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This article is cited in 4 scientific papers (total in 4 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
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.
Citation:
Alexander A. Gaifullin, “On the top homology group of the Johnson kernel”, Mosc. Math. J., 22:1 (2022), 83–102
Linking options:
https://www.mathnet.ru/eng/mmj817 https://www.mathnet.ru/eng/mmj/v22/i1/p83
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