Groupes de surface dans les r\'eseaux des groupes de Lie semi-simples [d'apr\`es J. Kahn, V. Markovi\'c, U. Hamenst\"adt, F. Labourie et S. Mozes]

TL;DR

This paper discusses the existence of surface subgroups within cocompact lattices of semisimple Lie groups, highlighting recent advances and the use of Anosov representations in this context.

Contribution

It presents ideas from recent work proving the presence of surface groups in broad classes of semisimple Lie groups, including complex simple groups, using Anosov representations.

Findings

Existence of surface groups in cocompact lattices of semisimple Lie groups.

Application of Anosov representations to construct surface subgroups.

Extension of previous results to complex simple Lie groups.

Abstract

A cocompact lattice in a semisimple Lie group $G$ is a discrete subgroup $\Gamma$ such that the quotient $G/\Gamma$ is compact. Does such a lattice always contain a surface group, i.e. a subgroup isomorphic to the fundamental group of a compact hyperbolic surface? If so, does it contain surface subgroups close (in a precise quantitative sense) to Fuchsian subgroups of $G$, i.e to discrete subgroups of $G$ contained in a copy of $\operatorname{(P)SL}(2,\mathbf{R})$ in $G$? The case $G=\operatorname{PSL}(2,\mathbf{C})$ corresponds to a famous conjecture of Thurston on 3-dimensional hyperbolic manifolds, and the quantitative version of the case $G=\operatorname{PSL}(2,\mathbf{R}) \times \operatorname{PSL}(2,\mathbf{R})$ implies a conjecture of Ehrenpreis on pairs of compact hyperbolic surfaces; these two conjectures were proved by Kahn and Markovi\'c around ten years ago. Motivated by a question of Gromov, Hamenst\"adt solved the case that $G$ has real rank one, except for $G=\operatorname{SO}(2n,1)$. In a recent preprint (arXiv:1805.10189), Kahn, Labourie, and Mozes treat the case of a large class of semisimple Lie groups, including in particular all complex simple Lie groups; the surface groups they obtain are images of representations that are Anosov in the sense of Labourie. We present some of the ideas of their proof.