Word Hyperbolic Semigroups

A.J. Duncan and R.H. Gilman


Math. Proc. Camb. Phil. Soc., 136 (2004) no. 3, 513--524.


The study of automatic and word hyperbolic groups is a prominent topic in geometric group theory. The definition of automatic groups extends naturally to semigroups but word hyperbolic groups are defined by a geometric condition, the thin triangle condition, whose extension is not immediate. The problem is that for semigroups the metric properties of the Cayley diagram are not as closely tied to the algebraic structure as they are in the case of groups. We propose a language--theoretic definition of word hyperbolic semigroup which for groups is equivalent to the original definition. We also consider briefly word problems of semigroups. The standard word problem of a group is the language of all words representing the identity. As it stands this definition makes no sense for semigroups, but there is a variation which does.

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Andrew Duncan
28th April 2005