Bounded Normal Generation and Invariant Automatic Continuity

TL;DR

This paper investigates how quickly products of a fixed conjugacy class in the projective unitary group of a II${}_1$-factor cover the entire group, establishing bounds related to the 1-norm and exploring automatic continuity and topological uniqueness.

Contribution

It establishes bounds on the covering time of conjugacy class products in projective unitary groups of II${}_1$-factors and proves automatic continuity and topological uniqueness results.

Findings

Products of conjugacy classes cover the group quickly, as bounded by the 1-norm.

Every homomorphism to a polish SIN group is continuous.

The projective unitary group has a unique polish group topology.

Abstract

We study the question how quickly products of a fixed conjugacy class in the projective unitary group of a II${}_1$-factor von Neumann algebra cover the entire group. Our result is that the number of factors that are needed is essentially as small as permitted by the $1$-norm - in analogy to a result of Liebeck-Shalev for non-abelian finite simple groups. As an application of the techniques, we prove that every homomorphism from the projective unitary group of a II${}_1$-factor to a polish SIN group is continuous. Moreover, we show that the projective unitary group of a II${}_1$-factor carries a unique polish group topology.