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Small cancellation theory and automatic groups. (English) Zbl 0714.20016

Let G be a group with generating set X, and let \(\Gamma\) be the Cayley graph of G with respect to X. We can regard \(\Gamma\) as a metric space by giving each edge unit length. We can then consider imposing conditions on this metric space. The most well-known example of this is the hyperbolicity condition imposed by M. Gromov [in: Essays in Group Theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)]. Another condition is the automaticity condition. This requires a constant \(k>0\) and a path in the Cayley graph 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 \(X\cup X^{-1}\). As the authors show, hyperbolic groups are automatic.
The main aim of this wide-ranging paper is to show that if G is given by a finite presentation satisfying the small cancellation conditions C(p), T(q) \(((p,q)=(6,3),(4,4),(3,6))\), and if all pieces have length 1 and no relator is a proper power, then G is automatic. In addition, the authors give some new examples of groups satisfying the C(3), T(6) conditions (for other examples see M. El-Mosalamy and S. J. Pride [Math. Proc. Camb. Philos. Soc. 102, 443-451 (1987; Zbl 0654.20032)]; M. Edjvet and J. Howie [Proc. Lond. Math. Soc., III. Ser. 57, 301-328 (1988; Zbl 0627.20020)]; J. Howie [Forum Math. 1, 251-272 (1989; Zbl 0676.20018)]). These examples are groups of isometries of certain Bruhat-Tits buildings. The groups are of cohomological dimension 2 (being torsion-free small cancellation groups), and neither they nor their subgroups of finite index can act on a tree without fixing a point (this follows from the fact that the groups have Kazhdan’s property T). (For other examples of finitely presented groups of cohomological dimension 2 with no non-trivial action on a tree, see S. J. Pride [J. Pure Appl. Algebra 29, 167-168 (1983; Zbl 0513.20019)].)
In an appendix to the paper the authors give a proof of the “if” part of the following characterization of hyperbolic groups due to Gromov: A group is hyperbolic if and only if it is finitely presented and satisfies a linear isoperimetric inequality.
Reviewer: S.J.Pride

MSC:

20F06 Cancellation theory of groups; application of van Kampen diagrams
20F05 Generators, relations, and presentations of groups
20E08 Groups acting on trees
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References:

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