Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 068, 14 pages      arXiv:1311.3880      https://doi.org/10.3842/SIGMA.2014.068
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

Groupoid Actions on Fractafolds

Marius Ionescu a and Alex Kumjian b
a) Department of Mathematics, United States Naval Academy, Annapolis, MD, 21402-5002, USA
b) Department of Mathematics, University of Nevada, Reno, NV, 89557, USA

Received February 04, 2014, in final form June 21, 2014; Published online June 28, 2014

Abstract
We define a bundle over a totally disconnected set such that each fiber is homeomorphic to a fractal blowup. We prove that there is a natural action of a Renault-Deaconu groupoid on our fractafold bundle and that the resulting action groupoid is a Renault-Deaconu groupoid itself. We also show that when the bundle is locally compact the associated $C^*$-algebra is primitive and has a densely defined lower-semicontinuous trace.

Key words: Renault-Deaconu groupoids; fractafolds; iterated function systems.

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