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The Euler characteristic of $\operatorname{Out}(F_n)$

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arxiv 1907.03543 v2 pith:CBMRVP7P submitted 2019-07-08 math.GR math-phmath.MPmath.QA

Michael Borinsky , Karen Vogtmann This is my paper · ORCID
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keywords characteristiceulerfunctionoperatornamealwaysasymptoticauthorconjecture
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abstract

We prove that the rational Euler characteristic of $\operatorname{Out}(F_n)$ is always negative and its asymptotic growth rate is $\Gamma(n- \frac32)/\sqrt{2\pi} \log^2 n$. This settles a 1987 conjecture of J. Smillie and the second author. We establish connections with the Lambert $W$-function and the zeta function.

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Cited by 1 Pith paper

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  1. On the top-dimensional $\ell^2$-Betti numbers

    math.GR 2019-09 conditional novelty 7.0 of 10

    A transfer trick yields non-vanishing top ℓ2-Betti numbers for Out(Fn), Aut(Fn) and Torelli groups, vanishing results for subgroups of 3-manifold groups, and ergodic dimension d+1 for F2^d × Z.

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