It is well known that the Ping Pong Lemma can be applied to many two-generated subgroups of xxx 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.

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.

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.

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.

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.