3 problems

Let $D_n$ be a dihedral group, let $S$ be a generating set, and let $l_S(g)$ denote the length of an element $g$ with respect to $S$. For an integer $m$ with $0<m<n$, consider the…

Let $D_n$ be a dihedral group with an order $n$ cyclic subgroup, and let $S$ be a generating set composed of three involutions, none of which belongs to that cyclic subgroup. Write…

Let $G$ be a simply connected nilpotent Lie group and let $\Gamma$ be a discrete co-compact subgroup. For a finite symmetric generating set $S\subset\Gamma$, let $\rho_S$ be the as…