AT-groups that are not AT-subgroups: Transition from $AT_{\omega}$-groups to $AT_{\Omega}$-groups

Periodic nonlocally finite (Burnside) groups of infinite period are studied. The first explicit example of such groups was proposed by S.V. Aleshin in 1972. His construction was generalized to AT-groups, i.e., tree automorphism groups. A number of known problems have been solved with the help of AT-groups. Up to now, in reality, only the class of $AT_{\omega}$-groups, i.e., the class of AT-groups over a sequence of cyclic groups of prime order, has been studied. In this paper, the class of $AT_{\Omega}$-groups, i.e., of AT-groups over a sequence of cyclic groups of arbitrary finite order, is studied. The difference between $AT_{\omega}$-groups and true $AT_{\Omega}$-groups was revealed by the solution of the Kourovka Problem 16.79. The study of the class of $AT_{\Omega}$-groups has allowed us to introduce a number of new notions. Now the $AT_{\omega}$-groups can be considered as elementary AT-groups by which the AT-groups over a sequence of periodic groups are saturated. We propose a strategy for studying such AT-groups and give promising directions of this kind of research.