Iterated identities and iterational depth of groups

Given word on $n$ letters, we study groups which satisfiy "iterated identity" $w$, meaning that for all $x_1, \dots, x_n$ there exists $m$ such that $m$-the iteration of $w$ of Engel type, applied to $x_1, \dots, x_n$, is equal to the identity. We define bounded groups and groups which are fractal with respect to identities. This notion of being fractal can be viewed as a self-similarity conditions for the set of identities, satisfied by a group. In contrast with torsion groups and Engel groups, groups which are fractal with respect to identities appear among finitely generated elementary amenable groups. We prove that any polycyclic, as well as any metabelian group is bounded and we compute the iterational depth for various wreath products. We study the set of iterated identities, satisfied by a given group, which is not necessarily a subgroup of a free group and not necessarily invariant under conjugation, in contrast with usual identities. Finally, we discuss another notion of iterated identities of groups, which we call solvability type iterated identities, and its relation to elementary classes of varieties of groups.