Topologically 2-generated groups

TL;DR

This paper demonstrates that automorphism groups of certain ultrahomogeneous structures, including those with free amalgamation, are topologically 2-generated and possess cyclically dense conjugacy classes, expanding known examples of such groups.

Contribution

It establishes that powers of automorphism groups and groups of measurable functions over these structures are topologically 2-generated, with new examples provided.

Findings

Automorphism groups have cyclically dense conjugacy classes.

These groups are topologically 2-generated.

Pairs generating the free group can cyclically generate these 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 ${\rm{Aut}}(M)^n$, $n=1,2,\ldots$, and the group $L_0({\rm{ Aut}}(M))$ of measurable functions with values in ${\rm{Aut}}(M)$, 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.