# Workshop on Groups, Generalisations and Applications

## 16th January 2019.

The next meeting of the network "Groups, Generalisations and Applications" will be held on Wednesday 16th January 2019, in Room FN156 of the Fraser Noble Building, University of Aberdeen.The Department of Mathematics is at the Old Aberdeen campus. For travel information see here, and for a campus map see here.

### Funding

Thanks to generous support from the Glasgow Mathematical Trust Learning and Research Support Fund, we expect to be aable to pay reasonable travel costs for all participants from a Scottish institution (depending on numbers). If more people come than expected, priority will be given to postgraduate students and those without other sources of funding.### Lunch

There are no formal arrangements for lunch, but a group of us will probably go to the main university cafeteria at around noon.### Schedule

1.30pm **Feyishayo Olukoya** (Aberdeen) *Outer automorphisms of the Higman-Thompson groups G_{n,r} and T_{n,r}*

** Abstract: **
In a seminal paper in 1996, Brin characterises the automorphism and outer automorphism groups of the Thompson groups F and T. In particular he shows that their outer automorphism groups are 'small' (cyclic of order 2 for T). In a follow up paper Brin and Guzman study automorphisms of generalisations of F and T, including the Higman-Thompson groups T_{n,n-1}. Their techniques yield no information about the groups T_{n,r} (for r not equal to n-1) and G_{n,r} (r general). In this talk I will describe recent work which bridges this gap by showing that automorphisms of G_{n,r} and T_{n,r} can be described by finite state machines called transducers. This yields a description in terms of transducers of their outer automorphism groups and results in some perhaps surprising results.
This is joint work with Peter Cameron, Collin Bleak, Yonah Maissel and Andres Navas.

2.45pm **Tara Brendle** (Glasgow) *The Steinberg module and level structures for surfaces with marked points*

** Abstract: ** It is known by work of Harer and Ivanov that the mapping class group of a surface is a virtual duality group, and that its dualizing module is the Steinberg module, that is, the unique nonzero homology group of the complex of curves associated to the surface. In this talk, we will give a new description of the Steinberg module for surfaces with marked points and explain applications to finding nontrivial cohomology in the level L congruence subgroups of the corresponding mapping class groups. This is work in progress with Andy Putman and Nathan Broaddus; it builds on work of Fullarton-Putman in the case of closed surfaces.

3.45pm Tea and Coffee

4.15pm **Martyn Quick** (St Andrews) *Presentations for Brinâ€™s higher-dimensional Thompson groups nV*

** Abstract: **
In 2004, Brin introduced, for each integer n >= 2, a group nV consisting of certain homeomorphisms of n-dimensional Cantor space. This generalises the group V defined by Richard Thompson in 1965 and provides further examples of finitely presented infinite simple groups. In this talk, I will describe recent work that gives new presentations for each of the group nV obtained by exploiting transpositions of disjoint basic open sets. The first presentation is infinite, is analogous to the Coxeter presentation for the finite symmetric group and reflects the self-similar structure of Cantor space. The second is a reduction to a finite presentation that is considerably smaller than those previously known.

### Dinner

We'll eat at 6pm at a restaurant near the train station. Please email Ben Martin by 5pm on Monday 14th January if you'd like to come to dinner.The Groups, Generalisations and Applications Network is organised by Jim Howie (Heriot-Watt), Ben Martin (Aberdeen) and Colva Roney-Dougal (St Andrews).

This is the seventh meeting in the series. Details of previous meetings can be found here: Meeting 1, Meeting 2, Meeting 3, Meeting 4, Meeting 5, Meeting 6