Combinatorial methods and algorithmic questions in group theory

We are planning to describe all amenable and non-amenable subgroup as well as all amenable normal subgroups of each n-torsion group for odd exponents n≥1003, to find out whether the inner automorphism group Inn(T) for an arbitrary relatively free n-torsion group T is a characteristic subgroup of automorphism group Aut(T), to investigate normal automorphisms of relatively free n-torsion groups and to find out if they are inner. We also plan to construct an algorithm which allows us to embed a given group from a large class of recursively presented groups into some finitely presented group (existence of such embedding follows from Higman's well-known theorem), to describe new classes of groups A_1,A_2 and B_1,B_2, for which from the equalities var (A_1) = var (A_2) and var (B_1) = var (B_2) follows the equality var (A_1 Wr B_1) = var (A_2 Wr B_2).