16.73 (2006)

Let $G$ be a group generated by a finite set $S$, and let $l(g)$ denote the word length function of $g \in G$ with respect to $S$. The group $G$ is said to be contracting if there exist a faithful action of $G$ on the set $X^*$ of finite words over a finite alphabet $X$ and constants $0 < \lambda < 1$ and $C > 0$ such that for every $g \in G$ and $x \in X$ there exist $h \in G$ and $y \in X$ such that $l(h) < \lambda l(g) + C$ and $g(xw) = yh(w)$ for all $w \in X^*$.
$\qquad$ a) Can a contracting group have a non-abelian free subgroup?
$\qquad$ b) Do there exist non-amenable contracting groups?