Frédéric Haglund

Hyperplane sections in arithmetic hyperbolic manifolds

In this paper, we prove that the homology groups of immersed totally geodesic hypersurfaces of compact arithmetic hyperbolic manifolds virtually inject in the homology group of the

Commensurability and separability of quasiconvex subgroups

We show that two uniform lattices of a regular right-angled Fuchsian building are commensurable, provided the chamber is a polygon with at least six edges. We show that in an arbitrary Gromov-hyperbolic regular right-angled building associated to a graph product of finite groups, a uniform lattice is commensurable with the graph product provided all of its quasiconvex subgroups are separable. We obtain a similar result for uniform lattices of the Davis complex of Gromov-hyperbolic two-dimensional Coxeter groups. We also prove that every extension of a uniform lattice of a CAT.0/ square complex by a finite group is virtually trivial, provided each quasiconvex subgroup of the lattice is separable. 20F55, 20F67, 20F65; 20E26, 51E24, 20E22, 20J06