Brandon Seward

A coloring property for countable groups

Motivated by research on hyperfinite equivalence relations we define a coloring property for countable groups. We prove that every countable group has the coloring property. This implies a compactness theorem for closed complete sections of the free part of the shift action of G on 2 G . Our theorems generalize known results about Z.

Borel structurability on the 2-shift of a countable group

Abstract We show that for any infinite countable group G and for any free Borel action G ↷ X there exists an equivariant class-bijective Borel map from X to the free part Free ( 2 G ) of the 2-shift G ↷ 2 G . This implies that any Borel structurability which holds for the equivalence relation generated by G ↷ Free ( 2 G ) must hold a fortiori for all equivalence relations coming from free Borel actions of G. A related consequence is that the Borel chromatic number of Free ( 2 G ) is the maximum among Borel chromatic numbers of free actions of G. This answers a question of Marks. Our construction is flexible and, using an appropriate notion of genericity, we are able to show that in fact the generic G-equivariant map to 2 G lands in the free part. As a corollary we obtain that for every ϵ > 0 , every free p.m.p. action of G has a free factor which admits a 2-piece generating partition with Shannon entropy less than ϵ. This generalizes a result of Danilenko and Park.

Group Colorings and Bernoulli Subflows

In this paper we study the dynamics of Bernoulli flows and their subflows over general countable groups from the symbolic and topological perspectives. We study free subflows (subflows in which every point has trivial stabilizer), minimal subflows, disjointness of subflows, the problem of classifying subflows up to topological conjugacy, and the differences in dynamical behavior between pairs of points which disagree on finitely many coordinates. We call a point hyper aperiodic if the closure of its orbit is a free subflow and we call it minimal if the closure of its orbit is a minimal subflow. We prove that the set of all (minimal) hyper aperiodic points is always dense but also 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 obtain a dichotomy for the complexity of classifying 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. A primary focus of the paper is to develop constructive methods for the notions studied. To construct hyper aperiodic points, a fundamental method of construction of multi-layer marker structures is developed with great generality. 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.

Burnside’s Problem, spanning trees and tilings

In this paper we study geometric versions of Burnside’s Problem and the von Neumann Conjecture. This is done by considering the notion of a translation-like action. Translation-like actions were introduced by Kevin Whyte as a geometric analogue of subgroup containment. Whyte proved a geometric version of the von Neumann Conjecture by showing that a finitely generated group is nonamenable if and only if it admits a translation-like action by any (equivalently every) nonabelian free group. We strengthen Whyte’s result by proving that this translation-like action can be chosen to be transitive when the acting free group is finitely generated. We furthermore prove that the geometric version of Burnside’s Problem holds true. That is, every finitely generated infinite group admits a translation-like action by Z. This answers a question posed by Whyte. In pursuit of these results we discover an interesting property of Cayley graphs: every finitely generated infinite group G has some locally finite Cayley graph having a regular spanning tree. This regular spanning tree can be chosen to have degree 2 (and hence be a bi-infinite Hamiltonian path) if and only if G has finitely many ends, and it can be chosen to have any degree greater than 2 if and only if G is nonamenable. We use this last result to then study tilings of groups. We define a general notion of polytilings and extend the notion of MT groups and ccc groups to the setting of polytilings. We prove that every countable group is poly-MT and every finitely generated group is poly-ccc. 20F65; 05C25, 05C63

Følner tilings for actions of amenable groups

We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G (“shapes”) with prescribed approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz–Huczek–Zhang tiling theorem for countable amenable groups and strengthens the Ornstein–Weiss Rokhlin lemma. As an application we prove that, for every countably infinite amenable group G, the crossed product of a generic free minimal action of G on the Cantor set is

Krieger's Finite Generator Theorem for Ergodic Actions of Countable Groups.

{\mathcal Z}

Arbitrarily large residual finiteness growth

Z-stable.

A subgroup formula for f-invariant entropy

A residually nilpotent group is k-parafree if all of its lower central series quotients match those of a free group of rank k. Magnus proved that k-parafree groups of rank k are themselves free. In this note we mimic this theory with finite extensions of free groups, with an emphasis on free products of the cyclic group C p , for p an odd prime. We show that for n ≤ p Magnus’ characterization holds for the n-fold free product within the class of finite-extensions of free groups. Specifically, if n ≤ p and G is a finitely generated, virtually free, residually nilpotent group having the same lower central series quotients as , then . We also show that such a characterization does not hold in the class of finitely generated groups. That is, we construct a rank 2 residually nilpotent group G that shares all its lower central series quotients with C p *C p , but is not C p *C p .