Borel asymptotic dimension and hyperfinite equivalence relations

A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we prove that this question always has a positive answer when the acting group is polycyclic, and we obtain a positive answer for all free actions of a large class of solvable groups including the Baumslag--Solitar group BS(1,2) and the lamplighter group. This marks the first time that a group of exponential volume-growth has been verified to have this property. In obtaining this result we introduce a new tool for studying Borel equivalence relations by extending Gromov's notion of asymptotic dimension to the Borel setting. We show that countable Borel equivalence relations of finite Borel asymptotic dimension are hyperfinite, and more generally we prove under a mild compatibility assumption that increasing unions of such equivalence relations are hyperfinite. As part of our main theorem, we prove for a large class of solvable groups that all of their free Borel actions have finite Borel asymptotic dimension (and finite dynamic asymptotic dimension in the case of a continuous action on a zero-dimensional space). We also provide applications to Borel chromatic numbers, Borel and continuous Folner tilings, topological dynamics, and $C^*$-algebras.

[1]  A. Tikuisis,et al.  Nuclear dimension of simple $$\mathrm {C}^*$$-algebras , 2019, Inventiones mathematicae.

[2]  T. Downarowicz,et al.  Symbolic Extensions of Amenable Group Actions and the Comparison Property , 2019, Memoirs of the American Mathematical Society.

[3]  David Kerr,et al.  Almost Finiteness and the Small Boundary Property , 2018, Communications in Mathematical Physics.

[4]  Su Gao,et al.  Continuous Combinatorics of Abelian Group Actions , 2018, 1803.03872.

[5]  T. Downarowicz,et al.  The comparison property of amenable groups , 2017, 1712.05129.

[6]  David Kerr.  Dimension, comparison, and almost finiteness , 2017, 1710.00393.

[7]  Clinton T. Conley,et al.  Følner tilings for actions of amenable groups , 2017, 1704.00699.

[8]  Clinton T. Conley,et al.  Hyperfiniteness and Borel combinatorics , 2016, Journal of the European Mathematical Society.

[9]  Guoliang Yu,et al.  Dynamic asymptotic dimension: relation to dynamics, topology, coarse geometry, and $$C^*$$C∗-algebras , 2015, 1510.07769.

[10]  A. Tikuisis,et al.  Quasidiagonality of nuclear C*-algebras , 2015, 1509.08318.

[11]  Su Gao,et al.  Countable abelian group actions and hyperfinite equivalence relations , 2015 .

[12]  Su Gao,et al.  FORCING CONSTRUCTIONS AND COUNTABLE BOREL EQUIVALENCE RELATIONS , 2015, The Journal of Symbolic Logic.

[13]  T. Downarowicz,et al.  Tilings of amenable groups , 2015, Journal für die reine und angewandte Mathematik (Crelles Journal).

[14]  S. Schneider,et al.  Locally Nilpotent Groups and Hyperfinite Equivalence Relations , 2013, 1308.5853.

[15]  G. Szabó.  The Rokhlin dimension of topological ℤm‐actions , 2013, 1308.5418.

[16]  Andrew S. Marks.  A determinacy approach to Borel combinatorics , 2013, 1304.3830.

[17]  G. Bell,et al.  Asymptotic Dimension , 2007, math/0703766.

[18]  Alain Louveau,et al.  Countable Borel Equivalence Relations , 2002, J. Math. Log..

[19]  Jean-Louis Tu.  La conjecture de Baum-Connes pour les feuilletages moyennables , 1999 .

[20]  Stevo Todorcevic,et al.  BOREL CHROMATIC NUMBERS , 1999 .

[21]  Benjamin Weiss,et al.  Entropy and isomorphism theorems for actions of amenable groups , 1987 .

[22]  M. Gromov.  Groups of polynomial growth and expanding maps , 1981 .

[23]  J. Feldman,et al.  Ergodic equivalence relations, cohomology, and von Neumann algebras , 1975 .

[24]  Hyman Bass,et al.  The Degree of Polynomial Growth of Finitely Generated Nilpotent Groups , 1972 .

[25]  Mikael Olsson.  Dimension , 2019, CSS3 Quick Syntax Reference.

[26]  Clinton T. Conley,et al.  A bound on measurable chromatic numbers of locally finite Borel graphs , 2016 .

[27]  Benjamin Naumann,et al.  Classical Descriptive Set Theory , 2016 .

[28]  Derek J. S. Robinson,et al.  The Theory of Infinite Soluble Groups , 2004 .

[29]  R. Dougherty,et al.  The structure of hy-per nite Borel equivalence relations , 1994 .

[30]  Graham A. Niblo,et al.  Asymptotic invariants of infinite groups , 1993 .

[31]  Theodore A. Slaman,et al.  Definable functions on degrees , 1988 .

[32]  Benjamin Weiss,et al.  Ergodic theory of amenable group actions. I: The Rohlin lemma , 1980 .

[33]  Y. Guivarc’h.  Croissance polynomiale et périodes des fonctions harmoniques , 1973 .

[34]  J. Wolf.  Growth of finitely generated solvable groups and curvature of Riemannian manifolds , 1968 .