15 problems

Let $G$ be a Kac–Moody group over an algebraically closed field of characteristic $0$. An automorphism of $G$ may be inner, diagonal, graph, field, or sign, where these are the sta…

Supplement conjecture. There exists a subgroup $\Omega\leq\widetilde{\Sigma}_S(W)$ such that, with $\Pi:=\Omega\cap\operatorname{Ref}_S(W)$, one has $\Pi\leq\Gamma_S(W)$ and…

Sharp-angled automorphism conjecture. For every Coxeter generating set $R\subseteq S^W$, there exists an automorphism $\alpha$ of $W$ such that $\alpha(S^W)=S^W$ and $\alpha(R)$ is…

Let $\mathcal P_{\mathrm{fin},0}(\mathbb Z)$ be the monoid of finite subsets of the additive group $\mathbb Z$ containing $0$, with setwise addition. Tringali–Wen's conjecture. The…

Let $G$ be a $\det$-finite group, meaning that the relevant Fuglede–Kadison determinants are finite, and let $\Phi\colon G\to G$ be an automorphism. Assume that $G$ has subexponent…

Bounded Fitting-height subgroup conjecture. $G$ has a characteristic subgroup $H$ such that:

Semicompleteness characterization. The graph product $G_\Gamma$ is semicomplete if and only if $\Gamma$ has no separating star.

Let $G$ be a virtually cocompact special group. An acylindrically hyperbolic automorphism-group conjecture. If $G$ is acylindrically hyperbolic, then its automorphism group … is ac…

Let $N$ be a normal subgroup of the extended mapping class group $\operatorname{MCG}(S_g)$, and let $\hat g(N)$ denote the invariant used in the source. Genus-bound conjecture. If…

Let $S$ be a surface whose genus is finite and at least $4$. The groups $\operatorname{PMap}(S)$, $\operatorname{Map}(S)$, and $\operatorname{Map}^\pm(S)$ denote the pure mapping c…

Completeness conjecture. For every austere graph $\Gamma$, the group $\operatorname{Aut}(A_\Gamma)$ is complete.

Let $G=\operatorname{UT}(n,F)$ be a group of upper unitriangular matrices over a field $F$, where $n\in\mathbb N\cup\{\infty\}$, and let $e$ denote the identity matrix. A bijection…

The Euler totient–automorphism conjecture. One should have

Let $G$ be a finitely generated group such that $h_{alg}(id_G)=\infty$, equivalently, $G$ has exponential growth. Infinite algebraic entropy conjecture. Then $h_{alg}(\phi)=\infty$…

Let $A_\Gamma$ be a right-angled Artin group, and let $\Gamma$ be austere, meaning that its defining graph has no nontrivial graph symmetries, no dominated vertices, and no separat…