# Maximal amenable subalgebras of von Neumann algebras associated with hyperbolic groups

@article{Boutonnet2013MaximalAS, title={Maximal amenable subalgebras of von Neumann algebras associated with hyperbolic groups}, author={R'emi Boutonnet and Alessandro Carderi}, journal={Mathematische Annalen}, year={2013}, volume={367}, pages={1199-1216} }

We prove that for any infinite, maximal amenable subgroup H in a hyperbolic group G, the von Neumann subalgebra LH is maximal amenable inside LG. It provides many new, explicit examples of maximal amenable subalgebras in II$$_1$$1 factors. We also prove similar maximal amenability results for direct products of relatively hyperbolic groups and orbit equivalence relations arising from measure-preserving actions of such groups.

## 19 Citations

### Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups

- Mathematics
- 2014

We provide a general criterion to deduce maximal amenability of von Neumann subalgebras LΛ ⊂ LΓ arising from amenable subgroups Λ of discrete countable groups Γ. The criterion is expressed in terms…

### Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups

- MathematicsGeometric and Functional Analysis
- 2015

We provide a general criterion to deduce maximal amenability of von Neumann subalgebras LΛ ⊂ LΓ arising from amenable subgroups Λ of discrete countable groups Γ. The criterion is expressed in terms…

### Maximal amenability of the generator subalgebra in q-Gaussian von Neumann algebras

- MathematicsJournal of Operator Theory
- 2018

In this article, we give explicit examples of maximal amenable subalgebras of the $q$-Gaussian algebras, namely, the generator subalgebra is maximal amenable inside the $q$-Gaussian algebras for real…

### On invariant von Neumann subalgebras rigidity property

- MathematicsJournal of Functional Analysis
- 2022

### Singular subgroups in $\tilde{A}_2$-groups and their von Neumann algebras

- Mathematics
- 2018

We show that certain amenable subgroups inside Ã2-groups are singular in the sense of Boutonnet and Carderi. This gives a new family of examples of singular group von Neumann subalgebras. We also…

### Maximal von Neumann subalgebras arising from maximal subgroups

- MathematicsScience China Mathematics
- 2021

Ge (2003) asked the question whether LF∞ can be embedded in to LF2 as a maximal subfactor. We answer it affirmatively in three different approaches, all containing the same key ingredient: the…

### ON THE NON-COMMUTATIVE MARGULIS PROPERTY

- Mathematics
- 2022

. We say that a countable discrete group Γ satisﬁes the non-commutative Margulis (NCM) property if every Γ - invariant von Neumann subalgebra M in L (Γ) is of the form L (Λ) for some normal subgroup…

### Singular Subgroups in $$ \widetilde {A}_2$$ -Groups and Their von Neumann Algebras

- Mathematics
- 2020

We show that certain amenable subgroups inside \(\widetilde {A}_2\)-groups are singular in the sense of Boutonnet and Carderi. This gives a new family of examples of singular group von Neumann…

### The cup subalgebra has the absorbing amenability property

- Mathematics
- 2015

Consider an inclusion of diffuse von Neumann algebras A c M . We say that A c M has the absorbing amenability property if for any diffuse subalgebra B c A and any amenable intermediate algebra B c D…

### Equivalence relations that act on bundles of hyperbolic spaces

- MathematicsErgodic Theory and Dynamical Systems
- 2017

Consider a measured equivalence relation acting on a bundle of hyperbolic metric spaces by isometries. We prove that every aperiodic hyperfinite subequivalence relation is contained in a unique…

## References

SHOWING 1-10 OF 22 REFERENCES

### The cup subalgebra of a II1 factor given by a subfactor planar algebra is maximal amenable

- Mathematics
- 2012

To every subfactor planar algebra was associated a II1 factor with a canonical abelian subalgebra generated by the cup tangle. Using Popa's approximative orthogonality property, we show that this cup…

### Elementary Subgroups of Relatively Hyperbolic Groups and Bounded Generation

- MathematicsInt. J. Algebra Comput.
- 2006

It is shown that if an element g ∈ G has infinite order and is not conjugate to an element of some Hλ, λ ∈ Λ, then the (unique) maximal elementary subgroup containing g is hyperbolically embedded into G, which allows us to prove that if G is boundedly generated, then G is elementary or Hλ = G for some λ → Λ.

### On hyperbolic groups

- Mathematics
- 2006

Abstract We prove that a δ-hyperbolic group for δ < ½ is a free product F * G 1 * … * Gn where F is a free group of finite rank and each Gi is a finite group.

### Maximal injective and mixing masas in group factors

- Mathematics
- 2010

We present families of pairs of finite von Neumann algebras $A\subset M$ where $A$ is a maximal injective masa in the type $\mathrm{II}_1$ factor $M$ with separable predual. Our results make use of…

### Relatively hyperbolic Groups

- MathematicsInt. J. Algebra Comput.
- 2012

This paper defines the boundary of a relatively hyperbolic group, and shows that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary, and generalizes a result of Tukia for geometRically finite kleinian groups.

### Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces

- Mathematics
- 2011

We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a…

### Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems

- Mathematics
- 2004

We suggest a new approach to the study of relatively hyperbolic groups based on relative isoperimetric inequalities. Various geometric, algebraic, and algorithmic properties are discussed.