We define and study the class of positively finitely related (PFR) profinite groups. Positive finite relatedness is a probabilistic property of profinite groups which provides a first step to defining higher finiteness properties of profinite groups which generalize the positively finitely generated groups introduced by Avinoam Mann. We prove many asymptotic characterisations of PFR groups, for instance we show the following: a finitely presented profinite group is PFR if and only if it has at most exponential representation growth, uniformly over finite fields (in other words: the completed group algebra has polynomial maximal ideal growth). From these characterisations we deduce several structural results on PFR profinite groups.

Let G be a group with a finite subgroup H. We define the L 2multiplicity of an irreducible representation of H in the L 2 -homology of a proper G-CW-complex. These invariants generalize the L 2 -Betti numbers. Our main results are approximation theorems for L 2 -multiplicities which extend the approximation theorems for L 2 -Betti numbers of Lück, Farber and Elek-Szabó respectively. The main ingredient is the theory of characters of infinite groups and a method to induce characters from finite subgroups. We discuss applications to the cohomology of (arithmetic) groups.

Let G be a profinite group. A strongly admissible smooth representation ̺ of G over C decomposes as a direct sum ̺ ∼ = π∈Irr(G) m π (̺) π of irreducible representations with finite multiplicities m π (̺) such that for every positive integer n the number r n (̺) of irreducible constituents of dimension n is finite. Examples arise naturally in the representation theory of reductive groups over non-archimedean local fields. In this article we initiate an investigation of the Dirichlet generating functionOur primary focus is on representations ̺ = Ind G H (σ) of compact p-adic Lie groups G that arise from finite-dimensional representations σ of closed subgroups H via the induction functor. In addition to a series of foundational results -including a description in terms of p-adic integrals -we establish rationality results and functional equations for zeta functions of globally defined families of induced representations of potent pro-p groups. A key ingredient of our proof is Hironaka's resolution of singularities, which yields formulae of Denef-type for the relevant zeta functions.In some detail, we consider representations of open compact subgroups of reductive p-adic groups that are induced from parabolic subgroups. Explicit computations are carried out by means of complementing techniques: (i) geometric methods that are applicable via distance-transitive actions on spherically homogeneous rooted trees and (ii) the p-adic Kirillov orbit method. Approach (i) is closely related to the notion of Gelfand pairs and works equally well in positive defining characteristic.

Abstract. We calculate the Lefschetz number of a Galois automorphism in the cohomology of certain arithmetic congruence groups arising from orders in quaternion algebras over number fields. As an application we give a lower bound for the first Betti number of a class of arithmetically defined hyperbolic 3-manifolds and we deduce the following theorem: Given an arithmetically defined cocompact subgroup Γ ⊂ SL 2 (C), provided the underlying quaternion algebra meets some conditions, there is a decreasing sequence {Γ i } i of finite index subgroups of Γ such that the first Betti number satisfies

We study the existence of lattices in a higher rank simple Lie group which are profinitely but not abstractly commensurable. We show that no such examples exist for the complex forms of type E 8 , F 4 , and G 2 . In contrast, there are arbitrarily many such examples for all other higher rank Lie groups, except possibly SL 2n+1 (R), SL 2n+1 (C), SL n (H), and any form of type E 6 .