Detecting quasiconvexity: Algorithmic aspects

The main result of this paper states that for any group G with an automatic structure L with unique representatives one can construct a uniform partial algorithm which detects L-rational subgroups and gives their preimages in L. This provides a practical, not just theoretical, procedure for solving the occurrence problem for such subgroups. 1. Generalized word problem and rational structures on groups The goal of this paper is to highlight connections between the theory of automatic groups and the generalized word problem and to demonstrate certain additional advantages of the class of automatic groups over the class of combable groups. We assume that the reader is familiar with the theory of automatic groups, regular languages and combable groups. Although some of the important definitions will be given, the reader is referred to [ECHLPT] for further details. A good overview of the theory of automatic groups can also be found in [BGSS]. We take for granted some basic facts about word hyperbolic groups and their connections with the theory of automatic groups. Here our main references are [Gr], [ABCFLMSS], [ECHLPT], [BGSS] and [GS]. An important discussion about combable groups can also be found in [A], [AB] and [N]. The author is grateful to the referee for greatly simplifying the proof of Proposition 1 and to Gilbert Baumslag for his help in writing this paper. Recall that if G is a recursively presented finitely generated group given by a presentation G =< x1, .., xn|r1, .., rm, .. > (1) and H is a subgroup of G then we say that G has solvable generalized word problem with respect to H if there is an algorithm which, for any word w in the generators x1, .., xn, decides whether or not w represents an element of H . Equivalently, G has solvable generalized word problem with respect to H if the set φ(H) is a recursive subset of the free group F (x1, .., xn) where φ:F (x1, .., xn) → G is the natural epimorphism associated with the presentation (1). It is not hard to see that this definition does not depend on the choice of a presentation of G with a finite number of generators. If G has solvable generalized word problem with respect to the trivial subgroup H = 1 then G is said to have solvable word problem . The concept of generalized word problem goes back to the work of W.Magnus [M] where he proved that if M is a subgroup of a one-relator group G =< x1, .., xn|R = 1 >, generated by any subset of the generating set, then G has solvable generalized word problem with respect to M . We should stress that the mere knowledge that G has solvable generalized word problem with respect to H does not yet give one an effective procedure for determining whether or not a particular word in the generators of G represents an element of H . That is, for practical purposes it is not enough to know that the required algorithm exists, it is necessary to be able to find it. 1991 Mathematics Subject Classification. Primary 20F10; Secondary 20F32. Typeset by AMS-TEX 1