Cohomology of hyperfinite Borel actions

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.

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.

Locally Nilpotent Groups and Hyperfinite Equivalence Relations

A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we show that this question has a positive answer when the acting group is locally nilpotent. This extends previous results obtained by Gao-Jackson for abelian groups and by Jackson-Kechris-Louveau for finitely generated nilpotent-by-finite groups. Our proof is based on a mixture of coarse geometric properties of locally nilpotent groups together with an adaptation of the Gao-Jackson machine…

Classification of regular subalgebras of the hyperfinite II_1 factor

We prove that the regular von Neumann subalgebras $B$ of the hyperfinite II_1 factor $R$ satisfying the condition $B'\cap R=Z(B)$ are completely classified (up to conjugacy by an automorphism of $R$) by the associated discrete measured groupoid $G$. We obtain a similar classification result for triple inclusions $A\subset B \subset R$, where $A$ is a Cartan subalgebra in $R$ and the intermediate von Neumann algebra $B$ is regular in $R$. A key step in proving these results is to show the vanishing cohomology for the associated cocycle actions of $G$ on $B$. We in fact prove two very general va…

有限生成群中阿基米德序的同构关系的Borel复杂度

Borel Complexity of the Isomorphism Relation of Archimedean Orders in Finitely Generated Groups

本文证明了有限生成群中阿基米德序的同构关系的Borel复杂度，具体回答了Calderoni等人提出的问题，表明在$n\geq3$时非超有限，在$n\geq4$时非可树化。

Edge sliding and ergodic hyperfinite decomposition

We use edge slidings and saturated disjoint Borel families to give a conceptually simple proof of Hjorth's theorem on cost attained: if a countable p.m.p. ergodic equivalence relation $E$ is treeable and has cost $n \in \mathbb{N} \cup \{\infty\}$ then it is induced by an a.e. free p.m.p. action of the free group $\mathbb{F}_n$ on $n$ generators. More importantly, our techniques give a significant strengthening of this theorem: the action of $\mathbb{F}_n$ can be arranged so that each of the $n$ generators alone acts ergodically. The existence of an ergodic action for the first generator immed…

The complexity of the topological conjugacy problem for Toeplitz subshifts

In this paper, we analyze the Borel complexity of the topological conjugacy relation on Toeplitz subshifts. More specifically, we prove that topological conjugacy of Toeplitz subshifts with separated holes is hyperfinite. Indeed, we show that the topological conjugacy relation is hyperfinite on a larger class of Toeplitz subshifts which we call Toeplitz subshifts with growing blocks. This result provides a partial answer to a question asked by Sabok and Tsankov.

Measurable perfect matchings for acyclic locally countable Borel graphs

We characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree at least three generating $μ$-hyperfinite equivalence relations admit $μ$-measurable matchings. We establish the analogous result for Baire measurable matchings in the locally finite case, and provide a c…

Abelian group actions and hypersmooth equivalence relations

We show that a Borel action of a standard Borel group which is isomorphic to a sum of a countable abelian group with a countable sum of real lines and circles induces an orbit equivalence relation which is hypersmooth, i.e., Borel reducible to eventual agreement on sequences of reals, and it follows from this result along with the structure theory for locally compact abelian groups that Borel actions of Polish LCA groups induce orbit equivalence relations which are essentially hyperfinite, extending a result of Gao and Jackson and answering a question of Ding and Gao.

Borel图的分解与上同调

Decomposition of Borel graphs and cohomology

本文通过上同调准则研究Borel图的分解，并证明具有均匀有界度数的Borel图与无循环图Lipschitz等价，从而为树准等距Borel图的结果提供新证明。

交换幺半群作用的渐近维数

Asymptotic dimension of commutative monoid actions

本文证明有限生成交换幺半群的有界到一作用的渐近维数不超过其群完备化的无挠秩，并利用经典与Borel渐近维数的相等性，得到精确的Borel秩界及上超有限性。

可数Borel等价关系的实现

Realizations of countable Borel equivalence relations

本文研究可数Borel等价关系的拓扑实现，涵盖连续群作用、极小子移位及普适空间中的性质刻画，并探讨Turing等价等特殊关系的实现问题。

Wasserstein稳定性与奇异Borel提升问题

Wasserstein stability and the nonsingular Borel lifting problem

本文通过引入Wasserstein稳定性，证明了大多数标准Lévy群的奇异Borel作用只有平凡不动点，从而否定了它们存在非平凡奇异Borel作用的问题。