Geometry, Topology and Dynamics Seminar
Mikolaj Fraczyk
Renyi Institute
Growth of torsion in low homology groups of arithmetic lattices and mapping class groups
Abstract: Let $\Gamma$ be a finitely generated countable group
and let $\Gamma_n$ be a sequence of subgroups.
We are interested in the growth of $\log |H_k(\Gamma_n,\mathbb{Z})_{tors}|$ as the index $[\Gamma:\Gamma_n]$ goes to infinity.
We will focus on the cases when $\Gamma$ is either the mapping class group of a genus $g\geq 2$ surface or $\Gamma=SL(d,\mathbb{Z})$ with $d\geq 3$.
For these groups we are able to show that under some natural conditions on $(\Gamma_n)$ the logarithm of $|H_k(\Gamma_n,\mathbb{Z})_{tors}|$ grows as $o([\Gamma:\Gamma_n])$ when $k\leq 3g-4, d-2$ respectively.
To prove these estimates we consider the action of $\Gamma$ on the curve complex (in the mapping class group case) or the rational Tits complex (in the $SL(d,\mathbb{Z})$ case) and use them to build an explicit "low dimensional" projective resolution.
This is a joint work with Miklos Abert, Nicolas Bergeron and Damien Gaboriau.
Monday March 11, 2019 at 3:00 PM in 636 SEO