Spectral rigidity of group actions on homogeneous spaces

International audienceActions of a locally compact group G on a measure space X give rise to unitary representations of G on Hilbert spaces. We review results on the rigidity of these actions from the spectral point of view, that is, results about the existence of a spectral gap for associated averaging operators and their consequences. We will deal both with spaces X with an infinite measure as well as with spaces with an invariant probability measure. The spectral gap property has several striking applications to group theory, geometry, ergodic theory, operator algebras, graph theory, theoretical computer science, etc

Chromatic numbers of Cayley graphs of abelian groups: A matrix method

In this paper, we take a modest first step towards a systematic study of chromatic numbers of Cayley graphs on abelian groups. We lose little when we consider these graphs only when they are connected and of finite degree. As in the work of Heuberger and others, in such cases the graph can be represented by an $m\times r$ integer matrix, where we call $m$ the dimension and $r$ the rank. Adding or subtracting rows produces a graph homomorphism to a graph with a matrix of smaller dimension, thereby giving an upper bound on the chromatic number of the original graph. In this article we develop the foundations of this method. In a series of follow-up articles using this method, we completely determine the chromatic number in cases with small dimension and rank; prove a generalization of Zhu's theorem on the chromatic number of $6$-valent integer distance graphs; and provide an alternate proof of Payan's theorem that a cube-like graph cannot have chromatic number 3.Comment: 17 page

Poincar\'e profiles of Lie groups and a coarse geometric dichotomy

Poincar\'e profiles are a family of analytically defined coarse invariants, which can be used as obstructions to the existence of coarse embeddings between metric spaces. In this paper we calculate the Poincar\'e profiles of all connected unimodular Lie groups, Baumslag-Solitar groups and Thurston geometries, demonstrating two substantially different types of behaviour. In the case of Lie groups, we obtain a dichotomy which extends both the dichotomy separating rank one and higher rank semisimple Lie groups and the dichotomy separating connected solvable unimodular Lie groups of polynomial and exponential growth. We provide equivalent algebraic, quasi-isometric and coarse geometric formulations of this dichotomy. Our results have many consequences for coarse embeddings, for instance we deduce that for groups of the form $N\times S$, where $N$ is a connected nilpotent Lie group, and $S$ is a simple Lie group of real rank 1, both the growth exponent of $N$, and the Ahlfors-regular conformal dimension of $S$ are non-decreasing under coarse embeddings. These results are new even in the quasi-isometric setting and give obstructions to quasi-isometric embeddings which in many cases are stronger than those previously obtained by Buyalo-Schroeder.Comment: 49 pages. v2: the paper has been restructured, the main results are the same but have been presented differentl