The dimension of the Torelli group

Bestvina, Mladen; Bux, Kai-Uwe; Margalit, Dan

doi:10.1090/S0894-0347-09-00643-2

The dimension of the Torelli group
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by Mladen Bestvina, Kai-Uwe Bux and Dan Margalit;

J. Amer. Math. Soc. 23 (2010), 61-105

DOI: https://doi.org/10.1090/S0894-0347-09-00643-2

Published electronically: July 10, 2009

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Abstract:

We prove that the cohomological dimension of the Torelli group for a closed, connected, orientable surface of genus $g \geq 2$ is equal to $3g-5$. This answers a question of Mess, who proved the lower bound and settled the case of $g=2$. We also find the cohomological dimension of the Johnson kernel (the subgroup of the Torelli group generated by Dehn twists about separating curves) to be $2g-3$. For $g \geq 2$, we prove that the top dimensional homology of the Torelli group is infinitely generated. Finally, we give a new proof of the theorem of Mess that gives a precise description of the Torelli group in genus 2. The main tool is a new contractible complex, called the “complex of minimizing cycles”, on which the Torelli group acts.

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