16.74 (2006)

a) Let $G = \langle \alpha, \beta \rangle$ be the group generated by the following two permutations of $\mathbb{Z}$: $\alpha(n) = n + 1$; $\beta(0) = 0$, $\beta(2^k m) = 2^k(m + 2)$, where $m$ is odd and $k$ is a positive integer. Is $G$ amenable?
b) Is it true that all groups generated by automata of polynomial growth in the sense of S. Sidki (Geom. Dedicata, 108 (2004), 193–204) are amenable?