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arXiv · 1506.08549

Bounded Normal Generation and Invariant Automatic Continuity

Philip A. Dowerk·Andreas Thom
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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.

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Philip A. Dowerk, Andreas Thom. 2015-06-29. Bounded Normal Generation and Invariant Automatic Continuity. https://arxiv.org/abs/1506.08549

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