56 research outputs found

    Factor-of-iid balanced orientation of non-amenable graphs

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    We show that if a non-amenable, quasi-transitive, unimodular graph GG has all degrees even then it has a factor-of-iid balanced orientation, meaning each vertex has equal in- and outdegree. This result involves extending earlier spectral-theoretic results on Bernoulli shifts to the Bernoulli graphings of quasi-transitive, unimodular graphs. As a consequence, we also obtain that when GG is regular (of either odd or even degree) and bipartite, it has a factor-of-iid perfect matching. This generalizes a result of Lyons and Nazarov beyond transitive graphs.Comment: 24 pages, 1 figure. This is one of two papers that are replacing the shorter arXiv submission arXiv:2101.12577v1 Factor of iid Schreier decoration of transitive graph

    Spectral rigidity of group actions on homogeneous spaces

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    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

    Actions and Invariants of Residually Finite Groups: Asymptotic Methods

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    The workshop brought together experts in finite group theory, L2-cohomology, measured group theory, the theory of lattices in Lie groups, probability and topology. The common object of interest was residually finite groups, that each field investigates from a different angle

    Finite-dimensional Approximations of Discrete Groups

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    The main objective of this workshop was to bring together experts from various fields, which are all interested in finite and finite-dimensional approximations of infinite algebraic and analytic objects, such as groups, algebras, dynamical systems, group actions, or even von Neumann algebras

    Constructing highly regular expanders from hyperbolic Coxeter groups

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    A graph XX is defined inductively to be (a0,…,an−1)(a_0,\dots,a_{n-1})-regular if XX is a0a_0-regular and for every vertex vv of XX, the sphere of radius 11 around vv is an (a1,…,an−1)(a_1,\dots,a_{n-1})-regular graph. Such a graph XX is said to be highly regular (HR) of level nn if an−1≠0a_{n-1}\neq 0. Chapman, Linial and Peled studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders "globally and locally". They ask whether such HR-graphs of level 3 exist. In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can lead to such graphs. Given a Coxeter system (W,S)(W,S) and a subset MM of SS, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope PW,M\mathcal{P}_{W,M}, which form an infinite family of expander graphs when (W,S)(W,S) is indefinite and PW,M\mathcal{P}_{W,M} has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of (W,S)(W,S). The expansion stems from applying superapproximation to the congruence subgroups of the linear group WW. This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked by Chapman, Linial and Peled.Comment: 22 pages, 2 tables. Dedicated to the memory of John Conway and Ernest Vinber

    Thurston's Theorem: Entropy in Dimension One

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    In his paper, Thurston shows that a positive real number hh is the topological entropy for an ergodic traintrack representative of an outer automorphism of a free group if and only if its expansion constant λ=eh\lambda = e^h is a weak Perron number. This is a powerful result, answering a question analogous to one regarding surfaces and stretch factors of pseudo-Anosov homeomorphisms. However, much of the machinery used to prove this seminal theorem on traintrack maps is contained in the part of Thurston's paper on the entropy of postcritically finite interval maps and the proof difficult to parse. In this expository paper, we modernize Thurston's approach, fill in gaps in the original paper, and distill Thurston's methods to give a cohesive proof of the traintrack theorem. Of particular note is the addition of a proof of ergodicity of the traintrack representatives, which was missing in Thurston's paper.Comment: 37 pages, 13 figures. Comments welcome
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