15.19 (2002)

Let $p$ be a prime, and $\mathcal{F}_p$ the class of finitely generated groups acting faithfully on a $p$-regular rooted tree by finite automata. Any group in $\mathcal{F}_p$ is residually-$p$ (residually in the class of finite $p$-groups) and has word problem that is solvable in (at worst) exponential time. There exist therefore groups that are residually-$p$, have a solvable word problem, and do not belong to $\mathcal{F}_p$; though no concrete example is known. For instance:
$\qquad$ a) Is it true that some (or even all) the groups given in (R. I. Grigorchuk, Math. USSR–Sb., 54 (1986), 185–205) do not belong to $\mathcal{F}_p$ when the sequence $\omega$ is computable, but not periodic?
$\qquad$ b) Does $\mathbb{Z} \wr (\mathbb{Z} \wr \mathbb{Z})$ belong to $\mathcal{F}_2$? (Here the wreath products are restricted.)
See (A. M. Brunner, S. Sidki, J. Algebra, 257 (2002), 51–64) and (R. I. Grigorchuk, V. V. Nekrashevich, V. I. Sushchanskiĭ, in: Dynamical systems, automata, and infinite groups. Proc. Steklov Inst. Math., 231 (2000), 128–203).