arXiv subjects
Publications and source records attributed to Nicolas Monod.
The automorphism group of a regular locally finite tree is shown to admit irreducible Banach representations that are not admissible. The dense subspace of smooth vectors contains no algebraically irreducible component.
We develop a generalization of conditional probability for arbitrary ordered vector spaces. A related problem is that of assigning a numerical value to one vector relative to another. We characterize the groups for which these generalized probabilities can be stationary, respectively invariant. Our results deviate from the setting of classical probability and lead to a new criterion for amenability and for fixed points in cones.
We establish an eigenfunctional theorem for positive operators, evocative of the Krein--Rutman theorem. A more general version gives a joint eigenfunctional for commuting operators.
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. Consequences for various monoids of order-preserving transformations of $\mathbf{N}$ are discussed in an appendix.
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 conjecture raised in earlier work with Bucher. Vanishing and invariance theorems for arithmetic groups are derived.
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 is not compactly supported, and it implies that many characteristic classes of flat $\mathbb{R}^n$- and $S^n$-bundles are unbounded. We obtain the same result for the group of homeomorphisms of the disc that restrict to the identity on the boundary, and for the homeomorphism group of the non-compact Cantor set. In the appendix, Alexander Kupers proves a controlled version of the annulus theorem which we use to study the bounded cohomology of the homeomorphism group of the discs.
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 generally Lie groups with a linear Levi component. It follows that every amenable locally compact second countable group acts faithfully on a countable set.
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 explicit.
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, we can strengthen the conclusion and exhibit uncountably many "amenability gaps" between various piecewise-projective groups.
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 finite-dimensional Hilbert spaces equipped with submultiplicative norms. Namely, we show that for (most) high-rank lattices, every finite-dimensional unitary "almost-representation" of $\Gamma$ is a small deformation of a (true) unitary representation. This extends a result of Kazhdan (1983) for amenable groups and of Burger-Ozawa-Thom (2013) for SL(n,Z) (for n>2). Towards this goal, we first build an elaborate cohomological theory capturing the obstruction to such stability, and show that the vanishing of second cohomology implies uniform stability in this setting. This cohomology can be roughly thought of as an asymptotic version of bounded cohomology, and sheds light on a question raised in Monod (2006) about a possible connection between vanishing of second bounded cohomology and Ulam stability.
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 among others Property (T), CAT(0) cubical complexes, local embeddability, amenable actions, and the algebraic structure of $G \vee H$.
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, these groups are boundedly acyclic. Our method is ergodic and applies to large transformation groups where the Mather-Matsumoto-Morita method sometimes fails because not all are acyclic in the usual sense.
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 polynomial ring generated by the bounded Euler class. These seem to be the first examples of groups for which the entire bounded cohomology can be described without being trivial. We further prove that, contrary to ordinary cohomology, the diffeomorphisms groups of the circle and of the closed 2-disc have the same bounded cohomology as their homeomorphism groups and that both differ from the ordinary cohomology. Finally, we determine the low-dimensional bounded cohomology of homeo- and diffeomorphism of the spheres $S^n$ and of certain 3-manifolds. In particular, we answer a question of Ghys by showing that the Euler class in $H^4(Homeo_\circ(S^3))$ is unbounded.
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 discussed to illustrate the relevance of nilpotent groups for Ulam's problem.
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 \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.
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 projective space, or the hyperbolic ideal volume on spheres. In rank one, this leads to an isomorphism between the cohomology of the group and of this boundary model. In higher rank, additional classes appear, which we determine completely.
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.
Every Gelfand pair (G,K) admits a decomposition G=KP, where P<G is an amenable subgroup. In particular, the Furstenberg boundary of G is homogeneous. Applications include the complete classification of non-positively curved Gelfand pairs, relying on earlier joint work with Caprace, as well as a canonical family of pure spherical functions in the sense of Gelfand--Godement for general Gelfand pairs.