# Group Colorings and Bernoulli Subflows

### About this Title

**Su Gao**, **Steve Jackson** and **Brandon Seward**

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: http://dx.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

### 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.

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