Computational group theory and automorphisms of relatively free groups.

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MY RESEARCH PROPOSAL IS TO SEEK SOLUTIONS to the following fundamental questions about automorphisms. (1) Describe automorphisms of free solvable groups. This question is due to Mal'cev, Kargapolov, Remeslennikov (unsolved problems in The Kourovka Note Book). This question will require finding a solution: whether or not a given n x n derivation matrix (n greater than or equal to 3) is invertible over certain free solvable group ring? (2) Primitivity is another aspect of automorphisms. An element in F is primitive if it can be included in a basis of F. Our main question is: given a word w, can we effectively construct a sequence of automorphisms which will map w to x_1, that is, w -> (w)phi_1 -> (w)phi_1 phi_2 -> ... -> (w)phi_1 phi_2 ... phi_k = x_1. Here, I propose to find a criterion for a system of m elements to be primitive in F/V (for example, V = [F'',F'']) for certain relatively free groups. (3) Algorithmic problem: The problem of automorphic reducibility is decidable for a group G if there exists an algorithm which decides the following question that for each pair of elements g,h of G: is there an automorphism phi in Aut(G) such as (g)phi = h? [This is an analogue of the famous Whitehead problem for free groups (Whitehead (1936), Rappaport (1952), Lyndon, Higgens (1974)). In 1997, as a first step Timoshenko and I proved that the problem of automorphic reducibility is decidable for a free metabelian group of rank 2. I propose to find necessary and sufficient conditions for which the above property holds for certain relatively free groups. (4) Algorithmic problem: A test element in a group G is an element g which is fixed by an endomorphism phi of G, then phi must be an automorphism. I propose to find an algorithm to decide: whether or not a given word w is a test element in certain relatively free groups; I will also seek description of test elements. NEXT I WILL CONTINUE MY RESEARCH ON THE FOLLOWING TOPICS: (i) Is the universal theory of partially commutative metabelian groups decidable?; (ii) finite basis problem: just non-Specht varieties of groups and group representations; (iii) identification problem of certain n x n triangular matrices, n greater than or equal to 3 (analogous to a well-known Magnus embedding (1932) for n greater than or equal to 2); (iv) find new automorphisms of the algebra of two generic 2 x 2 matrices. In (1997, 2000, 2005), Drensky and I constructed new automorphisms of 2 x 2 generic matrices.