** "Geometry and Analysis on Groups" Research Seminar **

**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)