NBGGT

1 December 2022, University of Manchester (note new date)

There will be a North British Geometric Group Theory meeting on the afternoon of Thursday 1st December 2022, in the Alan Turing Building, University of Manchester.

We will have talks from Alice Kerr (Bristol), Rachel Skipper (ENS Paris) and Nóra Szakács (Manchester). As usual there will be coffee/tea in the afternoon, and an early dinner near the train station for those that can make it.

Some funding is available for Early Career Researchers.

This meeting is supported by the London Mathematical Society and The Heilbronn Institute for Mathematical Research.

In case of any questions, please contact the local organisers Alex Evetts (alex.evetts@manchester.ac.uk), Alex Levine (alex.levine@manchester.ac.uk), and Richard Webb (richard.webb@manchester.ac.uk).

Venue

The talks will take place in Frank Adams Room 1 on the first floor of the Alan Turing building, which is number 46 on the campus map here, and can be found on google maps here.

Programme

12:45-13:45 Nóra Szakács: Inverse semigroups as metric spaces, and their uniform Roe algebras

14:00-15:00 Alice Kerr: Product set growth in mapping class groups

Coffee break

15:30-16:30 Rachel Skipper: On some subgroups of big mapping class groups

Discussions, socialising, walking to dinner

17:30- Dinner at Zouk Tea Bar and Grill

Abstracts

• Nóra Szakács: Given any quasi-countable, in particular any countable inverse semigroup S, we introduce a way to equip S with a proper and right subinvariant extended metric. This generalizes the notion of proper, right invariant metrics for discrete countable groups. Such a metric is shown to be unique up to bijective coarse equivalence of the semigroup, and hence depends essentially only on S. This allows us to unambiguously define the uniform Roe algebra of S, which is a C*-algebra capturing the large scale geometry of the space.

Using this setting, we study those inverse semigroups with asymptotic dimension 0. Generalizing results known for groups, we show that these are precisely the locally finite inverse semigroups, and are further characterized by having strongly quasi-diagonal uniform Roe algebras. We show that, unlike in the group case, having a finite uniform Roe algebra is strictly weaker and is characterized by S being locally $\mathcal L$-finite, and equivalently sparse as a metric space.

This work is joint with Yeong Chyuan Chung and Diego Martínez.

• Alice Kerr: A standard question in group theory is to ask if we can categorise the subgroups of a group in terms of their growth. In this talk we will be asking this question for uniform product set growth, a property that is stronger than the more widely understood notion of uniform exponential growth. We will see how considering acylindrical actions on hyperbolic spaces can help us, and give a particular application to mapping class groups.

• Rachel Skipper: In this talk, we'll discuss some recent work to try to understand some naturally occurring subgroups of big mapping class groups and their connections to other well studied groups in geometric group theory (Thompson's groups, RAAGs, asymptotically rigid mapping class groups). The talk will include joint work with various authors.