AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Group Colorings and Bernoulli Subflows
About this Title
Su Gao, Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, Texas 76203-5017, Steve Jackson, Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, Texas 76203-5017 and Brandon Seward, Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109-1043
Publication: Memoirs of the American Mathematical Society
Publication Year:
2016; Volume 241, Number 1141
ISBNs: 978-1-4704-1847-2 (print); 978-1-4704-2875-4 (online)
DOI: https://doi.org/10.1090/memo/1141
Published electronically: December 10, 2015
Keywords: Colorings,
hyper aperiodic points,
orthogonal colorings,
Bernoulli flows,
Bernoulli shifts,
Bernoulli subflows,
free subflows,
marker structures,
tilings,
topological conjugacy
MSC: Primary 37B10, 20F99; Secondary 03E15, 37B05, 20F65
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Basic Constructions of $2$-Colorings
- 4. Marker Structures and Tilings
- 5. Blueprints and Fundamental Functions
- 6. Basic Applications of the Fundamental Method
- 7. Further Study of Fundamental Functions
- 8. The Descriptive Complexity of Sets of $2$-Colorings
- 9. The Complexity of the Topological Conjugacy Relation
- 10. Extending Partial Functions to $2$-Colorings
- 11. Further Questions
Abstract
In this paper we study the dynamics of Bernoulli flows and their subflows over general countable groups. One of the main themes of this paper is to establish the correspondence between the topological and the symbolic perspectives. From the topological perspective, we are particularly interested in free subflows (subflows in which every point has trivial stabilizer), minimal subflows, disjointness of subflows, and the problem of classifying subflows up to topological conjugacy. Our main tool to study free subflows will be the notion of hyper aperiodic points; a point is hyper aperiodic if the closure of its orbit is a free subflow. We show that the notion of hyper aperiodicity corresponds to a notion of $k$-coloring on the countable group, a key notion we study throughout the paper. In fact, for all important topological notions we study, corresponding notions in group combinatorics will be established. Conversely, many variations of the notions in group combinatorics are proved to be equivalent to some topological notions. In particular, we obtain results about the differences in dynamical properties between pairs of points which disagree on finitely many coordinates.
Another main theme of the paper is to study the properties of free subflows and minimal subflows. Again this is done through studying the properties of the hyper aperiodic points and minimal points. We prove that the set of all (minimal) hyper aperiodic points is always dense but meager and null. By employing notions and ideas from descriptive set theory, we study the complexity of the sets of hyper aperiodic points and of minimal points, and completely determine their descriptive complexity. In doing this we introduce a new notion of countable flecc groups and study their properties. We also obtain the following results for the classification problem of free subflows up to topological conjugacy. For locally finite groups the topological conjugacy relation for all (free) subflows is hyperfinite and nonsmooth. For nonlocally finite groups the relation is Borel bireducible with the universal countable Borel equivalence relation.
The third, but not the least important, theme of the paper is to develop constructive methods for the notions studied. To construct $k$-colorings on countable groups, a fundamental method of construction of multi-layer marker structures is developed with great generality. This allows one to construct an abundance of $k$-colorings with specific properties. Variations of the fundamental method are used in many proofs in the paper, and we expect them to be useful more broadly in geometric group theory. As a special case of such marker structures, we study the notion of ccc groups and prove the ccc-ness for countable nilpotent, polycyclic, residually finite, locally finite groups and for free products.
- Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486
- John D. Clemens, Isomorphism of subshifts is a universal countable Borel equivalence relation, Israel J. Math. 170 (2009), 113–123. MR 2506320, DOI 10.1007/s11856-009-0022-0
- Ching Chou, Elementary amenable groups, Illinois J. Math. 24 (1980), no. 3, 396–407. MR 573475
- A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems 1 (1981), no. 4, 431–450 (1982). MR 662736, DOI 10.1017/s014338570000136x
- Michel Coornaert and Athanase Papadopoulos, Symbolic dynamics and hyperbolic groups, Lecture Notes in Mathematics, vol. 1539, Springer-Verlag, Berlin, 1993. MR 1222644
- R. Dougherty, S. Jackson, and A. S. Kechris, The structure of hyperfinite Borel equivalence relations, Trans. Amer. Math. Soc. 341 (1994), no. 1, 193–225. MR 1149121, DOI 10.1090/S0002-9947-1994-1149121-0
- Alexander Dranishnikov and Viktor Schroeder, Aperiodic colorings and tilings of Coxeter groups, Groups Geom. Dyn. 1 (2007), no. 3, 311–328. MR 2314048, DOI 10.4171/GGD/15
- Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 578656, DOI 10.1090/S0002-9947-1977-0578656-4
- Damien Gaboriau, Coût des relations d’équivalence et des groupes, Invent. Math. 139 (2000), no. 1, 41–98 (French, with English summary). MR 1728876, DOI 10.1007/s002229900019
- Su Gao, Invariant descriptive set theory, Pure and Applied Mathematics (Boca Raton), vol. 293, CRC Press, Boca Raton, FL, 2009. MR 2455198
- Su Gao and Steve Jackson, Countable abelian group actions and hyperfinite equivalence relations, Invent. Math. 201 (2015), no. 1, 309–383. MR 3359054, DOI 10.1007/s00222-015-0603-y
- Su Gao, Steve Jackson, and Brandon Seward, A coloring property for countable groups, Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 3, 579–592. MR 2557144, DOI 10.1017/S0305004109002655
- Eli Glasner and Vladimir V. Uspenskij, Effective minimal subflows of Bernoulli flows, Proc. Amer. Math. Soc. 137 (2009), no. 9, 3147–3154. MR 2506474, DOI 10.1090/S0002-9939-09-09905-5
- E. Glasner and B. Weiss, Kazhdan’s property T and the geometry of the collection of invariant measures, Geom. Funct. Anal. 7 (1997), no. 5, 917–935. MR 1475550, DOI 10.1007/s000390050030
- Walter Gottschalk, Some general dynamical notions, Recent advances in topological dynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Springer, Berlin, 1973, pp. 120–125. Lecture Notes in Math., Vol. 318. MR 0407821
- S. Jackson, A. S. Kechris, and A. Louveau, Countable Borel equivalence relations, J. Math. Log. 2 (2002), no. 1, 1–80. MR 1900547, DOI 10.1142/S0219061302000138
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597
- Douglas Lind and Brian Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. MR 1369092
- W. Magnus, Residually finite groups, Bull. Amer. Math. Soc. 75 (1969), 305–316. MR 241525, DOI 10.1090/S0002-9904-1969-12149-X
- Marston Morse and Gustav A. Hedlund, Symbolic Dynamics, Amer. J. Math. 60 (1938), no. 4, 815–866. MR 1507944, DOI 10.2307/2371264
- A. Ju. Ol′šanskiĭ, On the question of the existence of an invariant mean on a group, Uspekhi Mat. Nauk 35 (1980), no. 4(214), 199–200 (Russian). MR 586204
- Derek J. S. Robinson, A course in the theory of groups, 2nd ed., Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996. MR 1357169
- Jean-Pierre Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell; Corrected 2nd printing of the 1980 English translation. MR 1954121
- Theodore A. Slaman and John R. Steel, Definable functions on degrees, Cabal Seminar 81–85, Lecture Notes in Math., vol. 1333, Springer, Berlin, 1988, pp. 37–55. MR 960895, DOI 10.1007/BFb0084969
- Benjamin Weiss, Monotileable amenable groups, Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Transl. Ser. 2, vol. 202, Amer. Math. Soc., Providence, RI, 2001, pp. 257–262. MR 1819193, DOI 10.1090/trans2/202/18
- Joseph A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differential Geometry 2 (1968), 421–446. MR 248688