The Casson invariant and the word metric on the Torelli group

Abstract

We bound the value of the Casson invariant of any integral homology 3-sphere $M$ by a constant times the distance-squared to the identity, measured in any word metric on the Torelli group $\T$, of the element of $\T$ associated to any Heegaard splitting of $M$. We construct examples which show this bound is asymptotically sharp.

5 pages, minor corrections; to appear in C. R. Math. Acad. Sci. Paris

Cited by in corpus (2)

• Geometry of the mapping class group II: A biautomatic structure
• Symplectic Heegaard splittings and linked abelian groups