A lattice in more than two Kac--Moody groups is arithmetic

Let $\Gamma$ be an irreducible lattice in a product of n infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n is at least 3, then each Kac-Moody groups is in fact a simple algebraic group over a local field and $\Gamma$ is an arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n is at least 2: either $\Gamma$ is an S-arithmetic (hence linear) group, or it is not residually finite. In that case, it is even virtually simple when the ground field is large enough. More general CAT(0) groups are also considered throughout.Comment: Subsection 2.B was modified and an example was added ther

Rank, combinatorial cost and homology torsion growth in higher rank lattices

We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of right angled groups. A group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute. Most non-uniform lattices in higher rank simple Lie groups are right angled. We provide the first examples of uniform (co-compact) right angled arithmetic groups in $\mathrm{SL}(n,\mathbb{R}),~n\geq 3$ and $\mathrm{SO}(p,q)$ for some values of $p,q$. This is a class of lattices for which the Congruence Subgroup Property is not known in general. Using rigidity theory and the notion of invariant random subgroups it follows that both the rank gradient and the homology torsion growth vanish for an arbitrary sequence of subgroups in any right angled lattice in a higher rank simple Lie group.Comment: 29 pages, to appear in Duke Mat

Relative Kazhdan Property

We perform a systematic investigation of Kazhdan's relative Property (T) for pairs (G,X), where G a locally compact group and X is any subset. When G is a connected Lie group or a p-adic algebraic group, we provide an explicit characterization of subsets X of G such that (G,X) has relative Property (T). In order to extend this characterization to lattices of G, a notion of "resolutions" is introduced, and various characterizations of it are given. Special attention is paid to subgroups of SU(2,1) and SO(4,1).Comment: 36 pages, no figure; to appear in Ann. Sci. Ecole Norm. Su