The homeomorphism group of the universal Knaster continuum

TL;DR

This paper constructs a projective Fraïssé family approximating the universal Knaster continuum, analyzes its automorphism group, and shows both automorphism and homeomorphism groups have a universal minimal flow linked to a free abelian group.

Contribution

It introduces a new projective Fraïssé family for the universal Knaster continuum and characterizes the universal minimal flows of its automorphism and homeomorphism groups.

Findings

Automorphism group is dense in the homeomorphism group.

Both groups have universal minimal flow homeomorphic to that of a free abelian group.

Identifies an open, normal, extremely amenable subgroup within both groups.

Abstract

We define a projective Fraiss\'e family whose limit approximates the universal Knaster continuum. The family is such that the group $\textrm{Aut}(\mathbb{K})$ of automorphisms of the Fraiss\'e limit is a dense subgroup of the group, $\textrm{Homeo}(K)$, of homeomorphisms of the universal Knaster continuum. We prove that both $\textrm{Aut}(\mathbb{K})$ and $\textrm{Homeo}(K)$ have universal minimal flow homeomorphic to the universal minimal flow of the free abelian group on countably many generators. The computation involves proving that both groups contain an open, normal subgroup which is extremely amenable.