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Tilings, fundamental cocycles and fundamental groups of symbolic ${\Bbb Z}^{d}$-actions

Published online by Cambridge University Press:  01 December 1998

KLAUS SCHMIDT
KLAUS SCHMIDT
Affiliation:
Mathematics Institute, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria (e-mail: klaus.schmidt@univie.ac.at)
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Abstract

We prove that certain topologically mixing two-dimensional shifts of finite type have a ‘fundamental’ $1$-cocycle with the property that every continuous $1$-cocycle on the shift space with values in a discrete group is continuously cohomologous to a homomorphic image of the fundamental cocycle. These fundamental cocycles are closely connected with representations of the shift space by Wang tilings and the tiling groups of Conway, Lagarias and Thurston, and they determine the projective fundamental groups of the shift spaces introduced by Geller and Propp.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 18 , Issue 6 , December 1998 , pp. 1473 - 1525
Copyright
1998 Cambridge University Press

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