6 problems

Let $\Gamma$ be a discrete group, and let $\rho_1$ and $\rho_2$ be left-invariant word metrics on $\Gamma$. Assume that … as $\gamma\to\infty$. Burago–Margulis bounded-distance con…

Let $G$ be an infinite group and let $S$ be a finite generating set. Define $\iota_S(G)$ as the involution probability associated with the word-ball exhaustion. Let…

Let $V$ be Thompson's group, and let $\left\|g\right\|$ denote word length with respect to the generating set used in the paper. A reduced diagram with $n$ leaves defines an elemen…

Let $G$ be a torsionfree nilpotent group with a finite symmetric generating set $S$. Write $\|g\|_S$ for the word metric and $\|g\|_{\mathrm{Stoll},S}$ for the Stoll metric. Breuil…

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 $G$ 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…