  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.
