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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:

[1] Baumslag, G., Gersten, S.M., Shapiro, M., Short, H.: Automatic groups and amalgams. (in preparation) · Zbl 0749.20006
[2] Borel, A., Harder, G.: Existence of discrete cocompact subgroups of reductive groups over local fields. J. Reine Angew. Math.298, 53-64 (1978) · Zbl 0385.14014
[3] Borel, A., Serre, J.-P.: Cohomologie d’immeubles et des groupesS-arithmetiques. Topology15, 211-232 (1976) · Zbl 0338.20055 · doi:10.1016/0040-9383(76)90037-9
[4] Brown, K.S.: Buildings. Berlin-Heidelberg-New York: Springer 1988
[5] Bruhat, F., Tits, J.: Groupes r?ductifs sur un corps local I. Donn?es radicielles valu?es. Publ. Math. Inst. Hautes Etud. Sci.41, 5-251 (1972) · Zbl 0254.14017 · doi:10.1007/BF02715544
[6] Cannon, J.W.: The combinatorial structure of cocompact discrete hyperbolic groups. Geom. Dedicata16, 123-148 (1984) · Zbl 0606.57003 · doi:10.1007/BF00146825
[7] Cannon, J.W.: Negatively Curved Spaces and Groups, Preliminary lecture notes from Topical meeting on hyperbolic geometry and ergodic theory. Trieste (April 1989)
[8] Cannon, J.W., Epstein, D.B.A., Holt, D.F., Paterson, M.S., Thurston, W.P.: Word processing and group theory. University of Warwick, 1990 (Preprint)
[9] Harpe, P. de la, Valette, A.: La propri?t? (T) de Kazhdan pour les groupes localement compacts. Ast?risgue175 (1989) · Zbl 0759.22001
[10] El-Mosalamy, M., Pride, S.: OnT(6) groups. Math. Proc. Camb. Philos. Soc.102, 443-451 (1987) · Zbl 0654.20032 · doi:10.1017/S0305004100067499
[11] Gersten, S.M.: Reducible diagrams and equations over groups. In: Gersten, S.M. (ed.) Essays in Group Theory (M.S.R.I. series, Vol. 8, pp. 15-73). Berlin-Heidelberg-New York: Springer 1987 · Zbl 0644.20024
[12] Gersten, S.M.: Dehn functions andl 1-norms of finite presentations. In: Proceedings of the Workshop on algorithmic Problems C.F. M?ller III and G. Baumslag (eds.), Springer-Verlag MSRI series, 1990 (to appear)
[13] Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in Group Theory (M.S.R.I. series, Vol. 8, pp. 75-263). Berlin-Heidelberg-New York: Springer 1987 · Zbl 0634.20015
[14] Hopcroft, J.E., Ullman, J.D.: Formal languages and their relation to automata. Reading MA: Addison-Wesley 1969 · Zbl 0196.01701
[15] Lyndon, R.C.: On Dehn’s algorithm. Math. Ann.166, 208-228 (1966) · Zbl 0138.25702 · doi:10.1007/BF01361168
[16] Lyndon, R.C., Schupp, P.E.: Combinatorial group theory. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0368.20023
[17] Pride, S.: Some finitely presented groups of cohomological dimension two with property (FA). J. Pure Appl. Algebra29, 167-168 (1983) · Zbl 0513.20019 · doi:10.1016/0022-4049(83)90105-6
[18] Thurston, W.P.: Geometry and topology of 3-manifolds. Preprint Princeton University · Zbl 0483.57007
[19] Thurston, W.P.: Oral communication
[20] 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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