11 problems

Let $G$ be a solvable Lie group. A complete left-invariant affine structure on $G$ is a left-invariant flat and torsion-free affine connection whose associated affine structure is…

Let $G$ be a simply connected, unimodular, solvable Lie group with a left-invariant complex structure $J$. Hasegawa's Steinness conjecture. The complex manifold $(G,J)$ is Stein. T…

Let $G\cong \prod_{i=1}^n \mathbf{N}_i \rtimes \mathbb{R}^k$ be nondegenerate with $n\ge 2$. Let $q:G\to G$ be a quasi-isometry for some, and hence any, left-invariant Riemannian m…

Correction to the volume-entropy monotonicity conjecture. The volume entropy of $\mathbb{R}^{2}\rtimes_A\mathbb{R}$ was conjectured to be a monotonically decreasing function of…

Let $G$ and $H$ be completely solvable Lie groups, meaning closed subgroups of the upper triangular real matrix group. They are quasiisometric if their metric spaces are quasiisome…

Let $G$ be a simply connected unimodular solvable Lie group equipped with a left invariant complex structure. A complex manifold is Stein if it is biholomorphic to a closed complex…

Let $S_1$ and $S_2$ be simply connected solvable Lie groups of real type, meaning that all eigenvalues of $\operatorname{ad}(X)$ are real for every $X$ in their Lie algebras. Cornu…

Let $S$ be a SOL-like group. Suppose $\operatorname{Isom}(S,g)$ does not admit a uniform lattice for any left invariant Riemannian metric $g$ on $S$. The non-lattice conjecture. Th…

A solvmanifold is a quotient $G/\Gamma$ of a solvable Lie group $G$ by a discrete subgroup $\Gamma$. A Lie group is at most 3-step solvable when its derived series reaches the triv…

Let $G$ be the solvable Lie group under consideration, and let $\Lambda_{k,i}$ denote the family of lattices in $G$ constructed above. Lattice classification conjecture. Every latt…

Let $G$ be a $2n$-dimensional simply connected unimodular solvable Lie group, or a nilpotent Lie group, equipped with a left-invariant complex structure. Stein conjecture. The resu…