Abstract
We introduce a series of interesting subgroups of finite index in Aut(F n ). One of them has index 42 in Aut(F 3) and infinite abelianization. This implies that Aut(F 3) does not have Kazhdan’s property (T); see [15] and [5] for other proofs. We prove also that every subgroup of finite index in Aut(F n ), n ⩾ 3, which contains the subgroup of IA-automorphisms has a finite abelianization.
We introduce a subgroup K(n) of finite index in Aut(F n ) and show, that its abelianization is infinite for n=3, and it is finite for n ⩾ 4. We ask, whether the abelianization of its commutator subgroup K(n)′ is infinite for n ⩾ 4. If so, then Aut(F n ) would not have Kazhdan’s property (T) for n ⩾ 4.
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Bogopolski, O., Vikentiev, R. (2010). Subgroups of Small Index in Aut(F n ) and Kazhdan’s Property (T). In: Bogopolski, O., Bumagin, I., Kharlampovich, O., Ventura, E. (eds) Combinatorial and Geometric Group Theory. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9911-5_1
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