11 problems

Let $\Gamma$ be a complex crystallographic group acting on $E_\mathbb{C}^n$, with linear part a finite complex reflection group. The orbit space $E_\mathbb{C}^n/\Gamma$ is consider…

Seven-neighbor conjecture. The upper bound for $pgg$ is seven even without requiring that the union of the two orbits be contained in an orbit of the normalizer of $G_0$ in the aff…

Let $\Gamma$ be a crystallographic $r$-group, let $T$ be its subgroup of translations, and set $A=\mathbb{C}^n/T$. Then $\mathbb{C}^n/\Gamma=A/K$, where $K$ is the induced finite l…

Let $X\subset\mathbb R^3$ be a Delone set with parameter $R$. For each point $x\in X$, let $S_x(2R)$ be its local group, and let $K$ be the subset of points whose local groups cont…

Let $N$ be a connected and simply connected nilpotent Lie group, and let $\Gamma\subseteq\operatorname{Aff}(N)$ act properly discontinuously and cocompactly on $N$; such an action…

Let $\mathbb{R}^{2,1}$ be Minkowski space, and let a discrete group act properly discontinuously and cocompactly on it. The Auslander conjecture. The group is solvable up to finite…

Petrosyan's spectral-sequence collapse conjecture. This spectral sequence should collapse at the $E_2$ page for every such coefficient module $A$, not only for integral coefficient…

Let $G$ be a finite cyclic group and $L$ a finitely generated $\mathbb{Z}G$-lattice. For the split crystallographic group $L\rtimes G$, consider its integral cohomology and the coh…

Let $\Gamma<\operatorname{Isom}(\mathbb{R}^k)$ be a torsion-free crystallographic group. Its quotient $\mathbb{R}^k/\Gamma$ is a closed aspherical manifold, and…

Let $G$ be a finite cyclic group and let $L$ be a finitely generated $\mathbb ZG$-lattice. For the semidirect product extension … consider its Lyndon–Hochschild–Serre spectral sequ…

Collapse conjecture. For $G$ any cyclic group, the spectral sequence associated to a semidirect product of the form $\mathbb Z^n\rtimes G$ should collapse at $E_2$. The conjecture…