8 problems

Let $\mathbf{B}_2(\mathbb{Z}[\sqrt{2}])$ be the indicated polycyclic group, which is a discrete cocompact subgroup of the three-dimensional solvable Lie group $\operatorname{Sol}$.…

Bounded packing conjecture for virtually polycyclic groups. Each subgroup of $P$ has bounded packing in $P$.

Let $M\in\operatorname{SL}(n,\mathbf{Z})$ have no eigenvalues on the unit circle, and let $G$ be a finitely generated group quasi-isometric to the associated group $\Gamma_M$. The…

Polycyclic cyclic escape conjecture. Every infinite finitely generated polycyclic group has the cyclic escape property.

Eskin–Fisher–Whyte's conjecture. The class of virtually polycyclic groups should be quasiisometrically complete within finitely generated groups.

Finite-Krull-dimension conjecture. The following assertions hold:

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…

For a polycyclic group $G$, consider algorithms for deciding whether two words represent conjugate elements, with input word length and the Hirsch length of $G$ as complexity param…