M.Sc Thesis | |

M.Sc Student | Gamliel Avihu Meir |
---|---|

Subject | Equations over Groups with Finite Order Solution |

Department | Department of Mathematics |

Supervisor | ASSOCIATE PROFESSOR Arie Juhasz |

Full Thesis text |

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 t ^{m}*).
There is a clear advantage for equations with finite order solution. Assume
that

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.