17 research outputs found
Amenable Invariant Random Subgroups
We show that an amenable Invariant Random Subgroup of a locally compact
second countable group lives in the amenable radical. This answers a question
raised in the introduction of the paper "Kesten's Theorem for Invariant Random
Subgroup" by Abert, Glasner and Virag. We also consider, in the opposite
direction, property (T), and prove a similar statement for this property. The
Appendix by Phillip Wesolek proves that the set of amenable subgroups is a
Borel subset in the Chabauty topology.Comment: We added an Appendix by Phillip Wesole
Combinatorial and group-theoretic compactifications of buildings
Let X be a building of arbitrary type. A compactification of the set
Res(X) of spherical residues of X is introduced. We prove that it coincides
with the horofunction compactification of Res(X) endowed with a natural
combinatorial distance which we call the root-distance. Points of
admit amenable stabilisers in Aut(X) and conversely, any amenable subgroup
virtually fixes a point in . In addition, it is shown that, provided
Aut(X)is transitive enough, this compactification also coincides with the
group-theoretic compactification constructed using the Chabauty topology on
closed subgroups of Aut(X). This generalises to arbitrary buildings results
established by Y. Guivarc'h and B. R\'emy in the Bruhat--Tits case
Amenability of actions on the boundary of a building
We prove that the action of the automorphism group of a building on its boundary is topologically amenable. The notion of boundary we use was defined in a previous paper \cite{CL}. It follows from this result that such groups have property (A), and thus satisfy the Novikov conjecture. It may also lead to applications in rigidity theory
Boundary maps and maximal representations on infinite dimensional Hermitian symmetric spaces
We define a Toledo number for actions of surface groups and complex hyper-bolic lattices on infinite dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite dimensional totally geodesic subspace on which the action is maximal. In the opposite direction we construct examples of geometrically dense maximal representation in the infinite dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, that we are able to construct in low ranks or under some suitable Zariski-density assumption, circumventing the lack of local compactness in the infinite dimensional setting
Proper proximality in non-positive curvature
37 pagesProper proximality of a countable group is a notion that was introduced by Boutonnet, Ioana and Peterson as a tool to study rigidity properties of certain von Neumann algebras associated to groups or ergodic group actions. In the present paper, we establish the proper proximality of many groups acting on nonpositively curved spaces. First, these include many countable groups acting properly nonelementarily by isometries on a proper space . More precisely, proper proximality holds in the presence of rank one isometries or when is a locally thick affine building with a minimal -action. As a consequence of Rank Rigidity, we derive the proper proximality of all countable nonelementary cubical groups, and of all countable groups acting properly cocompactly nonelementarily by isometries on either a Hadamard manifold with no Euclidean factor, or on a -dimensional piecewise Euclidean simplicial complex. Second, we establish the proper proximality of many hierarchically hyperbolic groups. These include the mapping class groups of connected orientable finite-type boundaryless surfaces (apart from a few low-complexity cases), thus answering a question raised by Boutonnet, Ioana and Peterson. We also prove the proper proximality of all subgroups acting nonelementarily on the curve graph. In view of work of Boutonnet, Ioana and Peterson, our results have applications to structural and rigidity results for von Neumann algebras associated to all the above groups and their ergodic actions
Amenable Invariant Random Subgroups
We added an Appendix by Phillip WesolekInternational audienceWe show that an amenable Invariant Random Subgroup of a locally compact second countable group lives in the amenable radical. This answers a question raised in the introduction of the paper "Kesten's Theorem for Invariant Random Subgroup" by Abert, Glasner and Virag. We also consider, in the opposite direction, property (T), and prove a similar statement for this property. The Appendix by Phillip Wesolek proves that the set of amenable subgroups is a Borel subset in the Chabauty topology
Tangent bundles of hyperbolic spaces and proper affine actions on spaces
We define the notion of a negatively curved tangent bundle of a metric measured space. We prove that, when a group acts on a metric measured space with a negatively curved tangent bundle, then acts on some space, and that this action is proper under suitable assumptions. We then check that this result applies to the case when is a CAT(-1) space or a quasi-tree
Tangent bundles of hyperbolic spaces and proper affine actions on spaces
We define the notion of a negatively curved tangent bundle of a metric measured space. We prove that, when a group acts on a metric measured space with a negatively curved tangent bundle, then acts on some space, and that this action is proper under suitable assumptions. We then check that this result applies to the case when is a CAT(-1) space or a quasi-tree
Coupling radiative, conductive and convective heat-transfers in a single Monte Carlo algorithm: A general theoretical framework for linear situations
It was recently shown that radiation, conduction and convection can be combined within a single Monte Carlo algorithm and that such an algorithm immediately benefits from state-of-the-art computer-graphics advances when dealing with complex geometries. The theoretical foundations that make this coupling possible are fully exposed for the first time, supporting the intuitive pictures of continuous thermal paths that run through the different physics at work. First, the theoretical frameworks of propagators and Green’s functions are used to demonstrate that a coupled model involving different physical phenomena can be probabilized. Second, they are extended and made operational using the Feynman-Kac theory and stochastic processes. Finally, the theoretical framework is supported by a new proposal for an approximation of coupled Brownian trajectories compatible with the algorithmic design required by ray-tracing acceleration techniques in highly refined geometry