5 problems

Let $n\geq 3$ and $m$ be a positive integer, and let $\Gamma_{n,m}$ denote the principal congruence subgroup of $\mathrm{SL}(n,\mathbb{Z})$ of level $m$. Lubotzky's conjecture. The…

A group is isolated if it is an isolated point in the space of marked groups, and a subgroup is non-isolated if it is not isolated there. Finite-index subgroup conjecture. There ex…

Let $B_n$ be the braid group on $n$ strands and let $K_n=[B_n,B_n]$ be its commutator subgroup. A subgroup has index $r$ if its coset action gives a transitive representation…

Let $F$ denote Thompson group $F$, and let a group be 2-generated if it has a generating set with two elements. The finite-index generation conjecture. Every finite-index subgroup…

F-A characterization conjecture. $G$ is F-A if and only if