Let S denote a closed, connected, oriented surface of genus g, and let T denote its Torelli group - the subgroup of the mapping class group that acts trivially on first homology. One year ago, Benson Farb announced that he had proved that every automorphism of T is induced by a homeomorphism of S, if g > 3. We prove that this result is true, using a uniform argument, for g > 2. Since the Torelli group is trivial for g < 2, this result also holds for g < 2. On the other hand, by Mess' result that the Torelli group is a free group of infinite rank for g = 2, this result does not hold for g = 2. Indeed, Mess' result implies that the group of automorphisms of the Torelli group is uncountable for g = 2, containing a copy of the group of permutations of any infinite basis for the Torelli group, whereas only countably many automorphisms of the Torelli group are induced by homeomorphisms, due to the finite generation of the extended mapping class group of S.
Contact: mccarthy@math.msu.edu
Last Revised 11/13/03