Greg Hjorth, Asger Tornquist
We show that the unitary conjugacy relation for unitary representations of a second countable locally compact group on a separable Hilbert space is a Borel equivalence relation.
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We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We construct a large family of II$_1$ factors whose Cartan subalgebras up to unitary conjugacy are not classifiable by countable structures, providing the first such examples. Additionally, we construct examples of II$_1$ factors whose Cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures. Finally, we show directly that the Cartan subalgebras of the hyperfinite II$_1$ factor up to unitary conjugacy are not classifiable by countable structures, and d…
The Conjugacy Relation on One-sided Subshifts is Non-treeable
本文研究了一位侧移映射的共轭关系，证明了其在字母表{0,1}情况下是非树形且非可换的。
In this paper, we study the descriptive set theoretic complexity of the equivalence relation of conjugacy of Toeplitz subshifts of a residually finite group $G$. On the one hand, we show that if $G = \mathbb{Z}$, then topological conjugacy on Toeplitz subshifts with separated holes is amenable. In contrast, if $G$ is non-amenable, then conjugacy of Toeplitz $G$-subshifts is a non-amenable equivalence relation. The results were motivated by a general question, asked by Gao, Jackson and Seward, about the complexity of conjugacy for minimal, free subshifts of countable groups.
We show that for any countable homogeneous ordered graph $G$, the conjugacy problem for automorphisms of $G$ is Borel complete. In fact we establish that each such $G$ satisfies a strong extension property called ABAP, which implies that the isomorphism relation on substructures of $G$ is Borel reducible to the conjugacy relation on automorphisms of $G$.
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.
The unbalanced graph of the conjugacy relation of ergodic measure-preserving transformations
本研究证明遍历保测变换的同构关系无法Borel归约到其满群自共轭作用诱导的关系，否定回答Le Maître的一个问题，支持Sabok的猜想，还证实该同构关系非平衡，正面回答Le Maître的另一问题。
Anti-classification results for conjugacy of diffeomorphisms on manifolds
该研究证明维数至少为2的流形上微分同胚拓扑共轭关系无法由可数结构分类，并给出二维环面极小微分同胚关系的$E_0$可归约性结果。
We consider the conjugacy problem for the automorphism groups of a number of countable homogeneous structures. In each case we find the precise complexity of the conjugacy relation in the sense of Borel reducibility.
Building on work of Popa, Ioana, and Epstein--Törnquist, we show that, for every nonamenable countable discrete group $Γ$, the relations of conjugacy, orbit equivalence, and von Neumann equivalence of free ergodic (or weak mixing) measure preserving actions of $Γ$ on the standard atomless probability space are not Borel, thus answering questions of Kechris. This is an optimal and definitive result, which establishes a neat dichotomy with the amenable case, since any two free ergodic actions of an amenable group on the standard atomless probability space are orbit equivalent by classical result…
In this note, we analyze the classification problem for compact metrizable $G$-ambits for a countable discrete group $G$ from the point of view of descriptive set theory. More precisely, we prove that the topological conjugacy relation on the standard Borel space of compact metrizable $G$-ambits is Borel for every countable discrete group $G$.
We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of the Hjorth's conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube hom…
We introduce a new, game-theoretic approach to anti-classification results for orbit equivalence relations. Within this framework, we give a short conceptual proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classification by orbits of CLI groups. We apply this criterion to the relation of equality of countable sets of reals, and the relations of unitary conjugacy of unitary and selfadjoint operators on the separable infinite-dimensional Hilbert space.
We decide the Borel complexity of the conjugacy problem for automorphism groups of countable homogeneous digraphs. Many of the homogeneous digraphs, as well as several other homogeneous structures, have already been addressed in previous articles. In this article we complete the program, and establish a dichotomy theorem that this complexity is either the minimum or the maximum among relations which are classifiable by countable structures. We also discuss the possibility of extending our results beyond graphs to more general classes of countable homogeneous structures.
Recursive permutations whose cycles are the classes of a decidable equivalence relation are studied; the set of these permutations is called $\mathrm{Perm}$, the group of all recursive permutations $\mathcal{G}$. Multiple equivalent computable representations of decidable equivalence relations are provided. $\mathcal{G}$-conjugacy in $\mathrm{Perm}$ is characterised by computable isomorphy of cycle equivalence relations. This result parallels the equivalence of cycle type equality and conjugacy in the full symmetric group of the natural numbers. Conditions are presented for a permutation $f \i…
On the Borel complexity of the space of left-orderings of nilpotent groups
本文首次构造非交换左序群，其序空间共轭轨道关系有无穷轨道但Borel光滑，并给出幂零群光滑性的充分条件，同时证明幂零性非充分条件。
An anti-classification theorem for minimal homeomorphisms on the torus
本文证明了二维环面上的极小同胚无法通过可数结构进行分类。
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