12 problems

Let a closed graph manifold be decomposed along incompressible tori into Seifert fibered JSJ components. Properties $(1a)$, $(1b)$, and $(2)$ denote, respectively, the existence of…

Let $M$ be a closed, oriented, irreducible 3-manifold with … A graph manifold is a 3-manifold assembled from Seifert-fibered pieces. Conjecture II. $M$ is a graph manifold satisfyi…

Let $M$ be an orientable graph manifold with infinite fundamental group. A circularly orderable manifold is one whose fundamental group admits a circular order; write $CO$ when thi…

Let $W$ be a rational homology sphere graph manifold. Circular orderability conjecture. If $\pi_1(W)$ is infinite, then $\pi_1(W)$ is circularly orderable. The conjecture isolates…

Let $W$ be a graph manifold. Circular orderability conjecture. If $\pi_1(W)$ is infinite, then $\pi_1(W)$ is circularly orderable. This conjecture proposes extending the techniques…

Let $N$ and $M$ be two closed graph manifolds. Let $S \looparrowright N$ and $S' \looparrowright M$ be two separable, horizontal surfaces. Quasi-isometry conjecture. There is a qua…

A CN2 manifold is a Riemannian manifold whose curvature operator has nullity of codimension at most two. Let $M$ be a compact CN2 manifold with finite volume and scalar curvature t…

Let $M^n$ be a complete CN2 manifold with finite volume, let $V$ be its set of nonflat points, and let an extension of $V$ be an open set carrying a complete flat totally geodesic…

Let $M$ be a non-geometric graph manifold, and let $G=\pi_1M$ be its fundamental group. A group is virtually compact special if it has a finite-index subgroup that is the fundament…

Let $W$ be a rational homology $3$-sphere graph manifold with JSJ tori, and let $m$ be the number of these tori. For each $i$, let $[alpha_i]$ be a rational slope associated to the…

Let $M$ be a closed graph manifold and let $\widetilde{M}$ denote its universal cover. The Assouad–Nagata dimension conjecture. The Assouad–Nagata dimension of $\widetilde{M}$ equa…

Let $M$ be a graph manifold, let $Z$ be a free piece of its torus decomposition, and let a singular orbit be homotopic into $Z$. Boundary homotopy conjecture. The singular orbit mu…