On the low-dimensional cohomology groups of the IA-automorphism group of the free group of rank three

Takao Satoh

doi:10.1017/s0013091521000171

Abstract

AbstractIn this paper, we study the structure of the rational cohomology groups of the IA-automorphism group$\mathrm {IA}_3$of the free group of rank three by using combinatorial group theory and representation theory. In particular, we detect a nontrivial irreducible component in the second cohomology group of$\mathrm {IA}_3$, which is not contained in the image of the cup product map of the first cohomology groups. We also show that the triple cup product of the first cohomology groups is trivial. As a corollary, we obtain that the fourth term of the lower central series of$\mathrm {IA}_3$has finite index in that of the Andreadakis–Johnson filtration of$\mathrm {IA}_3$.

Highlights

Let Fn be a free group of rank n ≥ 2 with basis x1, . . . , xn, and Aut Fn the automorphism group of Fn
In the 1980s, by introducing Outer spaces, Culler–Vogtmann [11] made a breakthrough in computation of homology groups of the outer automorphism groups of free groups
The following theorem shows that nontrivial elements in H3(IA3, Q) cannot be detected by the triple cup product of the first cohomology group of IA3

Summary

Introduction

Let Fn be a free group of rank n ≥ 2 with basis x1, . . . , xn, and Aut Fn the automorphism group of Fn. He showed that the stable rational homology groups Hq(Aut Fn, Q) are trivial for n ≥ 2q + 1 This result is quite a contrast to the case of the mapping class groups of surfaces. We should remark that in [10], Conant–Hatcher–Kassabov–Vogtmann gave a construction of many nontrivial unstable homology classes of Aut Fn and Out Fn, and studies the Morita classes. This shows that there is a possibility that the second homology group H2(IA3, Z) is not finitely generated This follows by a work of Bestvina–Bux–Margalit [5]. We remark that the arguments and techniques which we use in this paper are applicable to study the cohomology groups of IAn for general n ≥ 4. We give the first combinatorial group theoretic approach to the study of the low-dimensional cohomology groups of the IA-automorphism groups of free groups

Preliminaries

IA-automorphism groups

Free Lie algebra generated by H

Andreadakis–Johnson filtration