Charney, Ruth Geodesic automation and growth functions for Artin groups of finite type. (English) Zbl 0813.20042 Math. Ann. 301, No. 2, 307-324 (1995). For an Artin group (or generalized braid group) associated to a finite Coxeter group, we exhibit an automatic structure whose language is symmetric, geodesic and one-to-one. Using this automatic structure, we show how to explicitly compute rational growth functions for these groups. Reviewer: R.Charney (Columbus) Cited in 2 ReviewsCited in 43 Documents MSC: 20F36 Braid groups; Artin groups 20F65 Geometric group theory 20F05 Generators, relations, and presentations of groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20F55 Reflection and Coxeter groups (group-theoretic aspects) Keywords:Artin group; generalized braid group; finite Coxeter group; automatic structure; rational growth functions PDF BibTeX XML Cite \textit{R. Charney}, Math. Ann. 301, No. 2, 307--324 (1995; Zbl 0813.20042) Full Text: DOI EuDML References: [1] K.S Brown, Buildings. Springer 1989 [2] R. Charney, Artin groups of finite type are biautomatic. Math Ann.292 (1992), 671-683 · Zbl 0782.57001 · doi:10.1007/BF01444642 [3] P. Deligne, Les immeubles des groupes de tresses généralises. Invent. Math.17 (1972), 273-302 · Zbl 0238.20034 · doi:10.1007/BF01406236 [4] D. Epstein, J. Cannon, D. Holt, S. Levy, M. Paterson, W. Thurston, Word Processing in Groups. Jones and Bartlett 1992 · Zbl 0764.20017 [5] F.A. Garside, The braid groups and other groups. Oxford Q. J. Math.20 (1969), 235-254 · Zbl 0194.03303 · doi:10.1093/qmath/20.1.235 [6] M. Gromov, Hyperbolic groups, in Essays in Group Theory. ed. by S.M. Gersten, MSRI Publ. 8, Springer 1987 · Zbl 0634.20015 [7] S.M. Gersten, H. Short, Small cancellation theory and automatic groups. Invent. Math.102 (1990), 305-334 · Zbl 0714.20016 · doi:10.1007/BF01233430 [8] S.M. Gersten, H. Short, Small cancellation theory and automatic groups, Part II. Invent. Math.105 (1991), 641-662 · Zbl 0734.20014 · doi:10.1007/BF01232283 [9] S.M. Gersten, H. Short, Rational subgroups of biautomatic groups. Ann. Math.134 (1991), 125-158 · Zbl 0744.20035 · doi:10.2307/2944334 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.