Double cosets for $\SU(2)\times...\times\SU(2)$ and outer automorphisms of free groups

TL;DR

This paper studies the structure of double cosets in a product of special unitary groups, providing parametrizations, measure descriptions, and explicit automorphism actions relevant to algebraic and geometric group theory.

Contribution

It introduces a parametrization of double cosets for multiple SU(2) groups and derives explicit formulas for automorphism and braid group actions.

Findings

Parametrization of double cosets in SU(2)^n

Explicit formulas for automorphism group actions

Description of the Haar measure's radial part

Abstract

Consider the space of double cosets of the product of $n$ copies of $\SU(2)$ with respect to the diagonal subgroup. We get a parametrization of this space, the radial part of the Haar measure, and explicit formulas for the actions of the group of outer automorphisms of the free group $F_{n-1}$ and of the braid group of $n-1$ strings.