2 Order Automatic Mapping Class

We prove that the mapping class group of a compact surface with a nite number of punctures and non-empty boundary is order automatic. More precisely, the group is right-orderable, has an automatic structure as described by Mosher, and there exists a nite state automaton that decides, given the Mosher normal forms of two elements of the group, which of them represents the larger element of the group. Moreover, the decision takes linear time in the length of the normal forms. 0. Introduction This paper is a sequel to 5] and contains the proof of the linear time algorithm which was announced in 5, remark 5.2]. In addition to giving the full proofs, this paper extends the results of 5] to a considerably larger class of mapping class groups. Let S be a compact surface, not necessarily orientable, with boundary @S and a nite number of distinguished points in its interior, called the punctures. The mapping class group MCG(S) is deened to be the group of homeomorphisms of S mapping @S identically and permuting the punctures, up to isotopies xing @S and the punctures. The group multiplication is given by composition and written algebraically, ie given , 2 MCG(S) deene := : S ! S. For instance, if S is a disk with n punctures then MCG(S) is isomorphic to B n , the braid group on n strings 1]. In 2, 3] P Dehornoy deened a total ordering on the braid group B n which is right invariant, ie for , , 2 B n , > implies >. This ordering was reinterpreted in 5] in more geometrical terms. Right invariant orderings are important because of a long-standing conjecture in group theory: that the group ring of a torsion-free group has no zero divisors. The conjecture is true for the smaller class of right-orderable groups (groups admitting a right invariant order). For more detail see 8, chapter 2]. The mapping class group of a surface S is torsion-free if and only if @S is non-empty. 1 In section 1 we generalise the construction in 5] to prove that for any surface S with non-empty boundary, MCG(S) is right-orderable. 2 In 6, 7] L Mosher constructed an automatic structure (in the sense of 4]) for mapping class groups of surfaces. More precisely, suppose that a surface S with boundary and possibly some punctures is equipped with a triangulation whose vertex set …