## Algebraic Generalizations of Discrete Groups: A Path to Combinatorial Group Theory Through One-Relator ProductsA survey of one-relator products of cyclics or groups with a single defining relation, extending the algebraic study of Fuchsian groups to the more general context of one-relator products and related group theoretical considerations. It provides a self-contained account of certain natural generalizations of discrete groups. |

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### Contents

Preface | 8 |

OneRelator Groups | 34 |

Discrete Groups | 77 |

OneRelator Products | 111 |

OneRelator Products of Cyclics | 137 |

Linear Properties of OneRelator Products | 159 |

Groups of FType | 221 |

Related Generalizations of Discrete Groups | 253 |

291 | |

309 | |

### Common terms and phrases

abelian algebraic assume conjugate contains a non-abelian cyclic group cyclically pinched one-relator cyclically reduced word discrete groups epimorphism equation essential representation Euler characteristic factor group faithful representation free group free product Freiheitssatz Fuchsian group Further G contains G is virtually geometric group G group of F-type group theory hence HNN group hyperbolic groups infinite cyclic infinite order integer isometries isomorphic Lemma length at least Let G locally indicable Magnus NEC groups Nielsen transformation non-abelian free subgroup non-elementary non-trivial free product normal closure normal subgroup one-relator products pinched one-relator groups polynomial product of cyclics product on a1 product with amalgamation proof of Theorem proper power quotient representation into PSL2(C residually finite result Riemann-Hurwitz formula solvable subgroup of finite subgroup of G subgroup of PSL2(C subgroup of rank Suppose G surface group syllable length tetrahedron group Tits alternative tr(AB triangle group vertex virtually torsion-free zero