131 research outputs found
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
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
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
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 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 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
has a finite and positive abscissa of convergence s_0.
Moreover, the function satisfies a remarkable functional
equation involving for e=1,...,|X|. As a consequence of this,
we exhibit some properties of the function, in particular that
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
Irreducibly represented groups
A group is irreducibly represented if it has a faithful irreducible unitary
representation. For countable groups, a criterion for irreducible
representability is given, which generalises a result obtained for finite
groups by W. Gasch\"utz in 1954. In particular, torsionfree groups and infinite
conjugacy class groups are irreducibly represented. We indicate some
consequences of this for operator algebras. In particular, we charaterise up to
isomorphism the countable subgroups of the unitary group of a
separable infinite dimensional Hilbert space \Cal H of which the bicommutants
(in the sense of the theory of von Neumann algebras) coincide with
the algebra of all bounded linear operators on \Cal H.Comment: 18 page
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