A group with Property (T) acting on the circle

TL;DR

This paper constructs a topological group with property (T) acting non-elementarily on the circle, revealing new insights into group actions, unitary duals, and minimal flows, with implications for geometric and dynamical systems.

Contribution

It introduces a novel uncountable totally disconnected group with property (T) acting on the circle, derived from dendrites and hyperbolic laminations, and analyzes its dynamical properties.

Findings

The group has a large unitary dual and separates points.

The action on the circle is unique up to conjugation.

The universal minimal flow of the group is determined.

Abstract

We exhibit a topological group $G$ with property (T) acting non-elementarily and continuously on the circle. This group is an uncountable totally disconnected closed subgroup of $\operatorname{Homeo}^+(\mathbf{S}^1)$. It has a large unitary dual since it separates points. It comes from homeomorphisms of dendrites and a kaleidoscopic construction. Alternatively, it can be seen as the group of elements preserving some specific geodesic lamination of the hyperbolic disk. We also prove that this action is unique up to conjugation and that it can't be smoothened in any way. Finally, we determine the universal minimal flow of the group $G$.