11 problems

Let $A_\Gamma$ be an Artin group and let $\chi:A_\Gamma\to\mathbb{R}$ be a character. Let $\mathrm{Liv}^\chi$ denote the living subgraph associated to $\chi$, and let…

Let $G$ be a finitely generated metabelian group, so its commutator subgroup $G'$ is Abelian, and let $S(G)$ denote the character sphere. For each $n$, write $\Sigma^n(G)^c$ for th…

For $n\geq 3$, let $\mathrm{PB}_n$ be the pure braid group and let $Z(\mathrm{PB}_n)$ be its center. The quotient $\mathrm{PB}_n/Z(\mathrm{PB}_n)$ has character spheres and BNSR-in…

Let $F_n$ be the free group of rank $n$ with a fixed basis, and let $\mathrm{PSAut}_n$ be the group of automorphisms sending each basis element to a conjugate of itself. For…

Let $A_\Gamma$ be an Artin group satisfying the $K(\pi,1)$-conjecture, and let $\chi:A_\Gamma\to\mathbb{R}$ be a character. Write $\mathrm{Liv}^\chi$ for the living subgraph, and c…

Let $A_\Gamma$ be an Artin group and let $\chi:A_\Gamma\to\mathbb{R}$ be a character. The living subgraph $\mathrm{Liv}^\chi\subset\Gamma$ is obtained by deleting vertices $v$ with…

BNSR-invariant conjecture. For any $S$ and $r$,

Conjecture I. Under this identification,

Bieri's $\Sigma^m$-conjecture. Every metabelian group of type $\operatorname{F}_m$ has the Bieri $\Sigma^m$-property. This would give a polyhedral description of the BNSR-invariant…

Let $n\geq 2$, let $P\!\operatorname{\Sigma Aut}_n$ be the pure symmetric automorphism group of the free group of rank $n$, and let $S(P\!\operatorname{\Sigma Aut}_n)$ denote its c…

Let $m\in\mathbb{N}\cup\{\infty\}$, let $\psi$ be the character considered in the surrounding discussion, and let the ambient group be one of the Lodha--Moore groups $G$, $G_y$,…