Leonov Y. Growth and representation of self-similar groups

Thesis is devoted to study self-similar groups of isometries of regular trees. The theory of triangular representations for self-similar groups is developed for the first time. The analitic theory of self-similar groups is developed: the new methods of growth estimation is obtained. The new ways of investigations of self-similar groups is opened in this work: the language of triangular matrices and the language of labelled polynomials. Also, the language of Kaloujnine's tablices was developed in the case of algebraic wreath powers for cyclic groups of the prime order. In particular, the faithful trianfular representations of the Sylow -subgroup symmetric group (Kaloujnine groups) is build and studied. Triangular representations of groups of isometries of the regular trees , - prime number, and triangular representations of self-similar groups is build and sdudied. The new methods of embedding of arbitrary wreath powers of cyclic groups of prime order to the Kaloujnine groups is obtained. The problem of estimation of growth in the class of self-similar groups is investigated. Sufficient conditions for subexponential growth of self-similar groups is obtained. The estimation of growth from bellow for self-similar groups is developed. This method is applied to the first Grirorchuk group, well-known Gupta-Sidki -groups and Gupta group. Reidemeister number for residually finite groups is studied. We prove that this number is infinite for the first Grigorchuk group. The closure of the first Grigorchuk group in topology of isometries of binary tree is described. The new finite generated self-similar group with the same closure is constructed.