# 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} }

. 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…

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## References

SHOWING 1-10 OF 84 REFERENCES

Amenable hyperbolic groups

- Mathematics
- 2012

We give a complete characterization of the locally compact groups that are nonelementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous…

Continuous bounded cohomology and applications to rigidity theory

- Mathematics
- 2001

We present a theory of continuous bounded cohomology of locally compact groups with coefficients in Banach modules. A central role is played by amenable actions, as they give rise to relatively…

Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras

- MathematicsCommentarii Mathematici Helvetici
- 2019

We address the problem to characterise closed type I subgroups of the automorphism group of a tree. Even in the well-studied case of Burger-Mozes' universal groups, non-type I criteria were unknown.…

Reduction of cocycles with hyperbolic targets

- MathematicsErgodic Theory and Dynamical Systems
- 1996

Abstract We show that any cocycle from an ergodic, finite measure preserving action of a higher-rank group to a closed subgroup of the isometry group of a proper, geodesic hyperbolic, ‘at most…

On spectra of Koopman, groupoid and quasi-regular representations

- Mathematics
- 2015

In this paper we investigate relations between Koopman, groupoid and quasi-regular representations of countable groups. We show that for an ergodic measure class preserving action of a countable…

Piecewise strongly proximal actions, free boundaries and the Neretin groups

- Mathematics
- 2021

A closed subgroup H of a locally compact group G is confined if the closure of the conjugacy class of H in the Chabauty space of G does not contain the trivial subgroup. We establish a dynamical…

Cocompact amenable closed subgroups: weakly inequivalent representations in the left-regular representation

- Mathematics
- 2015

We show that if $H \leq G$ is a closed amenable and cocompact subgroup of a unimodular locally compact group, then the reduced group C*-algebra of $G$ is not simple. Equivalently, there are unitary…

Weak Containment and Induced Representations of Groups

- MathematicsCanadian Journal of Mathematics
- 1962

Let G be a locally compact group and G† its dual space, that is, the set of all unitary equivalence classes of irreducible unitary representations of G. An important tool for investigating the group…

A Theorem on Unitary Representations of Semisimple Lie Groups

- Mathematics
- 1950

We show that a connected semisimple Lie group G none of whose simple constituents is compact (in particular, any connected complex semisimple group) has no nontrivial measurable unitary…

Certain topological groups are type I

- Mathematics
- 1970

The purpose of this paper is to prove a theorem which roughly states that if G is a topological group with a "sufficiently large" Type I subgroup H, then G itself is also Type I. The main tool in the…