Small cancellation theory and automatic groups. II.

*(English)*Zbl 0734.20014A group G is said to be automatic if the Cayley graph \(\Gamma\) with respect to some generating set X (where \(\Gamma\) is regarded as a metric space by giving each edge unit length) has the following property: there exists a constant \(k>0\) and a path in \(\Gamma\) for each \(g\in G\) (starting at 1 and ending at g) such that paths which end a distance 1 apart stay within a distance k of each other, and such that the words defined by the paths constitute a regular language in the free monoid on \(X\cup X^{-1}\). In their previous paper [Part I, ibid. 102, 305-334 (1990; Zbl 0714.20016)] the authors showed that the fundamental group of a piecewise Euclidean 2-complex of nonnegative curvature of type \(A_ 1\times A_ 1\) or \(A_ 2\) is automatic. \((A_ 1\times A_ 1\) corresponds to the Euclidean planar tesselation by unit squares, and \(A_ 2\) to the tesselation by equilateral triangles.) In the present paper the authors prove an analogous result for 2-complexes of types \(B_ 2\) and \(G_ 2\) corresponding to the Euclidean tesselations by (\(\pi\) /2,\(\pi\) /4,\(\pi\) /4) and (\(\pi\) /2,\(\pi\) /3,\(\pi\) /6) triangles respectively. They give some applications of their theorem to discrete groups of isometries of certain Euclidean buildings, and to connected sums of alternating links. In their previous paper the authors showed that a group given by a finite presentation satisfying the small cancellation conditions C(p), T(q) \(((p,q)=(6,3),(4,4),(3,6))\) is automatic, under the additional assumptions that all pieces have length 1 and no relator is a proper power. In the present paper the authors show that these additional assumptions can be dropped. Finally, the authors introduce the notion of a biautomatic group, and they show that such groups have solvable conjugacy problem.

Reviewer: S.J.Pride (Glasgow)

##### MSC:

20F06 | Cancellation theory of groups; application of van Kampen diagrams |

20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |

57M07 | Topological methods in group theory |

20F05 | Generators, relations, and presentations of groups |

20E08 | Groups acting on trees |

##### Keywords:

automatic groups; Cayley graph; generating set; words; regular language; free monoid; fundamental group; Euclidean 2-complex; Euclidean tesselations; finite presentation; small cancellation conditions; biautomatic group; solvable conjugacy problem##### References:

[1] | [B] Brown, K.: Buildings. Berlin Heidelberg New York: Springer 1988 |

[2] | [BGSS] Baumslag, G., Gersten, S.M., Shapiro, M., Short, H.: Automatic groups and amalgams. J. Pure Appl. Algebra (to appear) · Zbl 0749.20006 |

[3] | [BT] Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. I. Données radicales valuées. Publ. Math., Inst. Hautes Étud. Sci.41 5-251 (1972) · Zbl 0254.14017 · doi:10.1007/BF02715544 |

[4] | [CEHPT] Cannon, J.W., Epstein, D.B.A., Holt, D.F., Patterson, M. S., Thurston, W.P.: Word processing and group theory. (University of Warwick preprint Jan. 1990) |

[5] | [G] Gromov, M.: Hyperbolic groups. In: Essays in group theory Gersten, S.M. (ed.) M.S.R.I series, vol. 8, pp. 75-263, Berlin Heidelberg New York: Springer 1987 |

[6] | [GS] Gersten, S.M., Short, H.: Small cancellation theory and automatic groups. Invent. Math.102, 305-334 (1990) · Zbl 0714.20016 · doi:10.1007/BF01233430 |

[7] | [GS2] Gersten, S.M., Short, H.: Rational subgroups of biautomatic groups. (to appear) Ann. Math. · Zbl 0744.20035 |

[8] | [HU] Hopcroft, J.E., Ullman, J.D.: Introduction to automata theory, languages and computation. Reading, MA: Addison-Wesley 1979 · Zbl 0426.68001 |

[9] | [LS] Lyndon, R.C., Schupp, P.E.: Combinatorial group theory, Berlin Heidelberg New York: Springer 1977 · Zbl 0368.20023 |

[10] | [W] Weinbaum, C.M.: The word and conjugacy problem for the knot group of any prime alternating knot. Proc. Am. Math. Soc.22, 22-26 (1971) · Zbl 0228.55004 · doi:10.1090/S0002-9939-1971-0279169-X |

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