain kinds of topological spaces, and groups of dynamical origin provide new and interesting example classes of groups. One particularly striking example is the result by Juschenko and Monod: They proved that
the topological full groups attached to certain etale groupoids over the Cantor set provided the first examples of finitely generated simple amenable (infinite) groups [JM].

To each expanding self-covering map of a compact path-connected metric space, Nekrashevych associates a finitely presented group that encodes the the initial dynamical system up to conjugacy [N]. This project is concerned with studying group-theoretic properties of such groups in the special case where the space is itself a compact group and the self-covering is a group endomorphism. Typically examples are the expanding toral endomorphisms. Such groups are known to be non-amenable, but it is not known they have, e.g., the Haagerup property (i.e., are a-T-amenable in the sense of Gromov). A natural and elementary approach to this problem would be to (try to) construct a zipper action for these groups as was done by Matui for the topological full
groups arising from irreducible shifts of finite type [M].

A simple example is the doubling map on the unit circle, and the project will be begin with a careful look at this example, which requires only basic prerequisites. This will give the student an opportunity to work hands on with one of these groups, and it will give me the opportunity to assess the students’s background and capabilities. The specific problems that we tackle and the level of generality for the project can then be adjusted to match the student’s level and interests.

[JM] K. Juschenko and N. Monod, Cantor systems, piecewise translations and simple amenable groups. Ann. of Math. (2) 178 (2013), no. 2, 775–787.

[M] H. Matui, Topological full groups of one-sided shifts of finite type. J. Reine Angew. Math. 705 (2015), 35–84.

[N] V. Nekrashevych, Finitely presented groups associated with expanding maps. Geometric and cohomological group theory, 115--171, London Math. Soc. Lecture Note Ser., 444, Cambridge Univ. Press, Cambridge, 2018.