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Dimension of the Torelli group for Out(Fn)

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Let \(\mathcal{T}_{n}\) be the kernel of the natural map Out(Fn)→GLn(ℤ). We use combinatorial Morse theory to prove that \(\mathcal{T}_{n}\) has an Eilenberg–MacLane space which is (2n-4)-dimensional and that \(H_{2n-4}(\mathcal{T}_{n},\mathbb{Z})\) is not finitely generated (n≥3). In particular, this shows that the cohomological dimension of \(\mathcal{T}_{n}\) is equal to 2n-4 and recovers the result of Krstić–McCool that \(\mathcal{T}_3\) is not finitely presented. We also give a new proof of the fact, due to Magnus, that \(\mathcal{T}_{n}\) is finitely generated.

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Authors and Affiliations

  1. Department of Mathematics, University of Utah, 155 S 1440 East, Salt Lake City, 84112-0090, UT, USA

    Mladen Bestvina & Dan Margalit

  2. Department of Mathematics, University of Virginia, Kerchof Hall 229, Charlottesville, VA, 22903-4137, USA

    Kai-Uwe Bux

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  1. Mladen Bestvina
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  2. Kai-Uwe Bux
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  3. Dan Margalit
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Correspondence to Mladen Bestvina.

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Bestvina, M., Bux, KU. & Margalit, D. Dimension of the Torelli group for Out(Fn). Invent. math. 170, 1–32 (2007). https://doi.org/10.1007/s00222-007-0055-0

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  • DOI: https://doi.org/10.1007/s00222-007-0055-0

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