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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 latticealways contain a surface group, i.e. a subgroup isomorphic to the fundamentalgroup of a compact hyperbolic surface? If so, does it contain surface subgroupsclose (in a precise quantitative sense) to Fuchsian subgroups of $G$, i.e todiscrete 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 ofThurston on 3-dimensional hyperbolic manifolds, and the quantitative version ofthe case $G=\operatorname{PSL}(2,\mathbf{R}) \times\operatorname{PSL}(2,\mathbf{R})$ implies a conjecture of Ehrenpreis on pairsof compact hyperbolic surfaces; these two conjectures were proved by Kahn andMarkovi\'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 theyobtain are images of representations that are Anosov in the sense of Labourie.We present some of the ideas of their proof.

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]