Quasi-automatic groups are asynchronously automatic

Abstract

Classes of groups and semigroups have been defined in regards to their computation complexity in the past, notably by Epstein et al. with their early work on automatic groups, see [1]. More general classes have followed. First, the asynchronous case for groups was defined and studied, notably by the same group of authors. Afterwards, monoids and semigroups have been investigated, notably by Campbell et al., see [4], and Duncan et al., see [5]. However, a key feature of automatic groups does not hold for automatic semigroups; while the notion of automaticity for finitely generated groups does not depend on the choice of generators, it does for finitely generated semigroups.Showing there exists no automatic structure for a given generating set is doable, but doing so for any generating set can be tedious, see for example [6], where Hoffmann and Thomas build a finitely generated commutative semigroup which is not automatic under any choice of generating set. This makes the feature extremely valuable. Recently, Blanchette et al. introduced quasi-automatic semigroups, a more general class of semigroups for which the feature actually holds, see [2].Several relations between these classes have been established. For instance, we know that the class of automatic groups sits strictly inside the class of asynchronously automatic groups, which itself is contained into the class of quasi-automatic groups. Geometric characterizations have also been made. Automatic groups are exactly those having the so-called Lipschitz property. Asynchronously automatic groups are characterized by the Lipschitz Hausdorff property of Epstein et al., along with a special function named a departure function. For quasi-automatic groups, we have a different weak form of the Lipschitz property that again characterizes the class geometrically.Our main result here is to establish a new link between those classes; we will show that for groups, being quasi-automatic is equivalent to being asynchronously automatic. This is particularly useful because quasi-automatic structures are somewhat natural, as opposed to the more complex asynchronously automatic structures. Context and definitionsLet A be a generating set of a semigroup S and p : A + ։ S be the canonical map which maps a word to the element of S it represents. We write u ∼ v if u and v represent the same element of S. A rational language L ⊆ A + that maps onto S is called a dictionary. A dictionary is called quasi-automatic if for each a in A ∪ {ε}, the relation R L a = {(u, v) ∈ L × L|ua ∼ v} is rational. If these relations are also recognizable by two-tape automata, this dictionary is called asynchronously automatic. We use a theorem of Nivat (see [3], proposition 4) to work with these rational relations.The theorem states that a relation R is rational if and only if there exist some finite B, some ra-1