$\operatorname{Aut}(\mathbb{F}_5)$ has property $(T)$

Marek Kaluba; Narutaka Ozawa; Piotr W. Nowak

$\operatorname{Aut}(\mathbb{F}_5)$ has property $(T)$

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arxiv 1712.07167 v2 pith:GV3AQOC7 submitted 2017-12-19 math.GR math.FAmath.OA

Marek Kaluba , Piotr W. Nowak , Narutaka Ozawa This is my paper · ORCID

classification math.GRmath.FAmath.OA

keywords groupmathbboperatornamepropertyautomorphismcomputer-assistedconstructivefree

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abstract

We give a constructive, computer-assisted proof that $\operatorname{Aut}(\mathbb{F}_5)$, the automorphism group of the free group on $5$ generators, has Kazhdan's property $(T)$.

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Abelianizations of finite-index subgroups of the handlebody group
math.GT 2026-07 accept novelty 6.0 of 10

For genus ≥ 4, meridian multitwists vanish in H_1 of any finite-index subgroup of the handlebody group, and subgroups containing the Torelli group, twist group, or Johnson kernel have trivial rational abelianization.

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