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Actions of Finite Groups on Substitution Tilings and Their Associated C*algebras
"... The goal of this thesis is to examine the actions of finite symmetry groups on aperiodic tilings. To an aperiodic tiling with finite local complexity arising from a primitive substitution rule one can associate a metric space, transformation groupoids, and C*algebras. Finite symmetry groups of the ..."
Abstract

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The goal of this thesis is to examine the actions of finite symmetry groups on aperiodic tilings. To an aperiodic tiling with finite local complexity arising from a primitive substitution rule one can associate a metric space, transformation groupoids, and C*algebras. Finite symmetry groups of the tiling act on each of these objects and we investigate appropriate constructions on each, namely the orbit space, semidirect product groupoids, and crossed product C*algebras respectively. Of particular interest are the crossed product C*algebras; we derive important structure results about them and compute their Ktheory. ii Acknowledgements First and foremost, I would like to thank my wife and the love of my life Anna. This would not have been possible without her love, support, encouragement and friendship. I am also eternally grateful to my wonderful supervisor Thierry Giordano for being generous with excellent advice (both mathematical and otherwise) and financial support throughout my PhD studies. I would also like to thank my family: my mother Susan, my brother Bob, my grandparents Noeline (Nana) and Peter, and my stepfather TJ. In particular I must thank my Nana, who taught me at a young age the importance of continual learning, inquisitiveness, curiosity, and compassion. Mathematically, I would like to thank Ian Putnam, Michael Whittaker, Daniel Gonçalves, David Handelman, and Siegfried Echterhoff for many extremely helpful conversations about this thesis. I would also like to thank NSERC for financial support for the first half of my PhD studies. iii Dedication In memory of my father Ray, who was taken from us before I could finish this degree. We love you, dad. iv
The Künneth Theorem in equivariant Ktheory for actions of a cyclic group of order 2
 Algebr. Geom. Topol
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Tduality for circle bundles via noncommutative geometry, in preparation
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