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Transactions of the American Mathematical Society

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Amenability of groupoids arising from partial semigroup actions and topological higher rank graphs


Authors: Jean N. Renault and Dana P. Williams
Journal: Trans. Amer. Math. Soc. 369 (2017), 2255-2283
MSC (2010): Primary 22A22, 46L55, 46L05
DOI: https://doi.org/10.1090/tran/6736
Published electronically: April 8, 2016
MathSciNet review: 3592511
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Abstract: We consider the amenability of groupoids $ G$ equipped with a group valued cocycle $ c:G\to Q$ with amenable kernel $ c^{-1}(e)$. We prove a general result which implies, in particular, that $ G$ is amenable whenever $ Q$ is amenable and if there is countable set $ D\subset G$ such that $ c(G^{u})D=Q$ for all $ u\in G^{(0)}$.

We show that our result is applicable to groupoids arising from partial semigroup actions. We explore these actions in detail and show that these groupoids include those arising from directed graphs, higher rank graphs and even topological higher rank graphs. We believe our methods yield a nice alternative groupoid approach to these important constructions.


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Additional Information

Jean N. Renault
Affiliation: Départment de Mathématiques, Université d’Orléans et CNRS (UMR 7349 et FR 2964), 45067 Orléans Cedex 2, France
Email: jean.renault@univ-orleans.fr

Dana P. Williams
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551
Email: dana.williams@Dartmouth.edu

DOI: https://doi.org/10.1090/tran/6736
Keywords: Groupoids, amenable groupoids, cocycle, semigroup actions, higher rank graphs, topological higher rank graphs.
Received by editor(s): January 12, 2015
Received by editor(s) in revised form: March 17, 2015
Published electronically: April 8, 2016
Additional Notes: The second author was supported by a grant from the Simons Foundation.
Article copyright: © Copyright 2016 American Mathematical Society

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