9 problems

Let $F_k$ be a free group of rank $k$, and let $G$ be a group. A subgroup $N\leq F_k$ with $F_k/N\cong G$ is a $G$-defining subgroup; write $\Sigma_k(G)$ for the set of all such su…

Let $G=\operatorname{SL}_2(\mathbb{F}_p)$, and let $(A,B)$ and $(A',B')$ be generating pairs of $G$. The Higman invariant of $(A,B)$ is the extended conjugacy class of the commutat…

Let $G$ be a finite group, let $\mu(G)$ denote the minimum number of generators of $G$, and let $n$ be an integer with $n>\mu(G)$. A generating $n$-vector of $G$ is an $n$-tuple of…

Let $q$ be a prime power and let an essential triple $(x,y,z)\in\mathbb{F}_q^3$ be the image under $(A,B)\mapsto(\operatorname{tr}A,\operatorname{tr}B,\operatorname{tr}(AB))$ of a…

Wiegold's free-factor conjecture. If $n\geq 3$, then $F_n$ has a proper free factor $H$ such that $\Psi(H)=G$.

One-stabilization conjecture. The tuples $(t_1,\ldots,t_n,1)$ and $(s_1,\ldots,s_n,1)$ are connected in $\bar X_{n+1}(G)$.

Kapovich–Weidmann higher-stabilization conjecture. For every $1\le t<m$, the generating $(k+t)$-tuple $(a_1,\ldots,a_k,1,\ldots,1)$ is not equivalent to a $(k+t)$-tuple of type…

Kapovich–Weidmann stabilization conjecture. In the above construction, at least $k$ stabilizations are needed to make the tuples $(a_1,\ldots,a_k)$ and $(b_1,\ldots,b_k)$ Nielsen e…

Let $G$ be a finite simple group with $d(G)=2$, where $d(G)$ is the minimum size of a generating set of $G$. For $k\geq 1$, a $T_k$-system is an orbit of generating $k$-tuples unde…