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Moscow Mathematical Journal, 2022, Volume 22, Number 1, Pages 83–102
DOI: https://doi.org/10.17323/1609-4514-2022-22-1-83-102
(Mi mmj817)
 

This article is cited in 3 scientific papers (total in 3 papers)

On the top homology group of the Johnson kernel

Alexander A. Gaifullinabcd

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
b Skolkovo Institute of Science and Technology, Skolkovo, Russia
c Lomonosov Moscow State University, Moscow, Russia
d Institute for Information Transmission Problems (Kharkevich Institute), Moscow, Russia
Full-text PDF Citations (3)
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Abstract: The action of the mapping class group $\mathrm{Mod}_g$ of an oriented surface $\Sigma_g$ on the lower central series of $\pi_1(\Sigma_g)$ defines the descending filtration in $\mathrm{Mod}_g$ called the Johnson filtration. The first two terms of it are the Torelli group $\mathcal{I}_g$ and the Johnson kernel $\mathcal{K}_g$. By a fundamental result of Johnson (1985), $\mathcal{K}_g$ is the subgroup of $\mathrm{Mod}_g$ generated by all Dehn twists about separating curves. In 2007, Bestvina, Bux, and Margalit showed that the group $\mathcal{K}_g$ has cohomological dimension $2g-3$. We prove that the top homology group $H_{2g-3}(\mathcal{K}_g)$ is not finitely generated. In fact, we show that it contains a free abelian subgroup of infinite rank, hence, the vector space $H_{2g-3}(\mathcal{K}_g,\mathbb{Q})$ is infinite-dimensional. Moreover, we prove that $H_{2g-3}(\mathcal{K}_g,\mathbb{Q})$ is not finitely generated as a module over the group ring $\mathbb{Q}[\mathcal{I}_g]$.
Key words and phrases: johnson kernel, Torelli group, homology of groups, complex of cycles, Casson invariant, abelian cycle.
Bibliographic databases:
Document Type: Article
MSC: Primary 20F34; Secondary 57M07, 20J05
Language: English
Citation: Alexander A. Gaifullin, “On the top homology group of the Johnson kernel”, Mosc. Math. J., 22:1 (2022), 83–102
Citation in format AMSBIB
\Bibitem{Gai22}
\by Alexander~A.~Gaifullin
\paper On the top homology group of the Johnson kernel
\jour Mosc. Math.~J.
\yr 2022
\vol 22
\issue 1
\pages 83--102
\mathnet{http://mi.mathnet.ru/mmj817}
\crossref{https://doi.org/10.17323/1609-4514-2022-22-1-83-102}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4407770}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85129027673}
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