arXiv subjects

Forte Shinko

Publications and source records attributed to Forte Shinko.

12 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 $\exp(O(r^\gamma))$ is hyperfinite, where $\gamma \approx 0.1523$ is the unique real root of $(1 - \gamma)^3 - 4\gamma = 0$. 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 $\exp(O(r^\gamma))$ for $\gamma$ as above. We also show that every bounded degree Borel graph of subexponential volume growth has Borel F{\o}lner 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

Hyperfiniteness of bounded-to-one actions of commutative monoids

A theorem of Dougherty--Jackson--Kechris states that any equivalence relation generated by a single Borel function is hypersmooth. A well-known open problem is whether this can be generalized to equivalence relations generated by countable families of pairwise commuting Borel functions. We give an affirmative answer in the case where the functions are bounded-to-one. This generalizes the theorem of Gao--Jackson on Borel actions of countable abelian groups.

math.LO↗

arXiv2024

Hyper-hyperfiniteness and complexity

We show that if there exists a countable Borel equivalence relation which is hyper-hyperfinite but not hyperfinite then the complexity of hyperfinite countable Borel equivalence relations is as high as possible, namely, $\Sigma^1_2$-complete.

math.LO↗

arXiv2024

Asymptotic dimension and hyperfiniteness of generic Cantor actions

We show that for a countable discrete group which is locally of finite asymptotic dimension, the generic continuous action on Cantor space has hyperfinite orbit equivalence relation. In particular, this holds for free groups, answering a question of Frisch-Kechris-Shinko-Vidny\'anszky.

math.LO↗

arXiv2023

Hyperfiniteness for group actions on trees

We identify natural conditions for a countable group acting on a countable tree which imply that the orbit equivalence relation of the induced action on the Gromov boundary is Borel hyperfinite. Examples of this condition include acylindrical actions. We also identify a natural weakening of the aforementioned conditions that implies measure hyperfinitenss of the boundary action. We then document examples of group actions on trees whose boundary action is not hyperfinite.

math.GR↗

arXiv2022

Lifts of Borel actions on quotient spaces

Given a countable Borel equivalence relation E and a countable group G, we study the problem of when a Borel action of G on X/E can be lifted to a Borel action of G on X.

math.LO↗

arXiv2021

A dichotomy for Polish modules

Let $R$ be a ring equipped with a proper norm. We show that under suitable conditions on $R$, there is a natural basis under continuous linear injection for the set of Polish $R$-modules which are not countably generated. When $R$ is a division ring, this basis can be taken to be a singleton.

math.LO↗

arXiv2021

Realizations of countable Borel equivalence relations

We study topological realizations of countable Borel equivalence relations, including realizations by continuous actions of countable groups, with additional desirable properties. Some examples include minimal realizations on any perfect Polish space, realizations as $K_\sigma$ relations, and realizations by continuous actions on the Baire space. We also consider questions related to realizations of specific important equivalence relations, like Turing and arithmetical equivalence. We focus in particular on the problem of realization by continuous actions on compact spaces and more specifically subshifts. This leads to the study of properties of subshifts, including universality of minimal subshifts, and a characterization of amenability of a countable group in terms of subshifts. Moreover we consider a natural universal space for actions and equivalence relations and study the descriptive and topological properties in this universal space of various properties, like, e.g., compressibility, amenability or hyperfiniteness.

math.LO↗

arXiv2020

Quotients by countable subgroups are hyperfinite

We show that for any Polish group $G$ and any countable normal subgroup $Γ\triangleleft G$, the coset equivalence relation $G/Γ$ is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite.

math.GR↗

arXiv2020

Equidecomposition in cardinal algebras

Let $Γ$ be a countable group. A classical theorem of Thorisson states that if $X$ is a standard Borel $Γ$-space and $μ$ and $ν$ are Borel probability measures on $X$ which agree on every $Γ$-invariant subset, then $μ$ and $ν$ are equidecomposable, i.e. there are Borel measures $(μ_γ)_{γ\inΓ}$ on $X$ such that $μ= \sum_γμ_γ$ and $ν= \sum_γγμ_γ$. We establish a generalization of this result to cardinal algebras.

math.LO↗

arXiv2017

Hyperfiniteness of boundary actions of cubulated hyperbolic groups

We show that if a hyperbolic group acts geometrically on a CAT(0) cube complex, then the induced boundary action is hyperfinite. This means that for a cubulated hyperbolic group the natural action on its Gromov boundary is hyperfinite, which generalizes an old result of Dougherty, Jackson and Kechris for the free group case.

math.GR↗

arXiv2014

Differential polynomial rings over rings satisfying a polynomial identity

Let $R$ be a ring satisfying a polynomial identity and let $δ$ be a derivation of $R$. We show that if $N$ is the nil radical of $R$ then $δ(N)\subseteq N$ and the Jacobson radical of $R[x;δ]$ is equal to $N[x;δ]$. As a consequence, we have that if $R$ is locally nilpotent then $R[x;δ]$ is locally nilpotent. This affirmatively answers a question of Smoktunowicz and Ziembowski.

math.RA↗