M.Sc Thesis

M.Sc StudentGamliel Avihu Meir
SubjectEquations over Groups with Finite Order Solution
DepartmentDepartment of Mathematics
Supervisor ASSOCIATE PROFESSOR Arie Juhasz
Full Thesis textFull thesis text - English Version


This work is devoted to a study of certain aspects of the known problem:

Given a group G, and an equation r(t) in G*<T>\G. Under what circumstances does r(t) has a solution over G?

Although the answer is known for a large families of groups, much less is known when r(t) is a singular equation (i.e. the sum of the t-exponents in r(t) equals 0).

In this work we will focus on singular equations over groups. Mostly, this problem wasn't studied because the existing methods couldn't been applied, and there is no general conjecture or method for the solvability of singular equations.

One of the main subjects of this research is to study finite order solution of equations over groups (i.e. simultaneously solution of the pair of equations r(t) and tm). There is a clear advantage for equations with finite order solution. Assume that r(t), tm has a solution over G. Considering every power m of t where there is a finite order solution to this pair of equations, we may have infinite number of solutions to the equation r(t) = 1. Moreover, we can distinguish two equations by their finite order solution (if exist).

In the first part of this work (chapter 1) we are considering singular equations over torsion free groups. Following A.Klyachko's method we solve families of singular equations, which are non-amenable equations.

The second part of this research (chapters 2 and 3) is devoted to the study of finite order solutions of equations over groups. Using bounded HNN extensions and small cancellation theory, we reduce the question whether there exist a finite order solution of an equation r(t) over G, to the question whether G embeds naturally in a bounded HNN-extension.