Title: Non-freeness of certain two-parabolic groups.
Speaker: Sang-hyun Kim (Korea Institute for Advanced Study (KIAS))
Abstract: For each positive rational number $$q$$, we study the (non-)freeness of the group $$G(q)$$ generated by $$2\times 2$$ matrices $$a = ( (1,0), (1,1) )$$ and $$b = ( (1,q), (0,1) )$$ in $$\mathrm{SL}(2,\mathbb{Q})$$. We give a computational criterion which allows us to prove that if $$q=s/r$$ for $$s\leq 27$$ then $$G(q)$$ is non-free, with the possible exception of $$s=24$$. In this latter case, we prove that the set of positive integers $$r$$ for which $$G(24/r)$$ is non-free has natural density 1. In the course of the proof, it will follow that for a fixed $$s$$, there are arbitrarily long sequences of consecutive denominators $$r$$ such that $$G(s/r)$$ is non-free. For the case $$s>27$$, we describe a density estimate. (Joint work with Thomas Koberda)