20 problems

Sabok's conjecture. The conjugacy relation of minimal compact systems is a complete orbit equivalence relation.

Let $F$ be a Borel equivalence relation, let $\mathbb E_1$ denote eventual agreement on sequences of reals, and let $E_G^X$ denote the orbit equivalence relation of a group action.…

Let $\mathcal{F}(M)$ and $\mathcal{F}(\mathcal{R})$ denote the fundamental groups of a $\mathrm{II}_1$ factor and an orbit equivalence relation, respectively. For a free ergodic pr…

Hjorth's reduction conjecture. Under these hypotheses, every such analytic orbit quotient can be represented as the orbit quotient of an analytic subset of a Polish $S_{\infty}$-sp…

Let $\phi^t$ be a pseudo-Anosov flow on a closed oriented $3$-manifold, let $c$ be a positive/negative horizontal surgery curve, and let $n$ be a positive/negative integer, respect…

Let $\phi_1^t$ and $\phi_2^t$ be transitive Anosov flows whose stable and unstable line bundles are orientable. Fried–Ghys conjecture. The flows $\phi_1^t$ and $\phi_2^t$ are almos…

Let $(X_{\mathfrak{G}},\sigma_{\mathfrak{G}})$ be a free $\mathbb{Z}^2$-odometer, and let $\mathbf{S}:\mathbb{Z}^2\curvearrowright X$ be a minimal speedup of…

Extension conjecture. There exists a smooth equivalence $\xi: \Omega_{1} \to \Omega_{2}$ that extends $\zeta$. This would establish the possibility of extending orbit equivalences…

Let a countable amenable group act continuously, minimally, and freely on the Cantor set. A Cantor minimal system is a minimal action of $\mathbb{Z}$ on the Cantor set. Giordano–Pu…

Spectral gap conjecture. The action $Gamma\curvearrowright G$ has spectral gap.

Let $\mathscr{G}_{\mathrm{dsc}}$ denote the class of all discrete countable groups. A group $G$ is Bernoulli $\mathscr{G}_{\mathrm{dsc}}$-superrigid if every Bernoulli action of…

Orbit-equivalence versus idealistic-equivalence conjecture. Orbit equivalence relations coincide with the idealistic equivalence relations.

The analytic non-orbit-equivalence conjecture. The equivalence $E_1$ is not the least analytic equivalence relation which is not Borel reducible to an orbit equivalence.

Consider an ergodic probability-measure-preserving action $G\curvearrowright (X,\mu)$ whose induced orbit-equivalence relation $E_G^X$ is not $\mu$-hyperfinite. Measurable von Neum…

Orbit-relation conjecture. Every $G$-orbit equivalence relation is Borel reducible to

Thomas's conjecture. The resulting orbit equivalence relations are pairwise incomparable with respect to Borel reducibility.

Thomas's conjecture. These orbit equivalence relations are neither Borel reducible to one another nor stably orbit equivalent.

Let $G$ be a transitive permutation group, and let $G^{*}$ be its orbit closure, the largest permutation group having the same orbits on subsets as $G$. The source notes that if…

Let $G\trianglelefteq H$ be transitive permutation groups acting on a finite set $\Omega$, and suppose that $G$ and $H$ are orbit equivalent, meaning that they have the same orbits…

Let two flows be related by an orbit-equivalence transformation, with the time reparameterization and invariant measures as in the paper. The invariance conjecture for information…