КОМБИНАТОРНАЯ ТЕОРИЯ ГРУПП В ИВАНОВСКОМ ГОСУДАРСТВЕННОМ УНИВЕРСИТЕТЕ

Outlines of the history of researches on the Combinatorial Group Theory in the Ivanovo State University and an overview of the results obtained from 60-s of the last century up to the present. The results that are presented concern mainly to the study of property of residual finiteness of groups and of its various generalizations as applied to free constructions of groups and to the one-relator groups. Изложение истории развития исследований по комбинаторной теории групп в Ивановском государственном университете и обзор результатов, полученных с 60-х годов прошлого столетия по настоящее время. Представляемые результаты относятся, главным образом, к изучению свойства финитной аппроксимируемости групп и его различных обобщений приме- нительно к свободным конструкциям групп и к группам, определяемым одним соотношением.

Free subgroups of one-relator relative presentations

Suppose that G is a nontrivial torsion-free group and w is a word over the alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\}. It is proved that for n\ge2 the group \~G= always contains a nonabelian free subgroup. For n=1 the question about the existence of nonabelian free subgroups in \~G is answered completely in the unimodular case (i.e., when the exponent sum of x_1 in w is one). Some generalisations of these results are discussed.Comment: V3: A small correction in the last phrase of the proof of Theorem 1. 4 page

Conjugacy Separability of Some One-Relator Groups

Conjugacy separability of any group of the class of one-relator groups given by the presentation ⟨,;[,]=1⟩(,>1) is proven. The proof made used of theoretical combinatorial group methods, namely the structure of amalgamated free products and some properties of the subgroups and quotients of any group of the class of one-relator groups given above