14.27 (1999)

Let $\Gamma$ be a group generated by a finite set $S$. Assume that there exists a nested sequence $F_1 \subset F_2 \subset \cdots$ of finite subsets of $\Gamma$ such that
$\qquad$ (i) $F_k \neq F_{k+1}$ for all $k \geqslant 1$,
$\qquad$ (ii) $\Gamma = \bigcup_{k \geqslant 1} F_k$,
$\qquad$ (iii) $\lim_{k \to \infty} |\partial F_k|/ |F_k| = 0$, where, by definition, $\partial F_k = \{ \gamma \in \Gamma \setminus F_k \mid$ there exists $s \in S$ such that $\gamma s \in F_k \}$, and
$\qquad$ (iv) there exist constants $c \geqslant 0$, $d \geqslant 1$ such that $|F_k| \leqslant ck^d$ for all $k \geqslant 1$.
Does it follow that $\Gamma$ has polynomial growth?