On the homology and cohomology of congruence subgroups

“…Considering the result of Lee and Szczarba [10] stated above, we see that for any odd prime p, the abelianization of I A n is isomorphic to (Z/ pZ) ⊕ 1 2 (n−1)(n 2 +2n+2) as an abelian group. Next we consider the outer automorphism group of a free group and the images of I A n and I A n,d by a natural projection.…”

Section: Introductionmentioning

confidence: 94%

The abelianization of the congruence IA-automorphism group of a free group

Satoh

2007

Math. Proc. Camb. Phil. Soc.

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We consider the abelianizations of some normal subgroups of the automorphism group of a finitely generated free group. Let F n be a free group of rank n. For d 2, we consider a group consisting the automorphisms of F n which act trivially on the first homology group of F n with Z/dZ-coefficients. We call it the congruence IA-automorphism group of level d and denote it by I A n,d . Let I O n,d be the quotient group of the congruence IA-automorphism group of level d by the inner automorphism group of a free group. We determine the abelianization of I A n,d and I O n,d for n 2 and d 2. Furthermore, for n = 2 and odd prime p, we compute the integral homology groups of I A 2, p for any dimension.

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“…The key idea is to identify deep lattices of a simple algebraic group G over Q that represents nonzero cycles in appropriate homology groups. Ideas of this sort appear in papers of Lee-Szczarba [34], Charney [19] and Church-Farb [21]. We also quickly show, using a Bockstein argument, that the elements constructed in [18] by strong approximation represents the zero element in its virtual structure set.…”

mentioning

confidence: 65%