Abstract. We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge-Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems.A basic idea is to adapt the notion of Benjamini-Schramm convergence (BSconvergence), originally introduced for sequences of finite graphs of bounded degree, to sequences of Riemannian manifolds, and analyze the possible limits. We show that BS-convergence of locally symmetric spaces Γ\G/K implies convergence, in an appropriate sense, of the normalized relative Plancherel measures associated to L 2 (Γ\G). This then yields convergence of normalized multiplicities of unitary representations, Betti numbers and other spectral invariants. On the other hand, when the corresponding Lie group G is simple and of real rank at least two, we prove that there is only one possible BS-limit, i.e. when the volume tends to infinity, locally symmetric spaces always BSconverge to their universal cover G/K. This leads to various general uniform results.When restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of BS-convergence which in turn implies upper estimates on the rate of convergence of normalized Betti numbers in the spirit of Sarnak-Xue.An important role in our approach is played by the notion of Invariant Random Subgroups. For higher rank simple Lie groups G, we exploit rigidity theory, and in particular the Nevo-Stück-Zimmer theorem and Kazhdan's property (T), to obtain a complete understanding of the space of IRSs of G.

An invariant random subgroup of the countable group Γ is a random subgroup of Γ whose distribution is invariant under conjugation by all elements of Γ.We prove that for a nonamenable invariant random subgroup H, the spectral radius of every finitely supported random walk on Γ is strictly less than the spectral radius of the corresponding random walk on Γ/H. This generalizes a result of Kesten who proved this for normal subgroups.As a byproduct, we show that for a Cayley graph G of a linear group with no amenable normal subgroups, any sequence of finite quotients of G that spectrally approximates G converges to G in Benjamini-Schramm convergence. In particular, this implies that infinite sequences of finite d-regular Ramanujan Schreier graphs have essentially large girth.

A proof is given that a permutation group in which different finite sets have different stabilizers cannot satisfy any group law. For locally compact topological groups with this property, almost all finite subsets of the group are shown to generate free subgroups. Consequences of these theorems are derived for: Thompson's group F , weakly branch groups, automorphism groups of regular trees, and profinite groups with alternating composition factors of unbounded degree.

Abstract. This paper investigates the asymptotic behaviour of the minimal number of generators of finite index subgroups in residually finite groups. We analyze three natural classes of groups: amenable groups, groups possessing an infinite soluble normal subgroup and virtually free groups. As a tool for the amenable case we generalize Lackenby's trichotomy theorem on finitely presented groups. Mathematics Subject Classification (2010). 20F69, 20E06.

Let Γ be a countable group and let f be a free probability measure preserving action of Γ. We show that all Bernoulli actions of Γ are weakly contained in f .It follows that for a finitely generated group Γ, the cost is maximal on Bernoulli actions for Γ and that all free factors of i.i.d. of Γ have the same cost.We also show that if f is ergodic, but not strongly ergodic, then f is weakly equivalent to f × I where I denotes the trivial action of Γ on the unit interval. This leads to a relative version of the Glasner-Weiss dichotomy.