Contents

Chapter 1. Introduction 1

1.1. Preliminary remarks 1

1.2. Main results 3

Chapter 2. Relative isoperimetric inequalities 9

2.1. Relative presentations and length functions 9

2.2. Geometry of van Kampen diagrams over relative presentations 11

2.3. Relative Dehn functions 18

2.4. Splitting Theorem for relatively finitely presented groups 25

2.5. Isoperimetric functions of Cayley graphs. 30

Chapter 3. Geometry of finitely generated relatively hyperbolic groups 35

3.1. Conventions and notation 35

3.2. Properties of quasi-geodesics 36

3.3. Geodesic triangles in Cayley graphs 45

3.4. Symmetric geodesies 51

Chapter 4. Algebraic properties 61

4.1. Elements of finite order 61

4.2. Relatively quasi-convex subgroups 63

4.3. Cyclic subgroups and translation numbers 70

Chapter 5. Algorithmic problems 77

5.1. The word and membership problems 77

5.2. The parabolicity problems 79

5.3. Algorithmic problems for hyperbolic elements 82

Open questions 89

Appendix. Equivalent definitions of relative hyperbolicity 91

Bibliography 97