Solução do problema da conjugação para algumas extensões de grupos

Abstract

This essay is a detailed introductory study of Combinatorial Group Theory and one of its three classical problems: the Conjugacy Problem. We studied its solution for several classes of group extensions, obtained in [1] and [2]. Among the results, we point out its solution for free-by-cyclic groups and for some free extensions of the form Znx^Z, or Z2xA1,...,AmFm, or F2X\Vi,..,VmFm (with A e GL,n(Z), Ax,...,Am e GL2(Z) and p1,...,pm e Aut(F2)), as well as ZnxAl,...,AmFm with (A1,..., Am) < GLn(Z) a finite index or virtually solvable subgroup, or, finally, for groups of the type Fnx^1,...,^mFm, with (p1,...,<£m) < Aut(Fn) a finite index subgroup. We also studied solutions of the Twisted Conjugacy Problem for finitely generated free, polycyclic and surface groups. In the course of the text, one tries to argue in a simple and detailed way, providing the reader a first contact with Combinatorial Group Theory and a certain level of familiarity with its decision problems and with the vastly used concept of algorithm.