2 problems

Let $w=[x,y]=x^{-1}y^{-1}xy$ be the commutator word, and for a group $G$ let $w(G)$ be its verbal set. The word $w$ has width $N$ in $G$ if the verbal subgroup generated by $w(G)$…

Let $S$ be a finite set of places, let $G$ be a semisimple group over the corresponding local factors, let … be an irreducible lattice in … and let $w\in F_n$ be a word in the free…