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

    6:30 pm, Monday, 8th Mar 2021
    , Online

    Schedule (Zoom):

    18.30-19.30: Charlotte Hoffmann
    19.30-20.00: Break
    20.00-21.00: David Kielak

    "Short words of high imprimitivity rank yield hyperbolic one-relator groups"
       — Ms Charlotte Hoffmann

    Abstract:
    It is a long standing question whether a group of type $F$ that does not contain Baumslag–Solitar subgroups is necessarily hyperbolic. One-relator groups are of type $F$ and Louder and Wilton showed that if the defining relator has imprimitivity rank greater than $2$, they do not contain Baumslag-Solitar subgroups, so they conjecture that such groups are hyperbolic. Cashen and I verified the conjecture computationally for relators of length at most $17$. In this talk I'll introduce hyperbolic groups and the imprimitivity rank of elements in a free group. I’ll also discuss how to verify hyperbolicity using versions of combinatorial curvature on van Kampen diagrams.
    "Recognising surface groups"
       — A/Prof David Kielak

    Abstract:
    I will address two problems about recognising surface groups. The first one is the classical problem of classifying Poincaré duality groups in dimension two. I will present a new approach to this, joint with Peter Kropholler. The second problem is about recognising surface groups among one-relator groups. Here I will present a new partial result, joint with Giles Gardam and Alan Logan.