School of Mathematics & Physics, University of Lincoln
On the 13th of November 2019, Matthew Conder (University of Cambridge) visited Charlotte Scott Centre for Algebra and gave a talk “Discrete and free two-generated subgroups of SL2 over non-archimedean local fields”.
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.