## Essays in group theoryEssays in Group Theorycontains five papers on topics of current interest which were presented in a seminar at MSRI, Berkeley in June, 1985. Special mention should be given to Gromov's paper, one of the most significant in the field in the last decade. It develops the theory of hyperbolic groups to include a version of small cancellation theory sufficiently powerful to recover deep results of Ol'shanskii and Rips. Each of the remaining papers, by Baumslag and Shalen, Gersten, Shalen, and Stallings contains gems. For example, the reader will delight in Stallings' explicit construction of free actions of orientable surface groups on R-trees. Gersten's paper lays the foundations for a theory of equations over groups and contains a very quick solution to conjugacy problem for a class of hyperbolic groups. Shalen's article reviews the rapidly expanding theory of group actions on R-trees and the Baumslag-Shalen article uses modular representation theory to establish properties of presentations whose relators are pth-powers. |

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### Contents

Affine Algebraic Sets and Some Infinite | 1 |

Reducible Diagrams and Equations | 15 |

Hyperbolic Groups | 75 |

Copyright | |

4 other sections not shown

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### Common terms and phrases

3-manifold 8-hyperbolic A-tree abelian group act freely algebraic argument aspherical boundary bounded called closed geodesic cocompact combinatorial 2-complex compact component conjugacy classes contains convex Corollary corresponding counting function curvature cyclic defined definition denote diagram diagrammatically reducible disk edge element equivalence relation example exists finitely generated group finitely presented fixed follows free abelian free group free product fundamental group Furthermore geo-Markov geodesic segment geometric given graph of groups hence homeomorphic homotopy hyperbolic geometry hyperbolic space incompressible surface inequality infinite integer invariant inversions IR-tree irj(M isometry group isomorphic korner Lemma length function manifold map f Markov Math metric space minimal MoSl obvious orbit oriented point xq polyhedron pregroup proof quasi-isometry quasiconvex quasigeodesic Question quotient quotient space reciprocity law result Riemannian sequence simply connected subdivision subgroup subset Theorem topological trivial unique vertex vertices word hyperbolic group word metric Z-tree