Is there any probabilistic characterization for generalized solvable groups?

References: This question is inspired by a conjecture of Alon Amit that is solved by Miklós Abért, Nikolay Nikolov and Dan Segal in the following papers:

They provided a probabilistic characterization for solvable groups as follows:

Theorem: A finite group $G$ is solvable if and only if there exists some positive constant $c>0$ (depending on $G$) such that, for every $r∈\mathbb{N}$ and every $w∈F_r$ (the free group on $r$ generators), the probability that $w(g_1,\cdots, g_r)=1$ is at least $c$, where the $g_i$ are uniformly, independently chosen elements of $G$.

This characterization interestingly relates the solvability of a group to the ease of solving equations in that group. (See Alon's explanations in his Quora post for more intuitive descriptions).

Now consider the notion of a hypoabelian group, that is a natural generalization of the notion of solvability by allowing the sequences of derivations of a group to be of a transfinite length. (See also Perfect Core).

Question: Is there any probabilistic characterization for hypoabelian groups similar to the above theorem for solvable groups? Is such a characterization related to the ordinal length of the sequence of derivations of the group? Do we need some extra set theoretic assumptions to get such a characterization for hypoabelian groups?

• @Rahman.M Thank you for your effort. The question is in fact just one question regarding the formulation of Alon's conjecture for hypoabelian groups instead of solvable groups. I saw that you suggested the "finite group" tag in your previous edit. Please note that the question is non-trivial in the case of "infinite groups" so this question have nothing to do with finite groups essentially. I added the "set theory" tag because it looks to me that depending on the length of the chain one may expect some set theoretic assumptions come to the story. – user82740 Nov 20 '15 at 12:51
• I immediately changed the tag, i think it's better now. – Rahman. M Nov 20 '15 at 13:24
• @Rahman.M Thanks a lot. I think your edit really improved the question. – user82740 Nov 20 '15 at 13:26