## mini-GAGTA

mini-GAGTA 2020

### 23 June 2020

ICMS, Bayes Centre, 47 Potterrow, Edinburgh EH8 9BT

### Scientific Committee

- Martin Bridson, University of Oxford
- Montserrat Casals-Ruiz, UPV Bilbao
- Jim Howie, Heriot-Watt University
- Sarah Rees, University of Newcastle

- Pascal Weil, LaBRI Bordeaux

**Organising committee**

- Laura Ciobanu, Heriot-Watt University
- Alan Logan, Heriot-Watt University
- Alexandre Martin, Heriot-Watt University
- Alina Vdovina, University of Newcastle
- Tim Riley, Cornell University

miniGAGTA was a gathering of GAGTA enthusiasts (and other interested parties) for an afternoon (UK time) of short talks by young researchers and informal discussions, during the week when GAGTA 2020 was supposed to take place. It was hosted online by the ICMS, Edinburgh, on Tuesday 23 June 2020.

Programme of speakers and links to videos of the talks are below:

**Matthew Conder **(Cambridge)

Title: *Discrete and free two-generated subgroups of SL2*

Abstract: It is well known that the Ping Pong Lemma can be applied to many two-generated subgroups of SL(2,ℝ) (using the action by Möbius transformations on the hyperbolic plane) in order to determine properties such as freeness and/or discreteness. In particular, there is a practical algorithm of Eick, Kirschmer and Leedham-Green which, given any two elements of SL(2,ℝ), will determine after finitely many steps whether or not the subgroup generated by these elements is both discrete and free of rank two. In this talk, I will show that a similar algorithm exists for two-generated subgroups of SL(2,K), where K is a non-archimedean local field (for instance, the p-adic numbers). Such groups act by isometries on a Bruhat-Tits tree, and the algorithm proceeds by computing and comparing various translation lengths, in order to determine whether or not a given two-generated subgroup of SL(2,K) is both discrete and free.

**Katie Vokes (**IHES)

Title: *Geometry of the separating curve graph*

Abstract: Given a compact, connected, orientable surface, we can define many associated graphs whose vertices represent curves or multicurves in the surface. A first example is the curve graph, which has a vertex for every simple closed curve in the surface and an edge joining two vertices if the corresponding curves are disjoint. We could alternatively restrict to those curves which separate the surface into two components. While the curve graph is known to always be Gromov hyperbolic, this is not the case for the separating curve graph. I will present joint work with Jacob Russell classifying for which surfaces the separating curve graph is hyperbolic, for which it is relatively hyperbolic, and for which it is neither of these.

**Simon André** (Vanderbilt)

Title: *Acylindrically hyperbolic groups and elementary equivalence*

**Abstract:** Zlil Sela proved that, given a non-cyclic torsion-free hyperbolic group G, the groups G and G*Z are elementarily equivalent. In my talk, I will present the following partial generalization of this result: if G is acylindrically hyperbolic, say with no non-trivial normal finite subgroup, then G and G*Z have the same \forall\exists - theory. It is an open question whether G and G*Z have the same elementary theory. Joint work with Jonathan Fruchter.

**Chris Natoli** (CUNY)

Title: *Two results in the model theory of free groups*

Abstract: In joint work with Olga Kharlampovich, we show that finitely generated free groups of rank at least 3 lack a model-theoretic property called universal homogeneity. We also show that countable groups that are elementarily equivalent to non-abelian free groups, with the extra condition that any finitely generated abelian subgroups are cyclic, are unions of chains of hyperbolic towers.

**Feyisayo Olukoya** (Aberdeen)

Title: *Simple overgroups of generalized Thompson groups Vn from asynchronous transducers*

Abstract: Groups generated by synchronous transducers contain groups with many interesting properties such as the first Grigorchuk group. A construction of Nekrashevych associates to each such group an overgroup of Vn, where the group Vn is a generalisation of Thompson's group V. This overgroup turns out to be simple in many cases, for instance the famous Rover group --- the overgroup of Vn associated with the first Girgorchuk group--- is finitely presented and simple. We generalise this construction and associate to every transducer, synchronous and asynchronous, an overgroup of Vn. In many cases the resulting overgroup is simple.

### Prize questions

The speakers at miniGAGTA each offered one problem as a "prize question".

Anyone answering one of these questions in time for GAGTA 2021 will be rewarded with a fantastic prize (possibly liquid in nature), which will be presented to them at GAGTA 2021. Please inform the organisers if you think you, or someone else, has solved one of these problems (we are, unfortunately, not omniscient, so we do need to be informed). The complete list of questions can be downloaded from here . Slides accompanying Diekert's questions can be downloaded from here .

Details regarding the invited speakers and other information about the conference can be found here.

**Arrangements**

** ****This conference will be held virtually - the registration form is now closed **** **