56 research outputs found
Factor-of-iid balanced orientation of non-amenable graphs
We show that if a non-amenable, quasi-transitive, unimodular graph 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
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
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
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
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
A graph is defined inductively to be -regular if
is -regular and for every vertex of , the sphere of radius
around is an -regular graph. Such a graph is said
to be highly regular (HR) of level if . 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 and a subset of , we construct highly regular quotients
of the 1-skeleton of the associated Wythoffian polytope ,
which form an infinite family of expander graphs when is indefinite and
has finite vertex links. The regularity of the graphs in
this family can be deduced from the Coxeter diagram of . The expansion
stems from applying superapproximation to the congruence subgroups of the
linear group .
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
In his paper, Thurston shows that a positive real number is the
topological entropy for an ergodic traintrack representative of an outer
automorphism of a free group if and only if its expansion constant 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
- …