Contents

1. Introduction

2. Golod-Shafarevich inequality
   Definition

3. Golod-Shafarevich groups
   Definition

4. Generalized Golod-Shafarevich groups
   Definition

6. Quotients of generalized Golod-Shafarevich groups

7. Free subgroups in generalized Golod-Shafarevich pro p groups
   Definition

9. Groups of positive power p-deficiency
   Definition

10. Applications in number theory

11. Applications in geometry and topology