On simplicity of reduced C*-algebras of groups

A countable group is C*-simple if its reduced C*-algebra is a simple algebra. Since Powers recognised in 1975 that non-abelian free groups are C*-simple, large classes of groups which appear naturally in geometry have been identified, including non-elementary Gromov hyperbolic groups and lattices in semisimple groups. In this exposition, C*-simplicity for countable groups is shown to be an extreme case of non-amenability. The basic examples are described and several open problems are formulated.Comment: 23 page

Expanding graphs, Ramanujan graphs, and 1-factor perturbations

We construct (k+-1)-regular graphs which provide sequences of expanders by adding or substracting appropriate 1-factors from given sequences of k-regular graphs. We compute numerical examples in a few cases for which the given sequences are from the work of Lubotzky, Phillips, and Sarnak (with k-1 the order of a finite field). If k+1 = 7, our construction results in a sequence of 7-regular expanders with all spectral gaps at least about 1.52

Amenability and ergodic properties of topological groups: from Bogolyubov onwards

The purpose of this expository article is to revisit the notions of amenability and ergodicity, and to point out that they appear for topological groups that are not necessarily locally compact in articles by Bogolyubov (1939), Fomin (1950), Dixmier (1950), and Rickert (1967).Comment: Terminology has been changed for reasons explained in Remark 3.15. Otherwise, minor corrections have been mad

Conjugacy growth series of some infinitely generated groups

It is observed that the conjugacy growth series of the infinite fini-tary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed, for other generating sets, for restricted permutational wreath products of finite groups by the finitary symmetric group, and for alternating groups. Similar methods are used to compute usual growth polynomials and conjugacy growth polynomials for finite symmetric groups and alternating groups, with respect to various generating sets of transpositions. Computations suggest a class of finite graphs, that we call partition-complete, which generalizes the class of semi-hamiltonian graphs, and which is of independent interest. The coefficients of a series related to the finitary alternating group satisfy congruence relations analogous to Ramanujan congruences for the partition function. They follow from partly conjectural "generalized Ramanujan congruences", as we call them, for which we give numerical evidence in Appendix C

Representation zeta functions of wreath products with finite groups

Let G be a group which has for all n a finite number r_n(G) of irreducible complex linear representations of dimension n. Let $\zeta(G,s) = \sum_{n=1}^{\infty} r_n(G) n^{-s}$ be its representation zeta function. First, in case G is a permutational wreath product of H with a permutation group Q acting on a finite set X, we establish a formula for $\zeta(G,s)$ in terms of the zeta functions of H and of subgroups of Q, and of the Moebius function associated with the lattice of partitions of X in orbits under subgroups of Q. Then, we consider groups W(Q,k) which are k-fold iterated wreath products of Q, and several related infinite groups W(Q), including the profinite group, a locally finite group, and several finitely generated groups, which are all isomorphic to a wreath product of themselves with Q. Under convenient hypotheses (in particular Q should be perfect), we show that r_n(W(Q)) is finite for all n, and we establish that the Dirichlet series $\zeta(W(Q),s)$ has a finite and positive abscissa of convergence s_0. Moreover, the function $\zeta(W(Q),s)$ satisfies a remarkable functional equation involving $\zeta(W(Q),es)$ for e=1,...,|X|. As a consequence of this, we exhibit some properties of the function, in particular that $\zeta(W(Q),s)$ has a singularity at s_0, a finite value at s_0, and a Puiseux expansion around s_0. We finally report some numerical computations for Q the simple groups of order 60 and 168.Comment: 35 pages, amstex sourc

Cubature formulas, geometrical designs, reproducing kernels, and Markov operators

Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way, several known results for spheres in Euclidean spaces, involving cubature formulas for polynomial functions and spherical designs, are shown to generalize to large classes of finite measure spaces $(\Omega,\sigma)$ and appropriate spaces of functions inside $L^2(\Omega,\sigma)$. The last section points out how spherical designs are related to a class of reflection groups which are (in general dense) subgroups of orthogonal groups