Character theory of infinite wreath product

The representation theory of infinite wreath product groups is developed by means of the relationship between their group algebras and conjugacy classes with those of the infinite symmetric group. Further, since these groups are inductive limits of finite groups, their finite characters can be classified as limits of normalized irreducible characters of prelimit finite groups. This identification is called the "asymptotic character formula." The K 0 -invariant of the group C -algebra is also determined. Preliminaries The purpose of this paper is to develop the representation theory of infinite wreath product groups (defined in Section 2) by exploiting the relationship between their group algebras and conjugacy classes with those of the infinite symmetric group The infinite wreath product groups are inductive limits of finite groups, so that their C -group algebra is, in fact, an AF-algebra; that is, an inductive limit of finite-dimensional C -algebras. Such algebras have been well studied by means of their K 0 -group invariant. For the wreath product groups, the K 0 -group has an order structure which is determined through the evaluation of both the finite and semifinite characters of the Calgebra as well as a natural multiplication which makes the K 0 -invariant into a special ordered ring, namely, a Riesz ring (see Section 4). Another special feature for infinite wreath products is that their finite characters can be described as limits of normalized irreducible characters of the prelimit groups which is sometimes called the "asymptotic character formula." The major principle of this paper is that all these important features of the representation theory of wreath products can be reduced to the known results fo

Equivariant properties of symmetric products

The filtration on the infinite symmetric product of spheres by the number of factors provides a sequence of spectra between the sphere spectrum and the integral Eilenberg-Mac Lane spectrum. This filtration has received a lot of attention and the subquotients are interesting stable homotopy types. While the symmetric product filtration has been a major focus of research since the 1980s, essentially nothing was known when one adds group actions into the picture. We investigate the equivariant stable homotopy types, for compact Lie groups, obtained from this filtration of infinite symmetric products of representation spheres. The situation differs from the non-equivariant case, for example the subquotients of the filtration are no longer rationally trivial and on the zeroth equivariant homotopy groups an interesting filtration of the augmentation ideals of the Burnside rings arises. Our method is by global homotopy theory, i.e., we study the simultaneous behavior for all compact Lie groups at once.Comment: 33 page

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

Bounded normal generation is not equivalent to topological bounded normal generation

We show that some derived $\mathrm{L}^1$ full groups provide examples of non simple Polish groups with the topological bounded normal generation property. In particular, it follows that there are Polish groups with the topological bounded normal generation property but not the bounded normal generation property.Comment: 11 page

On conjugacy separability of fibre products

In this paper we study conjugacy separability of subdirect products of two free (or hyperbolic) groups. We establish necessary and sufficient criteria and apply them to fibre products to produce a finitely presented group $G_1$ in which all finite index subgroups are conjugacy separable, but which has an index $2$ overgroup that is not conjugacy separable. Conversely, we construct a finitely presented group $G_2$ which has a non-conjugacy separable subgroup of index $2$ such that every finite index normal overgroup of $G_2$ is conjugacy separable. The normality of the overgroup is essential in the last example, as such a group $G_2$ will always posses an index $3$ overgroup that is not conjugacy separable. Finally, we characterize $p$-conjugacy separable subdirect products of two free groups, where $p$ is a prime. We show that fibre products provide a natural correspondence between residually finite $p$-groups and $p$-conjugacy separable subdirect products of two non-abelian free groups. As a consequence, we deduce that the open question about the existence of an infinite finitely presented residually finite $p$-group is equivalent to the question about the existence of a finitely generated $p$-conjugacy separable full subdirect product of infinite index in the direct product of two free groups.Comment: v2: 38 pages; this is the version accepted for publicatio

The orbifold transform and its applications

We discuss the notion of the orbifold transform, and illustrate it on simple examples. The basic properties of the transform are presented, including transitivity and the exponential formula for symmetric products. The connection with the theory of permutation orbifolds is addressed, and the general results illustrated on the example of torus partition functions

Equivariant K-theory, generalized symmetric products, and twisted Heisenberg algebra

For a space X acted by a finite group \G, the product space $X^n$ affords a natural action of the wreath product \Gn. In this paper we study the K-groups K_{\tG_n}(X^n) of \Gn-equivariant Clifford supermodules on $X^n$. We show that \tFG =\bigoplus_{n\ge 0}K_{\tG_n}(X^n) \otimes \C is a Hopf algebra and it is isomorphic to the Fock space of a twisted Heisenberg algebra. Twisted vertex operators make a natural appearance. The algebraic structures on \tFG, when \G is trivial and X is a point, specialize to those on a ring of symmetric functions with the Schur Q-functions as a linear basis. As a by-product, we present a novel construction of K-theory operations using the spin representations of the hyperoctahedral groups.Comment: 33 pages, latex, references updated, to appear in Commun. Math. Phy

Products of Geck-Rouquier conjugacy classes and the Hecke algebra of composed permutations

We show the q-analog of a well-known result of Farahat and Higman: in the center of the Iwahori-Hecke algebra $H_{n,q}$, if $(a_{\lambda\mu}^{\nu}(n,q))_\nu$ is the set of structure constants involved in the product of two Geck-Rouquier conjugacy classes $\Gamma_{\lambda,n}$ and $\Gamma_{\mu,n}$, then each coefficient $a_{\lambda\mu}^{\nu}(n,q)$ depends on $n$ and $q$ in a polynomial way. Our proof relies on the construction of a projective limit of the Hecke algebras; this projective limit is inspired by the Ivanov-Kerov algebra of partial permutations.Comment: 12 pages, published in the proceedings of FPSAC 201

Generating infinite symmetric groups

Let S=Sym(\Omega) be the group of all permutations of an infinite set \Omega. Extending an argument of Macpherson and Neumann, it is shown that if U is a generating set for S as a group, respectively as a monoid, then there exists a positive integer n such that every element of S may be written as a group word, respectively a monoid word, of length \leq n in the elements of U. Several related questions are noted, and a brief proof is given of a result of Ore's on commutators that is used in the proof of the above result.Comment: 9 pages. See also http://math.berkeley.edu/~gbergman/papers To appear, J.London Math. Soc.. Main results as in original version. Starting on p.4 there are references to new results of others including an answer to original Question 8; "sketch of proof" of Lemma 11 is replaced by a full proof; 6 new reference