The Casson invariant and the word metric on the Torelli group

Nathan Broaddus; Benson Farb; Andrew Putman

doi:10.1016/j.crma.2007.09.018

Topology/Group Theory

The Casson invariant and the word metric on the Torelli group

Presented by: Étienne Ghys

Nathan Broaddus ¹; Benson Farb ¹; Andrew Putman ¹

¹ Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA

Comptes Rendus. Mathématique, Volume 345 (2007) no. 8, pp. 449-452.

Abstract
Résumé

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 $I$ , of the element of $I$ associated to any Heegaard splitting of M. We construct examples which show this bound is asymptotically sharp.

Soit M une sphère d'homologie de dimension 3. Tout scindement de Heegaard de M définit un élément du groupe de Torelli $I$ . Nous montrons que l'invariant de Casson de M est borné par une constante fois le carré de la longueur de cet élément. Cette longueur est définie comme la longueur minimale d'un mot le représentant, écrit en utilisant un système générateur fini quelconque de $I$ . Nous construisons des exemples qui montrent que cette borne est asymptotiquement la meilleure possible.

Received: 2007-07-21
Accepted: 2007-09-19
Published online: 2007-10-15

DOI: 10.1016/j.crma.2007.09.018

Author's affiliations:

Nathan Broaddus ¹; Benson Farb ¹; Andrew Putman ¹

¹ Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA

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     author = {Nathan Broaddus and Benson Farb and Andrew Putman},
     title = {The {Casson} invariant and the word metric on the {Torelli} group},
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Nathan Broaddus; Benson Farb; Andrew Putman. The Casson invariant and the word metric on the Torelli group. Comptes Rendus. Mathématique, Volume 345 (2007) no. 8, pp. 449-452. doi : 10.1016/j.crma.2007.09.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.09.018/

References
Cited by

[1] S. Akbulut; J.D. McCarthy Casson's Invariant for Oriented Homology 3-Spheres, Princeton Univ. Press, Princeton, NJ, 1990

[2] D. Johnson An abelian quotient of the mapping class group $I_{g}$ , Math. Ann., Volume 249 (1980) no. 3, pp. 225-242

[3] D. Johnson The structure of the Torelli group. I. A finite set of generators for $I$ , Ann. of Math., Volume 118 (1983) no. 3, pp. 423-442

[4] D. McCullough; A. Miller The genus 2 Torelli group is not finitely generated, Topology Appl., Volume 22 (1986) no. 1, pp. 43-49

[5] S. Morita On the structure of the Torelli group and the Casson invariant, Topology, Volume 30 (1991) no. 4, pp. 603-621

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