13 problems

Let $P$ be a metric space, and let $\big((M_x),\,(\mu_{xy})\big)$ be an approximate action of the pair groupoid $P\times P$, with each $M_x$ a complete metric space and $\mu_{xy}$…

Let $\Sigma$ be a connected oriented marked surface other than a sphere with $4$ punctures or a projective plane with $2$ punctures, and let ${\bf F}:{\bf TSurf}^t\to{\bf Grp}$ be…

Let $\mathcal{G}$ be a Hausdorff étale groupoid. Weak amenability means that $\Lambda_{\text{cb}}(\mathcal{G})$ is finite, and the condition $\Lambda_{\text{cb}}(\mathcal{G})=1$ is…

Let $K$ be a field, let $\mathcal{G}$ be an ample groupoid, and let $u\in\mathcal{G}^{(0)}$. Write $A_K(\mathcal{G})$ for the Steinberg algebra, $\mathcal{G}_u$ for the isotropy gr…

Rational HK-conjecture. One should have

HK-realization conjecture. There should exist a possibly different groupoid $\widetilde{\mathcal{G}}$ such that

The paper considers an absolute free cocompletion of a point constructed from cubical sets, simplicial sets, or semisimplicial sets, together with three reflective localizations fo…

Cantor-action complexity conjecture. If $\Gamma$ has finite decomposition complexity, then $X\rtimes\Gamma$ has finite dynamical complexity for every free action of $\Gamma$ on the…

Equivalence conjecture. The categories $AC_A$ and $SFCG_A$ are equivalent. Moreover, this equivalence can be proved by extending the functor $C: Iso(SFCG_A) \rightarrow Iso(AC_A)$…

Let $\mathcal{G}$ be an effective minimal second countable Hausdorff ample groupoid whose unit space $\mathcal{G}^{(0)}$ is a Cantor space. Write $H_0(\mathcal{G})$ and…

Countability and continuity conjecture. For every finite groupoid $H$ and finite-dimensional commutative $H$-algebra $A$, $KK^H(A,B)$ is countable for all $B\in KK^H$ and commutes…

Translation-groupoid representation conjecture. Every orbifold groupoid can be represented by a translation groupoid with locally free group action.

The completion theorem. The completed twisted ${\mathcal G}$-equivariant $K$-theory of $X$ is isomorphic to the twisted equivariant $K$-theory after forming the homotopy quotient b…