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 prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite groups: let w be a 'locally finite' group word and d ∈

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.

Recently established rationality of correlation functions in a globally conformal invariant quantum field theory satisfying Wightman axioms is used to construct a family of soluble models in 4-dimensional Minkowski space-time. We consider in detail a model of a neutral scalar field φ of dimension 2 . It depends on a positive real parameter c, an analogue of the Virasoro central charge, and admits for all (finite) c an infinite number of conserved symmetric tensor currents. The operator product algebra of φ is shown to coincide with a simpler one, generated by a bilocal scalar field V (x 1 , x 2 ) of dimension (1, 1) . The modes of V together with the unit operator span an infinite dimensional Lie algebra L V whose vacuum (i.e. zero energy lowest weight) representations only depend on the central charge c . Wightman positivity (i.e. unitarity of the representations of L V ) is proven to be equivalent to c ∈ N .Mathematical Subject Classification. 81T40, 81R10, 81T10