Abstract of W. Vautaw's Thesis

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Abelian Subgroups and Automorphisms of the Torelli Group, Ph. D. Thesis, Michigan State University, 2002

Let S be a closed, connected, oriented surface ofgenus g > 2. The mapping class group M of Sis defined to be the group of isotopy classes of self-homeomorphisms ofS, while the Torelli group T of S is thesubgroup of M consisting of the isotopy classes of thoseself-homeomorphisms of S that induce the identity permutation of thefirst homology group of S.

This work considers two aspects of the Torelli group. The first is the Abeliansubgroups of T. This portion of the work, where graph theory is theprincipal tool, contains two primary theorems. One gives a complete descriptionof the multitwist subgroups of T, and the other states that anyAbelian subgroup of T has rank at most 2g-3.

The second subject of investigation is automorphisms of the Torelli group; specifically, we ask whether any automorphism Y: T ---> Tis induced by a homeomorphism of S. In several formal and informalannouncements made between October 2001 and March 2002, Benson Farb stated thathe was able to prove that this is indeed the case for g > 3. In this work,we lay the foundation for proving that it is also true for g = 3. This involvesthree basic steps. The first is to characterize algebraically certain elements of the Torelligroup, namely powers of Dehn twists about separating curves and powers ofbounding pair maps. The characterization given by Farb is valid for g > 3,while our's is valid for g > 2. The second step is to show that Yinduces an automorphism Y^* of C, the complex of curves of S. This difficult step remains incomplete at the time of this writing. In the laststep we use a theorem of Ivanov which states that Y^* is induced by a homeomorphism h of the surface S, and conclude, under the assumptionthat it is possible to complete the second step, that the automorphism of theTorelli group induced by the homeomorphism h agrees with our automorphismY.