• Corpus ID: 238252991

A type I conjecture and boundary representations of hyperbolic groups

@inproceedings{Caprace2021ATI,
  title={A type I conjecture and boundary representations of hyperbolic groups},
  author={Pierre‐Emmanuel Caprace and Mehrdad Kalantar and Nicolas Monod},
  year={2021}
}

• P. Caprace, Mehrdad Kalantar, N. Monod
• Published 1 October 2021
• Mathematics

. We establish new results on the weak containment of quasi-regular and Koopman representations of a second countable locally compact group G associated with non-singular G -spaces. We deduce that any two boundary representations of a hyperbolic locally compact group are weakly equivalent. We also show that non-amenable hyperbolic locally compact groups with a cocompact amenable subgroup are characterized by the property that any two proper length functions are homothetic up to an additive constant. Combining those results with the work of (cid:32)L. Garncarek on the irreducibility of boundary representations of discrete hyperbolic groups, we deduce that a type I hyperbolic group with a cocompact lattice contains a cocompact amenable subgroup. Specializing to groups acting on trees, we answer a question of C. Houdayer and S. Raum.