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Andrew S. Marks

Publications and source records attributed to Andrew S. Marks.

9 recordsLinked to original sources
arXiv2026

Borel dimension growth and hyperfiniteness

We show that every increasing union of Borel graphs of pointwise volume growth at most $\exp(O(r^γ))$ is hyperfinite, where $γ\approx 0.1523$ is the unique real root of $(1 - γ)^3 - 4γ= 0$. This implies a positive answer to the special case of Weiss's question on the hyperfiniteness of Borel actions of countable amenable groups, for countable amenable groups locally of volume growth $\exp(O(r^γ))$ for $γ$ as above. We also show that every bounded degree Borel graph of subexponential volume growth has Borel Følner tilings, improving a result of Downarowicz and Zhang for graphs generated by free Borel actions of groups of subexponential volume growth. Our main tools are the development of a Borel analogue of the two-parameter dimension growth function of a metric space as introduced by Dranishnikov and Sapir, and ball carving algorithms from theoretical computer science. We end with a pair of conjectures relating Borel amenability, Borel subexponential dimension growth, and hyperfiniteness, which would imply a positive answer to Weiss's question.

math.GR↗
arXiv2026

One-ended spanning subforests and treeability of groups

We show that several new classes of groups are measure strongly treeable. In particular, finitely generated groups admitting planar Cayley graphs, elementarily free groups, and the group of isometries of the hyperbolic plane and all its closed subgroups. This provides the first examples of one-ended nonamenable groups which are measure strongly treeable. In higher dimensions, we also prove a dichotomy that the fundamental group of a closed aspherical 3-manifold is either amenable or has strong ergodic dimension 2. Our main technical tool is a method for finding measurable treeings of Borel planar graphs by constructing one-ended spanning subforests in their planar dual. Our techniques for constructing one-ended spanning subforests also give a complete classification of the locally finite pmp graphs which admit Borel a.e. one-ended spanning subforests.

math.GR↗
arXiv2020

Measurable realizations of abstract systems of congruences

An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and $n$-divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the $2$-sphere. This answers a question of Wagon. We also construct Borel realizations of abstract systems of congruences for the action of $\mathsf{PSL}_2(\mathbb{Z})$ on $\mathsf{P}^1(\mathbb{R})$. The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings.

math.LO↗
arXiv2017

Borel Circle Squaring

We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If $k \geq 1$ and $A, B \subseteq \mathbb{R}^k$ are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than $k$, then $A$ and $B$ are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of $\mathbb{Z}^d$.

math.LO↗
arXiv2016

Topological generators for full groups of hyperfinite pmp equivalence relations

We give an elementary proof that there are two topological generators for the full group of every aperiodic hyperfinite probability measure preserving Borel equivalence relation. Our proof explicitly constructs topological generators for the orbit equivalence relation of the irrational rotation of the circle, and then appeals to Dye's theorem and a Baire category argument to conclude the general case.

math.DS↗
arXiv2016

Brooks's theorem for measurable colorings

We generalize Brooks's theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d \geq 3$ which contains no $(d+1)$-cliques, then $G$ admits a $μ$-measurable $d$-coloring with respect to any Borel probability measure $μ$ on $X$, and a Baire measurable $d$-coloring with respect to any compatible Polish topology on $X$. The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of IID $d$-colorings of Cayley graphs of degree $d$, except in two exceptional cases.

math.LO↗
arXiv2016

The universality of polynomial time Turing equivalence

We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees.

math.LO↗
arXiv2016

Jump operations for Borel graphs

We investigate the class of bipartite Borel graphs organized by the order of Borel homomorphism. We show that this class is unbounded by finding a jump operator for Borel graphs analogous to a jump operator of Louveau for Borel equivalence relations. The proof relies on a non-separation result for iterated Frechet ideals and filters due to Debs and Saint Raymond. We give a new proof of this fact using effective descriptive set theory. We also investigate an analogue of the Friedman-Stanley jump for Borel graphs. This analogue does not yield a jump operator for bipartite Borel graphs. However, we use it to answer a question of Kechris and Marks by showing that there is a Borel graph with no Borel homomorphism to a locally countable Borel graph, but each of whose connected components has a countable Borel coloring.

math.LO↗