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Homology of odometers

Part of: Selfadjoint operator algebras Topological and differentiable algebraic systems

Published online by Cambridge University Press:  13 March 2019

EDUARDO SCARPARO
EDUARDO SCARPARO*
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-970Florianópolis-SC, Brazil email [email protected]
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Abstract

We compute the homology groups of transformation groupoids associated with odometers and show that certain $(\mathbb{Z}\rtimes \mathbb{Z}_{2})$-odometers give rise to counterexamples to the HK conjecture, which relates the homology of an essentially principal, minimal, ample groupoid $G$ with the K-theory of $C_{r}^{\ast }(G)$. We also show that transformation groupoids of odometers satisfy the AH conjecture.

Keywords

odometersgroupoid homologyK-theory

MSC classification

Primary: 46L80: $K$-theory and operator algebras (including cyclic theory)
Secondary: 22A22: Topological groupoids (including differentiable and Lie groupoids)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 9 , September 2020 , pp. 2541 - 2551
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Copyright
© Cambridge University Press, 2019

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References

Blackadar, B.. K-theory for Operator Algebras (Mathematical Sciences Research Institute Publications, 5). Springer, New York, 1986.Google Scholar
Bratteli, O., Evans, D. E. and Kishimoto, A.. Crossed products of totally disconnected spaces by Z 2 ∗ Z 2. Ergod. Th. & Dynam. Sys. 13(3) (1993), 445–484.Google Scholar
Brown, K. S.. Cohomology of Groups (Graduate Texts in Mathematics, 87). Springer, New York, 1994; corrected reprint of the 1982 original.Google Scholar
Brownlowe, N., Mundey, A., Pask, D., Spielberg, J. and Thomas, A.. C ∗ -algebras associated to graphs of groups. Adv. Math. 316 (2017), 114–186.Google Scholar
Carrión, J. R.. Classification of a class of crossed product C ∗ -algebras associated with residually finite groups. J. Funct. Anal. 260(9) (2011), 2815–2825.Google Scholar
Cortez, M. A. I. and Medynets, K.. Orbit equivalence rigidity of equicontinuous systems. J. Lond. Math. Soc. (2) 94(2) (2016), 545–556.Google Scholar
Cortez, M. I. and Petite, S.. G-odometers and their almost one-to-one extensions. J. Lond. Math. Soc. (2) 78(1) (2008), 1–20.Google Scholar
de Cornulier, Y.. Groupes pleins-topologiques (d’après Matui, Juschenko, Monod, …). Astérisque 361 (2014), 183–223 , Exp. No. 1064, viii.Google Scholar
Dyer, J., Hurder, S. and Lukina, O.. The discriminant invariant of Cantor group actions. Topology Appl. 208 (2016), 64–92.Google Scholar
Farsi, C., Kumjian, A., Pask, D. and Sims, A.. Ample groupoids: equivalence, homology, and Matui’s HK conjecture. Preprint, 2018, arXiv:1808.07807.Google Scholar
Green, P.. The structure of imprimitivity algebras. J. Funct. Anal. 36(1) (1980), 88–104.Google Scholar
Ioana, A.. Cocycle superrigidity for profinite actions of property (T) groups. Duke Math. J. 157(2) (2011), 337–367.Google Scholar
Kumjian, A.. An involutive automorphism of the Bunce–Deddens algebra. C. R. Math. Rep. Acad. Sci. Canada 10(5) (1988), 217–218.Google Scholar
Matui, H.. Some remarks on topological full groups of Cantor minimal systems II. Ergod. Th. & Dynam. Sys. 33(5) (2013), 1542–1549.Google Scholar
Matui, H.. Étale groupoids arising from products of shifts of finite type. Adv. Math. 303 (2016), 502–548.Google Scholar
Nekrashevych, V.. Simple groups of dynamical origin. Ergod. Th. & Dynam. Sys. 39(3) (2019), 707–732.Google Scholar
Nekrashevych, V.. Palindromic subshifts and simple periodic groups of intermediate growth. Ann. of Math. (2) 187(3) (2018), 667–719.Google Scholar
Orfanos, S.. Generalized Bunce–Deddens algebras. Proc. Amer. Math. Soc. 138(1) (2010), 299–308.Google Scholar
Ortega, E.. Homology of the Katsura-Exel-Pardo groupoid. Preprint, 2018, arXiv:1806.09297.Google Scholar
Scarparo, E.. On the $C^{\ast }$-algebra generated by the Koopman representation of a topological full group. Preprint, 2017, arXiv:1705.07665.Google Scholar
Thomsen, K.. The homoclinic and heteroclinic C ∗ -algebras of a generalized one-dimensional solenoid. Ergod. Th. & Dynam. Sys. 30(1) (2010), 263–308.Google Scholar
Weibel, C. A.. An Introduction to Homological Algebra (Cambridge Studies in Advanced Mathematics, 38). Cambridge University Press, Cambridge, 1994.Google Scholar