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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The homology core of matchbox manifolds and invariant measures

Authors: Alex Clark and John Hunton
Journal: Trans. Amer. Math. Soc. 371 (2019), 1771-1793
MSC (2010): Primary 37C85; Secondary 28D15, 52C23, 37C70
Published electronically: October 11, 2018
MathSciNet review: 3894034
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Abstract: We consider the topology and dynamics associated with a wide class of matchbox manifolds, including spaces of aperiodic tilings and suspensions of higher rank (potentially nonabelian) group actions on zero-dimensional spaces. For such a space we introduce a topological invariant, the homology core, built using an expansion of it as an inverse sequence of simplicial complexes. The invariant takes the form of a monoid equipped with a representation, which in many cases can be used to obtain a finer classification than is possible with the previously developed invariants. When the space is obtained by suspending a topologically transitive action of the fundamental group $ \Gamma $ of a closed orientable manifold on a zero-dimensional compact space $ Z$, this invariant corresponds to the space of finite Borel measures on $ Z$ which are invariant under the action of $ \Gamma $. This leads to connections between the rank of the core and the number of invariant, ergodic Borel probability measures for such actions. We illustrate with several examples how these invariants can be calculated and used for topological classification and how it leads to an understanding of the invariant measures.

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Additional Information

Alex Clark
Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom
Address at time of publication: School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS

John Hunton
Affiliation: Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, United Kingdom

Keywords: Homology, matchbox manifold, invariant measure, aperiodic order
Received by editor(s): July 1, 2016
Received by editor(s) in revised form: June 28, 2017
Published electronically: October 11, 2018
Additional Notes: This work was supported by Leverhulme Trust for International Network grant IN-2013-045.
Article copyright: © Copyright 2018 American Mathematical Society