Hall universal group has ample generic automorphisms

Abstract

We show that the automorphism group of Philip Hall's universal locally finite group has ample generics, that is, it admits comeager diagonal conjugacy classes in all dimensions. Consequently, it has the small index property, is not the union of a countable chain of non-open subgroups, and has the automatic continuity property. Also, we discuss some algebraic and topological properties of the automorphism group of Hall universal group. For example, we show that every generic automorphism of Hall universal group is conjugate to all of its powers, and hence has roots of all orders.

Structural Ramsey Theory and the Extension Property for Partial Automorphisms

“…(1) The class of all finite groups is known to have EPPA [SS19,Son17]. It is not clear what the corresponding Ramsey results should be.…”