A Closed Subgroup of the Homeomorphism Group of the Circle with Property (T)

Abstract

Abstract We exhibit a topological group $G$ with property (T) acting nonelementarily and continuously on the circle. This group is an uncountable totally disconnected closed subgroup of ${\operatorname {Homeo}}^+(\textbf {S}^1)$. 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 cannot be smoothened in any way. Finally, we determine the universal minimal flow of the group $G$.