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]

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.