A GEOMETRIC CHARACTERIZATION OF AUTOMATIC MONOIDS by PEDRO V. SILVA (Centro de Matem´ atica, Faculdade de Ciˆ encias, Universidade do Porto, R. Campo Alegre, 687 4169-007 Porto, Portugal) and BENJAMIN STEINBERG (School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada) [Received 9 May 2001. Revised 21 December 2003] Abstract It is well known that automatic groups can be characterized using geometric properties of their Cayley graphs. Along the same line of thought, we provide a geometric characterization of automatic monoids. This involves working with a slightly strengthened definition of an automatic monoid which is still a proper generalization of the concept of an automatic group. The two definitions coincide in the case of right cancellative monoids for which a particularly simple characterization is obtained. 1. Introduction Automatic groups were introduced in [7] with great impact in both combinatorial group theory and automata theory. The general idea was to define a (large) class of groups combining two different approaches: automata-theoretic (the structure of such a group should be fully encoded by finitely many finite state automata) and geometric (the geometry of the Cayley graph should play a decisive role in the characterization of such groups). The combination of the two approaches led to an original and fruitful theory that deepened the connection between two areas of mathematics that had shared surprisingly little in the past. It should be said that these ideas were influenced by Gromov’s theory of hyperbolic groups [9] and subsequent developments where the geometry of the Cayley graph was used most effectively to produce beautiful results. The (automata-theoretic) definition of an automatic group was extended to semigroups (and monoids) in [2]. For an arbitrary semigroup, automaticity depends on the finite set of generators considered [2], a serious handicap for the theory. However, in the case of monoids, the existence of an automatic structure is independent of the generating set considered [5], making automatic monoids a promising field of research. However, we should note that this is true only if one considers semigroup generators [5], explaining our opting for semigroup homomorphisms in this paper. A geometric characterization was obtained in the particular case of completely simple semigroups [3], but at the time that this paper first appeared as a preprint (in February 2000), no general equivalent had yet been found. Hoffmann [10] has since then found (independently of this work) a condition that he terms geometric—we remark that his condition does not reduce to the usual fellow traveller property for groups. Quart. J. Math. 55 (2004), 333–356; doi: 10.1093/qmath/hah006 Quart. J. Math. Vol. 55 Part 3 c  Oxford University Press 2004; all rights reserved