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The Casson invariant and the word metric on the Torelli group

Nathan Broaddus, Benson Farb, Andrew Putman
Arxiv ID: 0707.2264Last updated: 6/8/2020
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.

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