Quantum isometry groups of duals of free powers of cyclic groups

TL;DR

This paper investigates the quantum isometry groups of duals of free powers of cyclic groups, revealing that for certain parameters, the main character's law follows a compound free Poisson distribution, and introduces a new quantum group as a limit case.

Contribution

It computes the law of the main character for quantum isometry groups of duals of free powers of cyclic groups, and constructs a new quantum group as a representation-theoretic limit.

Findings

Main character follows a compound free Poisson law for specific groups

Introduces a new quantum group as a limit at s approaching infinity

Provides technical variants and constructions related to quantum isometry groups

Abstract

We study the quantum isometry groups of the noncommutative Riemannian manifolds associated to discrete group duals. The basic representation theory problem is to compute the law of the main character of the relevant quantum group, and our main result here is as follows: for the group Z_s^{*n}, with s>4 and n>1, half of the character follows the compound free Poisson law with respect to the measure $\underline{\epsilon}$/2, where $\epsilon$ is the uniform measure on the s-th roots of unity, and $\epsilon\to\underline{\epsilon}$ is the canonical projection map from complex to real measures. We discuss as well a number of technical versions of this result, notably with the construction of a new quantum group, which appears as a "representation-theoretic limit", at s equal to infinity.