Topologically 2-generated groups

Abstract

We prove that for a number of ultrahomogeneous structures $M$, including those with the free amalgamation property, the powers of the automorphism group $\mathrm{Aut}(M{)}^n$, $n=1,2,\dots$, and the group $L_0(\mathrm{Aut}(M))$ of measurable functions with values in $\mathrm{Aut}(M)$, have a cyclically dense conjugacy class, in particular, are topologically 2-generated. This provides a number of new examples of groups with this property. Moreover, we will show that each of these groups is cyclically generated by a pair generating the free group.

arXiv admin note: substantial text overlap with arXiv:1612.03106