Abstract
Let F n be the free group of rank n, and let Aut+(F n ) be its special automorphism group. For an epimorphism π : F n → G of the free group F n onto a finite group G we call \({\Gamma^+(G,\pi)=\{ \varphi \in {\rm Aut}^+(F_n) \mid \pi\varphi = \pi \}}\) the standard congruence subgroup of Aut+(F n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ+(G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ+(G, π) ≤ Aut+(F 2) has infinite abelianization.
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The author is very thankful to F. Grunewald for proposing the topic. The results presented in this paper are a part of the author’s PhD Thesis [1], which was supervised by B. Klopsch.
The author would therefore like to express his gratitude for B. Klopsch’ great support and many inspiring conversations.
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Appel, D. On the abelianizations of congruence subgroups of Aut(F 2). Arch. Math. 99, 101–109 (2012). https://doi.org/10.1007/s00013-012-0415-x
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DOI: https://doi.org/10.1007/s00013-012-0415-x