Mapping class groups are automatic (Q5917718)

Let \(S\) be a compact surface, possibly with the extra structure of an orientation or a distinguished set of points called punctures. The mapping class group is \(\text{MCG} (S) = \text{Homeo}(S)/\text{Homeo}_0(S)\), where \(\text{Homeo} (S)\) is the group of structure preserving homeomorphisms and \(\text{Homeo}_0(S)\) the normal subgroup of those homeomorphisms isotopic to the identity through elements of \(\text{Homeo}(S)\). A group, \(G\), is automatic if there is a finite set of generators and a regular language (i.e. a set of words recognized by a finite automaton) over the generators such that each element in \(G\) is represented by at least one word in \(L\) and there is a finite automaton which will check whether two words in \(L\) represent the same element in \(G\) and whether they differ by a certain generator. In this paper ideal triangulations (vertex set = set of punctures) and elementary moves are used to give an automatic structure on \(\text{MSG} (S)\) for a punctured surface \(S\). This result along with results on ``combability'' of groups leads to a construction of an automatic structure for \(\text{MCG}(S)\) where \(S\) is a closed surface.