Birkbeck, University of London, Wednesday December 11th 2019
Venue: room G04 of 43 Gordon Square. On entering the building walk towards the coffee shop in front of you and immediately take a right. Confusingly, the crazy logic of the room numbering involves having to walk past G03 and G05 to get to G04. This is the same building as the past couple of years. (You’re warned that an accurate map of the building itself can be found here.) If you would like some directions then email b.fairbairn[usual funny AT symbol]bbk.ac.uk. Afterwards we will very likely go for some refreshments nearby and a curry afterwards.
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.
In this talk I will discuss a relative to Thompson’s group F, the group Fτ, which is the group of piecewise linear homeomorphisms of [0,1] with breakpoints in ℤ[τ] and slopes powers of τ, where τ = (√5 -1)/2 is the small Golden Ratio. This group was first considered by S. Cleary, who showed that the group was finitely presented and of type F∞. Here we take a combinatorial approach considering elements as tree-pair diagrams, where the trees are finite binary trees, but with two different kinds of carets. We use this representation to show that the commutator subgroup is simple and give a unique normal form for its elements. The surprising feature is that the T – and V – versions of these groups are not simple, however. This is joint work with J. Burillo and L. Reeves.
There is a bound, 2n-3, on the maximal number of factors in a direct product of non-elementary (equivalently, free) subgroups inside Aut(Fn), the automorphism group of a free group of rank n. I shall explain how we establish this bound, and explain the classification of the maximal products. The proof relies on a refined understanding of the action of Aut(Fn) on spaces of Fn-trees, and on the structure of auxiliary hyperbolic complexes. If time allows, I will also discuss applications to commensurator rigidity. (This is joint work with Ric Wade, Oxford.)