A type I conjecture and boundary representations of hyperbolic groups

Caprace, Pierre-Emmanuel;Kalantar, Mehrdad;Monod, Nicolas
(2023) Proceedings of the London Mathematical Society — Vol. 127, n° 2, p. 447-486 (2023)

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Abstract
We establish new results on the weak containment of quasi-regular and Koopman representations of a second countable locally compact group associated with nonsingular spaces. We deduce that any two boundary representations of a hyperbolic locally compact group are weakly equivalent. We also show that nonamenable 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 Ł. 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.
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Caprace, P.-E., Kalantar, M., & Monod, N. (2023). A type I conjecture and boundary representations of hyperbolic groups. Proceedings of the London Mathematical Society, 127(2), 447-486. https://doi.org/10.1112/plms.12547 (Original work published 2023)