# Prof Sang-Hyun Kim

(Korea Institute of Advanced Study)

# Optimal regularity of mapping class group actions on the circle

We prove that for each finite index subgroup $H$ of the mapping class group of a closed hyperbolic surface, and for each real number $r>0$ there does not exist a faithful $C^{1+r}$--action of $H$ on a circle. (Joint with Thomas Koberda and Cristobal Rivas)

(UC Louvain)

# Uniform discreteness of arithmetic groups and the Lehmer conjecture

The famous Lehmer problem asks whether there is a gap between 1 and the Mahler measure of algebraic integers which are not roots of unity. Asked in 1933, this deep question concerning number theory has since then been connected to several other subjects. After introducing the concepts involved, we will briefly describe a few of these connections with the theory of linear groups. Then, we will discuss the equivalence of a weak form of the Lehmer conjecture and the "uniform discreteness" of cocompact lattices in semisimple Lie groups (conjectured by Margulis). Joint work with Lam Pham.

# 65th Annual Meeting of the Australian Mathematical Society

## Tuesday, 7th Dec 2021 — Friday, 10th Dec 2021

The 65th Annual Meeting of the Australian Mathematical Society will be hosted, online, by CARMA and the School of Information and Physical Sciences. Please visit the conference web page for more information.