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The homology core of matchbox manifolds and invariant measures.

Clark, Alex and Hunton, John (2019) 'The homology core of matchbox manifolds and invariant measures.', Transactions of the American Mathematical Society., 371 (3). pp. 1771-1793.

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.

Item Type:Article
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution Non-commercial No Derivatives.
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1090/tran/7398
Publisher statement:© 2018 American Mathematical Society. First published in Transactions of the American Mathematical Society in Clark, Alex & Hunton, John (2019). The homology core of matchbox manifolds and invariant measures. Transactions of the American Mathematical Society 371(3): 1771-1793 published by the American Mathematical Society. Accepted manuscript available under a CC-BY-NC-ND licence.
Date accepted:16 December 2016
Date deposited:19 September 2017
Date of first online publication:11 October 2018
Date first made open access:11 October 2018

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