Published on *Max Planck Institute for Mathematics* (http://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

Speaker:

Misha Belolipetsky
Affiliation:

(IMPA Rio de Janeiro/MPIM Bonn)
Date:

Tue, 2020-03-03 10:15 - 12:00 We show that any cocompact Kleinian group $\Gamma$ has an exhaustive filtration by normal subgroups $\Gamma_i$ such that any subgroup of $\Gamma_i$ generated by $k_i$ elements is free, where $k_i \ge [\Gamma:\Gamma_i]^C$ and $C = C(\Gamma) > 0$. Together with this result we prove that $\log k_i \ge C_1 \mathrm{sys}_1(M_i)$, where $\mathrm{sys}_1(M_i)$ denotes the systole of $M_i$, thus providing a large set of new examples for a conjecture of Gromov. In the second theorem $C_1> 0$ is an absolute constant. We also consider a generalization of these results to non-cocompact Kleinian groups.

In the talk I am going to discuss the proofs of these theorems and some related open problems. This is a joint work with Cayo DÃ³ria.

**Links:**

[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] http://www.mpim-bonn.mpg.de/node/3444

[3] http://www.mpim-bonn.mpg.de/node/9809