Groupes de surface dans les réseaux des groupes de Lie semi-simples [d'après J. Kahn, V. Marković, U. Hamenstädt, 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 $(P)SL(2,\mathbf{R})$ in $G$? The case $G=\mathrm{PSL}(2,\mathbf{C})$ corresponds to a famous conjecture of Thurston on 3-dimensional hyperbolic manifolds, and the quantitative version of the case $G=\mathrm{PSL}(2,\mathbf{R})\times \mathrm{PSL}(2,\mathbf{R})$ implies a conjecture of Ehrenpreis on pairs of compact hyperbolic surfaces; these two conjectures were proved by Kahn and Marković around ten years ago. Motivated by a question of Gromov, Hamenstädt solved the case that $G$ has real rank one, except for $G=\mathrm{SO}(2n,1)$. 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.

66 pages, in French language, 12 figures. Bourbaki Seminar, 2 October 2021, Exposé 1181