Hall universal group has ample generic automorphisms

TL;DR

This paper proves that the automorphism group of Hall's universal locally finite group has ample generics, leading to several important algebraic and topological properties such as the small index property and automatic continuity.

Contribution

It establishes that the automorphism group of Hall's universal group has ample generics and explores its algebraic and topological properties, which was previously unknown.

Findings

Automorphism group has ample generics in all dimensions

Automorphism group possesses the small index property

Every generic automorphism has roots of all orders

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.