Nicolas Monod
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View article: Some comments on piecewise-projective groups of the line
Some comments on piecewise-projective groups of the line Open
We consider groups of piecewise-projective homeomorphisms of the line which are known to be non-amenable using notably the Carrière–Ghys theorem on ergodic equivalence relations. Replacing that theorem by an explicit fixed-point argument, …
View article: A fixed-point theorem for face maps, or deletion-tolerant random finite sets
A fixed-point theorem for face maps, or deletion-tolerant random finite sets Open
We establish a fixed-point theorem for the face maps that consist in deleting the $i$th entry of an ordered set. Furthermore, we show that there exists random finite sets of integers that are almost invariant under such deletions. Conseque…
View article: Flatmates and the bounded cohomology of algebraic groups
Flatmates and the bounded cohomology of algebraic groups Open
For all algebraic groups over non-Archimedean local fields, the bounded cohomology vanishes. This follows from the corresponding statement for automorphism groups of Bruhat--Tits buildings, which hinges on the solution to the flatmate conj…
View article: The bounded cohomology of transformation groups of Euclidean spaces and discs
The bounded cohomology of transformation groups of Euclidean spaces and discs Open
We prove that the groups of orientation-preserving homeomorphisms and diffeomorphisms of $\mathbb{R}^n$ are boundedly acyclic, in all regularities. This is the first full computation of the bounded cohomology of a transformation group that…
View article: A family of exotic group C*-algebras
A family of exotic group C*-algebras Open
We show that a large family of groups without non-abelian free subgroups satisfy the following strengthening of nonamenability: they each have a rich supply of irreducible representations defining exotic C* -algebras. The construction is e…
View article: A type I conjecture and boundary representations of hyperbolic groups
A type I conjecture and boundary representations of hyperbolic groups Open
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 …
View article: Lie groups in the symmetric group: reducing Ulam's problem to the simple case
Lie groups in the symmetric group: reducing Ulam's problem to the simple case Open
Ulam asked whether all Lie groups can be represented faithfully on a countable set. We establish a reduction of Ulam's problem to the case of simple Lie groups. In particular, we solve the problem for all solvable Lie groups and more gener…
View article: A family of exotic group C*-algebras
A family of exotic group C*-algebras Open
We show that a large family of groups without non-abelian free subgroups satisfy the following strengthening of non-amenability: they each have a rich supply of irreducible representations defining exotic C*-algebras. The construction is e…
View article: Some comments on piecewise-projective groups of the line
Some comments on piecewise-projective groups of the line Open
We consider groups of piecewise-projective homeomorphisms of the line which are known to be non-amenable using notably the Carriere--Ghys theorem on ergodic equivalence relations. Replacing that theorem by an explicit fixed-point argument,…
View article: Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups
Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups Open
It is, by now, classical that lattices in higher rank semisimple groups have various rigidity properties. In this work, we add another such rigidity property to the list: uniform stability with respect to the family of unitary operators on…
View article: Lie groups as permutation groups: Ulam’s problem in the nilpotent case
Lie groups as permutation groups: Ulam’s problem in the nilpotent case Open
Ulam asked whether every connected Lie group can be represented on a countable structure. This is known in the linear case. We establish it for the first family of non-linear groups, namely in the nilpotent case. Further context is discuss…
View article: The cohomology of semi-simple Lie groups, viewed from infinity
The cohomology of semi-simple Lie groups, viewed from infinity Open
We prove that the real cohomology of semi-simple Lie groups admits boundary values, which are measurable cocycles on the Furstenberg boundary. This generalises known invariants such as the Maslov index on Shilov boundaries, the Euler class…
View article: Between free and direct products of groups
Between free and direct products of groups Open
We investigate the group $G \vee H$ obtained by gluing together two groups $G$ and $H$ at the neutral element. This construction curiously shares some properties with the free product but others with the direct product. Our results address…
View article: Lamplighters and the bounded cohomology of Thompson's group
Lamplighters and the bounded cohomology of Thompson's group Open
We prove the vanishing of the bounded cohomology of lamplighter groups for a wide range of coefficients. This implies the same vanishing for a number of groups with self-similarity properties, such as Thompson's group F. In particular, the…
View article: On the bounded cohomology of certain homeomorphism groups
On the bounded cohomology of certain homeomorphism groups Open
We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional manifolds. In particular, for the group of orientation-preserving homeomorphisms of the circle and of the closed 2-disc, we show that the bounded…
View article: Bounded and unbounded cohomology of homeomorphism and diffeomorphism groups
Bounded and unbounded cohomology of homeomorphism and diffeomorphism groups Open
We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional manifolds. In particular, for the group of orientation-preserving homeomorphisms of the circle and of the closed 2-disc, it is isomorphic to the …
View article: Lie groups as permutation groups: Ulam's problem in the nilpotent case
Lie groups as permutation groups: Ulam's problem in the nilpotent case Open
Ulam asked whether every connected Lie group can be represented on a countable structure. This is known in the linear case. We establish it for the first family of non-linear groups, namely in the nilpotent case. Further context is discuss…
View article: A type I conjecture and boundary representations of hyperbolic groups
A type I conjecture and boundary representations of hyperbolic groups Open
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 hyp…
View article: The cohomology of semi-simple Lie groups, viewed from infinity
The cohomology of semi-simple Lie groups, viewed from infinity Open
We prove that the cohomology of semi-simple Lie groups admits boundary values, which are measurable cocycles on the Furstenberg boundary. This generalises known invariants such as the Maslov index on Shilov boundaries, the Euler class on p…
View article: Notes on functions of hyperbolic type
Notes on functions of hyperbolic type Open
Functions of hyperbolic type encode representations on real or complex hyperbolic spaces, usually infinite-dimensional. These notes set up the complex case. As applications, we prove the existence of a non-trivial deformation family of rep…
View article: Furstenberg boundaries for pairs of groups
Furstenberg boundaries for pairs of groups Open
Furstenberg has associated to every topological group $G$ a universal boundary $\unicode[STIX]{x2202}(G)$ . If we consider in addition a subgroup $H<G$ , the relative notion of $(G,H)$ -boundaries admits again a maximal object $\unicode[ST…
View article: Erratum and addenda to "Isometry groups of non-positively curved spaces: discrete subgroups"
Erratum and addenda to "Isometry groups of non-positively curved spaces: discrete subgroups" Open
We amend the statement of point~(i) in Theorem~1.3 in arxiv:0901.1022 and supply the additional arguments and minor changes for the results that depend on it. We also seize the occasion and generalize to non-finitely generated lattices.
View article: Self-representations of the Möbius group
Self-representations of the Möbius group Open
Contrary to the finite-dimensional case, the Möbius group admits interesting self-representations when infinite-dimensional. We construct and classify all these self-representations. The proofs are obtained in the equivalent setting of iso…
View article: Kaleidoscopic groups: permutation groups constructed from dendrite homeomorphisms
Kaleidoscopic groups: permutation groups constructed from dendrite homeomorphisms Open
Given a transitive permutation group, a fundamental object for studying its higher transitivity properties is the permutation action of its isotropy subgroup. We reverse this relationship and introduce a universal construction of infinite …