We study property (T) and the fixed point property for actions on $L^p$ and other Banach spaces. We show that property (T) holds when $L^2$ is replaced by $L^p$ (and even a subspace/quotient of $L^p$), and that in fact it is independent of $1\leq p<\infty$. We show that the fixed point property for $L^p$ follows from property (T) when $1

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.

We give a method for constructing dense and free subgroups in real Lie groups. In particular we show that any dense subgroup of a connected semisimple real Lie group G contains a free group on two generators which is still dense in G, and that any finitely generated dense subgroup in a connected non-solvable Lie group H contains a dense free subgroup of rank ≤ 2 · dim H. As an application, we obtain a new and elementary proof of a conjecture of Connes and Sullivan on amenable actions, which was first proved by Zimmer.

Abstract. We consider locally symmetric manifolds with a fixed universal covering, and construct for each such manifold M a simplicial complex R whose size is proportional to the volume of M . When M is noncompact, R is homotopically equivalent to M , while when M is compact, R is homotopically equivalent to M \ N , where N is a finite union of submanifolds of fairly smaller dimension. This reflects how the volume controls the topological structure of M , and yields concrete bounds for various finiteness statements which previously had no quantitative proofs. For example, it gives an explicit upper bound for the possible number of locally symmetric manifolds of volume bounded by v > 0, and it yields an estimate for the size of a minimal presentation for the fundamental group of a manifold in terms of its volume. It also yields a number of new finiteness results.