## The dimension of the Torelli group

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- by Mladen Bestvina, Kai-Uwe Bux and Dan Margalit PDF
- J. Amer. Math. Soc.
**23**(2010), 61-105 Request permission

## 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.## References

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## Additional Information

**Mladen Bestvina**- Affiliation: Department of Mathematics, University of Utah, 155 S 1400 East, Salt Lake City, Utah 84112-0090
- MR Author ID: 36095
- Email: bestvina@math.utah.edu
**Kai-Uwe Bux**- Affiliation: Department of Mathematics, University of Virginia, Kerchof Hall 229, Charlottesville, Virginia 22903-4137
- Email: kb2ue@virginia.edu
**Dan Margalit**- Affiliation: Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, Massachusetts 02155
- MR Author ID: 706322
- Email: dan.margalit@tufts.edu
- Received by editor(s): September 7, 2007
- Published electronically: July 10, 2009
- Additional Notes: The first and third authors gratefully acknowledge support by the National Science Foundation.
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**23**(2010), 61-105 - MSC (2000): Primary 20F34; Secondary 57M07
- DOI: https://doi.org/10.1090/S0894-0347-09-00643-2
- MathSciNet review: 2552249