## Date of Defense

16-4-2024 2:30 PM

## Document Type

Thesis Defense

## Degree Name

Master of Science in Mathematics

## College

College of Science

## Department

Mathematical Sciences

## First Advisor

Dr. Alexandr Zubkov

## Keywords

free group, equivalence classes, Mennicke subgroup, congruence subgroup, principal congruence subgroup.

## Abstract

This thesis is concerned with the matrix representation of a free non-abelian group by matrices of size β₯ 3. We proceed from defining an equivalence classes and then transitioning to free groups. We discuss in details the group πΊπ(π) which is the group generated by the matrices filled with first, (second, etc.) column, except for the intersection with the diagonal, and we have ones on the diagonal and zeros at the other places. The filled places are occupied by the same parameter π. An alternative proof for the known fact that πΊπ(3) is not free is provided. The main objective of this thesis is to find a lower bound for the parameter. An explicit value of the lower bound is found which is a refinement of a previous lower bound.

#### Included in

FAITHFUL REPRESENTATION OF FREE GROUPS AND CONGRUENT SUBGROUPS OF ππΏ3 (π)

This thesis is concerned with the matrix representation of a free non-abelian group by matrices of size β₯ 3. We proceed from defining an equivalence classes and then transitioning to free groups. We discuss in details the group πΊπ(π) which is the group generated by the matrices filled with first, (second, etc.) column, except for the intersection with the diagonal, and we have ones on the diagonal and zeros at the other places. The filled places are occupied by the same parameter π. An alternative proof for the known fact that πΊπ(3) is not free is provided. The main objective of this thesis is to find a lower bound for the parameter. An explicit value of the lower bound is found which is a refinement of a previous lower bound.