13 problems

Let $T$ be a finite simple group, and let $\operatorname{Out}(T)$ denote its group of outer automorphisms. Schreier's conjecture. The group $\operatorname{Out}(T)$ is solvable. Thi…

Quantitative dynamics conjectures. First, if $\phi$ is fully irreducible with expansion factor $\lambda_\phi>1$, then

Generic uniqueness conjecture. For a reasonable random walk on $\operatorname{Out}(F_r)$, having that $\mathcal{A}_\varphi$ has a unique artery is generic.

Finite-index conjecture. There exists a family $\mathcal{K}$ of cyclic subgroups of $A_\Gamma$ such that $\mathcal{U}(A_\Gamma;\{H\})$ is a subgroup of finite index in…

Let $\mathbb G=\mathbb F\rtimes_\phi\mathbb Z$ and $\mathbb G'=\mathbb F'\rtimes_\psi\mathbb Z$ be free-by-cyclic groups, where an outer automorphism is fully irreducible if it has…

Merging rarity conjecture. Merging outer automorphisms are rare, and most fully singular train track representatives have a proper full fold decomposition.

Existence and genericity conjecture. Every ageometric fully irreducible outer automorphism has a fully singular train track representative, and these representatives generically ha…

Let $F$ be a finitely generated free group, let $Gamma=F\rtimes_{alpha}langle trangle$ be a unipotent polynomial suspension, and let $\operatorname{Aut}(Gamma,F)$ denote the automo…

Let $F$ be a finitely generated free group, let $Gamma=F\rtimes_{alpha}langle trangle$ be a unipotent polynomial suspension, and let $Q$ be a finite quotient of $Gamma$. Minkowski…

Let $F_n$ be a free group of rank $n$, let $Gamma=F_n\rtimes_{alpha}langle trangle$ be a unipotent polynomial suspension, and let $F_m$ be a free group of rank $m$. Suppose that…

Thurston's conjecture. Every pair of weak Perron numbers greater than $1$ is a pair of expansion constants for an outer automorphism and its inverse.

Let $G$ be a special group. A DLS automorphism is one of the automorphisms appearing in Theorems and; those appearing in Theorem are the relevant ones for…

Garibaldi–Petersson conjecture. The Tits class obstruction is the only obstruction: every automorphism of $\Delta$ that leaves $t_G$ invariant lifts to an outer automorphism of $G$…