Artin groups, rewriting systems and three-manifolds

Abstract We construct finite complete rewriting systems for two large classes of Artin groups: those of finite type, and those whose defining graphs are based on trees. The constructions in the two cases are quite different; while the construction for Artin groups of finite type uses normal forms introduced through work on complex hyperplane arrangements, the rewriting systems for Artin groups based on trees are constructed via three-manifold topology. This construction naturally leads to the question: Which Artin groups are three-manifold groups? Although we do not have a complete solution, the answer, it seems, is “not many”.

[1]  A. Brunner.  Geometric quotients of link groups , 1992 .

[2]  P. Deligne,et al.  Les immeubles des groupes de tresses généralisés , 1972 .

[3]  G. P. Scott.  Finitely Generated 3‐Manifold Groups are Finitely Presented , 1973 .

[4]  S. M. Gersten,et al.  Small cancellation theory and automatic groups: Part II , 1991 .

[5]  Susan Hermiller,et al.  Tame combings, almost convexity and rewriting systems for groups , 1997 .

[6]  Craig C. Squier,et al.  Word problems and a homological niteness condition for monoids , 1987 .

[7]  Egbert Brieskorn,et al.  Artin-Gruppen und Coxeter-Gruppen , 1972 .

[9]  Nellie Clarke Brown.  Trees , 1896, Savage Dreams.

[10]  H. van der Lek,et al.  The homotopy type of complex hyperplane complements , 1983 .

[11]  Leonard Van Wyk.  Graph groups are biautomatic , 1994 .

[12]  W. Jaco.  Finitely presented subgroups of three-manifold groups , 1971 .

[13]  Morris W. Hirsch,et al.  Topology of 3-Manifolds and Related Topics. , 1964 .

[14]  Ruth Charney,et al.  The (,1)-problem for hyperplane complements associated to infinite reflection groups , 1995 .

[15]  B. H. Neumann,et al.  A Remark on Generalized Free Products , 1950 .

[16]  David Peifer.  Artin groups of extra-large type are biautomatic , 1996 .

[17]  S. Hermiller.  Rewriting systems for Coxeter groups , 1994 .

[18]  David B. A. Epstein,et al.  Word processing in groups , 1992 .

[19]  Ruth Charney,et al.  Geodesic automation and growth functions for Artin groups of finite type , 1995 .

[20]  Carl Droms,et al.  Graph groups, coherence, and three-manifolds , 1987 .

[21]  S. Gersten,et al.  Small cancellation theory and automatic groups , 1990 .

[22]  S. J. Pride.  On Tits' conjecture and other questions concerning Artin and generalized Artin groups , 1986 .

[23]  Susan Hermiller,et al.  Algorithms and Geometry for Graph Products of Groups , 1995 .

[24]  Geometric Invariants for Artin Groups , 1997 .

[25]  Leonard Evens,et al.  Cohomology of groups , 1991, Oxford mathematical monographs.

[26]  Deepak Kapur,et al.  An Overview of Rewrite Rule Laboratory (RRL) , 1989, RTA.

[27]  Ruth Charney,et al.  Artin groups of finite type are biautomatic , 1992 .

[28]  Daniel E. Cohen.  Geometric Group Theory: String Rewriting — A Survey for Group Theorists , 1993 .