34 problems

Let $X$ be a locally compact space and let $Gamma$ act on $X$, with $C_0(X)\rtimes\Gamma$ the associated crossed-product $C^*$-algebra. Assume the action is amenable and second cou…

Let $G=\prod^*_{\lambda\in\Lambda}G_\lambda$ and let $H$ be a subgroup of $G$. In the proofs of Higgins's theorem and Kuroš's theorem, let $\tau$ denote the tree used to obtain the…

Let $SP^n(X):=X^n/S_n$ be the $n$th symmetric product of a space $X$. A topological, commutative $(m+k,m)$-group is a commutative $(m+k,m)$-group $(M,\mu)$ whose multiplication map…

Let $A_\theta$ be the irrational rotation algebra, a non-Type I C-algebra, with canonical Cartan subalgebra $C(S^1)$ and associated transformation groupoid…

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 a general higher-rank graph be given, and consider the inverse semigroup constructed from it as in Farthing, Muhly, and Yeend (2005). Let $G_\mathrm{tight}$ denote the tight gr…

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…

Hypergraph generalisation of Lemma 3. There are analogous functions $\alpha$, $\beta$, and $\gamma$ in terms of hypergraphs such that the first sum tends to zero as $n$ tends to in…

Let $\mathcal M(2)$ act on the boundary $\partial T$ of the rooted tree $T$, and let $\mathcal G(\partial T,\mathcal M(2),\Phi)$ be the associated germinal groupoid. Hausdorffness…

Associativity conjecture. The structure $(A,*,^{-1})$ is an associative invertible magmoid. After adjoining the necessary units, $(A,*,^{-1})$ forms a groupoid.

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

Let $\bigl(\text{groupoids},\text{anafunctors}\bigr)$ denote the proposed left derivator whose objects are groupoids and whose morphisms are anafunctors. Let $\mathbb{Set}$,…

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 a locally compact Hausdorff étale groupoid. A unit $u\in\mathcal{G}^{(0)}$ is global fixed if every $\gamma\in\mathcal{G}$ with $s(\gamma)=u$ also satisfies…

Groupoid realization conjecture. The universal Hopf coalgebroid $kG^{par}$ is exactly the Hopf coalgebroid associated to the groupoid $\Gamma(G)$.

An orbifold is a space locally modelled on quotients $\tilde{U}/G$, where $G$ is a finite group acting on an open set $\tilde{U}\subseteq\mathbb{R}^n$. It is representable by a tra…

Ornstein–Weiss quasitiling conjecture. If $G$ is a topologically amenable groupoid, then tiles $T_1,\ldots,T_n$ satisfying the conditions of the corresponding quasitiling theorem s…

Closed-tower conjecture. The diameter condition in the definition of almost finiteness should also be achievable with closed towers in the groupoid setting.

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…

Let $\mathfrak G$ be a finite groupoid acting affinely on a vector space $V$, and let $\Phi$ denote the affine action. Suppose that all orbits of the action are finite, and let…