8 problems

Non-local exponentiality conjecture. The Lie group $C^\infty(M,{\mathbb{R}}) \rtimes_\alpha {\mathbb{R}}$ is not locally exponential. More specifically, there is a sequence…

Let $\pi$ be a discrete group with an action by a group $\Gamma$, let $\pi\rtimes\Gamma$ be the associated semidirect product, and let $\mathcal{F}$ and $\mathcal{G}$ denote the fa…

Let $\mathcal{X}$ and $\mathcal{C}$ be left closed monoidal categories, meaning that tensoring on the left has a right adjoint. Let there be a strong action of $\mathcal{X}$ on…

Let $\mathcal{PODI}_n$ be the monoid of partial order-preserving or order-reversing permutations, let $\mathcal{POI}_n$ be the monoid of partial order-preserving permutations, and…

Let $\Gamma$ be a finitely generated group, and let $\rho$ be a linear representation of $\Gamma$ on $\mathbf{R}^n$. Equip $\mathbf{R}^n\rtimes_\rho\Gamma$ and…

Let $G\cong \mathbb{Z}/m$ be a finite cyclic group, let $L\cong\mathbb{Z}^n$ be a finitely generated free abelian group, let $\rho\colon G\to\operatorname{aut}_{\mathbb{Z}}(L)$ be…

Let $M$ be a square matrix of size at least $m$ with nonzero determinant, let $k<m$, and write $\Gamma_M=\mathbb{Z}^m\rtimes_M\mathbb{Z}$. The subgroup $\mathbb{Z}^m$ is $k$-volume…

Let $M\in GL(k,\mathbb{Z})$ and let $k\geq 3$. Consider the semidirect product $\mathbb{Z}^k\rtimes_M\mathbb{Z}$, where $M$ acts on $\mathbb{Z}^k$. The subgroup $\mathbb{Z}^k$ is a…