The category of totally disconnected locally compact groups

Friday 13th April 2018, University of Lincoln Local organiser: Anitha Thillaisundaram

This is the fourth meeting of the Research Group Functor Categories for Groups (FCG). This meeting, which is more of a masterclass in nature, shall focus on the category of totally disconnected locally compact groups and how they intersect with other areas, such as permutation groups, operator algebras and model theory. The totally disconnected compact groups are of course the profinite groups, whose influence is far-reaching. The locally-compact case has received much attention since the ground-breaking result of Willis in 1994.

All talks will be held at the University of Lincoln. The first talk and the tea break will be held at the Sarah Swift Building in room SSB0203, and the remaining two talks will be in The Junxion Building in room JUN0001. The timetable is as follows:

1.30-2.30: Simon Smith (University of Lincoln), The interplay between permutation groups and tdlc groups
2.30-3.00: tea break
3.00-4.00: Corina Ciobotaru (University of Fribourg), Cartan limits of SL(n,Q_p)
4.00-5.00: Dugald Macpherson (University of Leeds), Model theory and tdlc groups

Travel information and maps can be found at https://www.lincoln.ac.uk/home/maps/

The FCG Research Group is supported by an LMS Joint Research Groups in the UK Scheme 3 grant. Limited funding is available for PhD students, allocated on a first come first served basis. In addition, the LMS administers a Childcare Supplementary Grant Scheme. Further information about this scheme can be found on the LMS website: www.lms.ac.uk/content/childcare-supplementary-grants.

To register for the event, please email the local organiser Dr Anitha Thillaisundaram (athillaisundaram@lincoln.ac.uk).

Abstracts

Simon Smith (University of Lincoln), The interplay between permutation groups and tdlc groups

Permutation groups can be used to prove deep results about tdlc groups. This talk is intended to be a masterclass on some common techniques for exploring tdlc groups via their permutation representation.

Corina Ciobotaru (University of Fribourg), Cartan limits of SL(n,Q_p)

For a locally compact group G the set of all its closed subgroups S(G) is endowed with the Chabauty topology, under which S(G) becomes a compact space. Given a family of closed subgroups of G satisfying some properties it is then natural to ask if its limit subgroups in S(G) preserve the same properties and if we can explicitly compute them.

In a recent joint work with Arielle Leitner we study the limits under the Chabauty topology of Cartan subgroups of SL(n,Q_p), the closed subgroups of SL(n,Q_p)-conjugated to the diagonal subgroup of SL(n,Q_p). When a limit subgroup contains only elliptic elements we prove that up to conjugacy it is contained in the unipotent radical of the Borel subgroup of SL(n,Q_p). The key idea is an explicit homeomorphism between the Chabauty closure of Cartan sub-algebras of sl(n,Q_p). By the Flat Torus Theorem when a limit subgroup contains hyperbolic elements we prove it preserves a flat, not necessary of maximal dimension, in the Bruhat-Tits building associated with SL(n,Q_p).

Dugald Macpherson (University of Leeds), Model theory and tdlc groups

I will discuss a number of issues around connections between model theory and tdlc groups. The main topic will be joint work with Cheryl Prager and Simon Smith on various permutation group-theoretic issues around tdlc groups, e.g. concerning multiply transitive tdlc groups (with model-theoretic constructions) and tdlc groups which are maximal with respect to various conditions (eg subdegree finiteness). I will also discuss joint work with Bodirsky on permutation groups on a countably infinite set (in some cases tdlc groups) which are maximal subject to being topologically closed in the symmetric group.