Abstract
We introduce dynamic asymptotic dimension, a notion of dimension for actions of discrete groups on locally compact spaces, and more generally for locally compact étale groupoids. We study our notion for minimal actions of the integer group, its relation with conditions used by Bartels, Lück, and Reich in the context of controlled topology, and its connections with Gromov’s theory of asymptotic dimension. We also show that dynamic asymptotic dimension gives bounds on the nuclear dimension of Winter and Zacharias for \(C^*\)-algebras associated to dynamical systems. Dynamic asymptotic dimension also has implications for K-theory and manifold topology: these will be drawn out in subsequent work.
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Notes
This means that if \(gx=x\) for some \(g\in \Gamma \) and \(x\in X\), then \(g=e\) is the identity element of \(\Gamma \).
As will be clear from the proof, one can replace nuclear dimension with decomposition rank here, but we will not need this distinction.
The cited paper only covers the second countable case, but the second countability assumption is unnecessary when the groupoid is étale: see [13].
If A is a \(C^*\)-algebra faithfully represented on a Hilbert space H, and I, J are non-zero orthogonal ideals in A, then \(I\cdot H\) and \(J\cdot H\) are A-invariant non-zero subspaces of H; in particular, the representation is reducible.
References
Bartels, A: On proofs of the Farrell-Jones conjecture. arXiv:1210.1044v1 (2012)
Bartels, A., Lück, W.: The Borel conjecture for hyperbolic and CAT(0)-groups. Ann. Math. 175(2), 631–689 (2012)
Bartels, A., Lück, W., Reich, H.: Equivariant covers for hyperbolic groups. Geom. Topol. 12, 1799–1882 (2008)
Bell, G., Dranishnikov, A.: Asymptotic dimension. Topology Appl. 155(12), 1265–1296 (2008)
Bosa, J., Brown, N., Sato, Y., Tikuisis, A., White, S., Winter, W.: Covering dimension of \({C}^*\)-algebras and 2-coloured classification. arXiv:1506.03974 (2015)
Brown, N., Ozawa, N.: \({C}^*\)-Algebras and Finite-Dimensional Approximations. Graduate Studies in Mathematics, vol. 88. American Mathematical Society, Providence (2008)
Dadarlat, M., Guentner, E.: Uniform embeddability of relatively hyperbolic groups. J. Reine Angew. Math. 612, 1–15 (2007)
Dixmier, J.: \({C^*}\)-Algebras. North Holland Publishing Company, Amsterdam (1977)
Elliott, G.: The classification problem for amenable \({C}^*\)-algebras. Proc. Int. Congr. Math. 1(2), 922–932 (1995)
Elliott, G., Gong, G., Lin, H., Niu, Z.: On the classification of simple amenable \({C^*}\)-algebras with finite decomposition rank, II. arXiv:1507.03437 (2015)
Elliott, G., Niu, Z.: The \({C}^*\)-algebra of a minimal homeomorphism of zero mean dimension. arXiv:1406.2383v2 (2014)
Farrell, T., Jones, L.: \({K}\)-theory and dynamics I. Ann. Math. 124(2), 531–569 (1986)
Felix, R.: Renault’s equivalence theorem for \({C}^*\)-algebras of étale groupoids. Master’s thesis, University of Hawai‘i at Mānoa (2014)
Gromov, M.: Asymptotic invariants of infinite groups. In: Niblo, G., Roller, M. (eds.) Geometric Group Theory, vol. 2. London Mathematical Society, London (1993)
Guentner, E., Tessera, R., Yu, G.: A notion of geometric complexity and its application to topological rigidity. Invent. Math. 189(2), 315–357 (2012)
Guentner, E., Willett, R., Yu, G.: Dynamic asymptotic dimension and controlled operator K-theory. Working draft (2015)
Higson, N.: Bivariant \({K}\)-theory and the Novikov conjecture. Geom. Funct. Anal. 10, 563–581 (2000)
Higson, N., Roe, J.: Amenable group actions and the Novikov conjecture. J. Reine Angew. Math. 519, 143–153 (2000)
Hirshberg, I., Winter, W., Zacharias, J.: Rokhlin dimension and \({C^*}\)-dynamics. Comm. Math. Phys. 335, 637–670 (2015)
Hirshberg, I., Wu, J.: The nuclear dimension of \({C^*}\)-algebras associated to homeomorphisms. arXiv:1509.01508. With an appendix by Gabor Szabó (2015)
Kirchberg, E., Winter, W.: Covering dimension and quasidiagonality. Intern. J. Math. 15, 63–85 (2004)
Massey, W.: Algebraic Topology: An Introduction. Springer, Berlin (1977)
Muhly, P., Renault, J., Williams, D.: Equivalence and isomorphism for groupoid \({C}^*\)-algebras. J. Oper. Theory 17, 3–22 (1987)
Nowak, P., Yu, G.: Large scale geometry. In: EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich (2012)
Ozawa, N.: Amenable actions and exactness for discrete groups. C. R. Acad. Sci. Paris Sér. I Math. 330, 691–695 (2000)
Pears, A.: Dimension Theory of General Spaces. Cambridge University Press, Cambridge (1975)
Putnam, I.: The \({C^*}\)-algebras associated with minimal homeomorphisms of the Cantor set. Pac. J. Math. 136(2), 329–353 (1989)
Renault, J.: A Groupoid Approach to \({C}^*\)-Algebras. Lecture Notes in Mathematics, vol. 793. Springer, Berlin (1980)
Renault, J.: The ideal structure of groupoid crossed product \({C^*}\)-algebras. J. Oper. Theory 25, 3–36 (1991)
Renault, J.: \({C}^*\)-algebras and dynamical systems. Publicações Mathemáticas do IMPA, 27\(^\circ \) Colóquio Brasilieiro de Mathemática. Instituto Nacional de Matemática Pura e Aplicada (2009)
Roe, J.: Lectures on Coarse Geometry. University Lecture Series, vol. 31. American Mathematical Society, Providence (2003)
Roe, J.: Hyperbolic groups have finite asymptotic dimension. Proc. Am. Math. Soc. 133(9), 2489–2490 (2005)
Rørdam, M., Sierakowski, A.: Purely infinite \({C}^*\)-algebras arising from crossed products. Ergod. Theory Dynam. Syst. 32, 273–293 (2012)
Ruiz, E., Sims, A., Sørenson, A.: UCT-Kirchberg algebras have nuclear dimension one. Adv. Math. 279, 1–28 (2015)
Skandalis, G., Tu, J.-L., Yu, G.: The coarse Baum–Connes conjecture and groupoids. Topology 41, 807–834 (2002)
Szabó, G.: The Rokhlin dimension of topological \(\mathbb{Z}^m\)-actions. Proc. Lond. Math. Soc. 110(3), 673–694 (2015)
G. Szabó, Zacharias, J., Wu, J.: Rokhlin dimension for actions of residually finite groups. arXiv:1408.6096v2 (2014)
Tikuisis, A., White, S., Winter, W.: Quasidiagonality of nuclear \({C^*}\)-algebras. arXiv:1509.08318 (2015)
Toms, A., Winter, W.: Minimal dynamics and \({K}\)-theoretic rigidity: Elliott’s conjecture. Geom. Funct. Anal. 23, 467–481 (2013)
Vick, J.: Homology Theory: An Introduction to Algebraic Topology. Academic Press, Cambridge (1973)
Winter, W.: Decomposition rank of subhomogeneous \({C^*}\)-algebras. Proc. Lond. Math. Soc. 89, 427–456 (2004)
Winter, W., Zacharias, J.: The nuclear dimension of \({C}^*\)-algebras. Adv. Math. 224(2), 461–498 (2010)
Wright, N.: Finite asymptotic dimension for CAT(0) cube complexes. Geom. Topol. 1, 527–554 (2012)
Yu, G.: The Novikov conjecture for groups with finite asymptotic dimension. Ann. Math. 147(2), 325–355 (1998)
Acknowledgments
This paper has been some time in gestation, and has benefited from conversations with several people. In particular, we would like to thank Arthur Bartels, Siegfried Echterhoff, David Kerr, Ian Putnam, Daniel Ramras, Wilhelm Winter, and Jianchao Wu for useful comments, penetrating questions, and/or patient explanations. E. Guentner, R. Willett would like to thank Texas A&M University and the Shanghai Center for Mathematical Sciences for their hospitality during some of the work on this paper. E. Guentner was partially supported by a grant from the Simons Foundation (#245398). R. Willett was partially supported by NSF Grant DMS-1401126. G. Yu was partially supported by NSF Grant DMS-1362772 and NSFC Grant NSFC11420101001.
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Guentner, E., Willett, R. & Yu, G. Dynamic asymptotic dimension: relation to dynamics, topology, coarse geometry, and \(C^*\)-algebras. Math. Ann. 367, 785–829 (2017). https://doi.org/10.1007/s00208-016-1395-0
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DOI: https://doi.org/10.1007/s00208-016-1395-0