The Johnson–Morita theory for the ring of Fricke characters of free groups

Takao Satoh

Vol. 275 (2015), No. 2, 443–461

DOI: 10.2140/pjm.2015.275.443

Abstract

For a free group $F_n$ of rank $n$, we consider the ring ${\mathfrak{X}}_{\mathbb{Q}}\left(F_n\right)$ of $SL\left(2,ℂ\right)$-characters of $F_n$ generated by all Fricke characters $trx$ for $x\in F_n$. Its ideal $J$ generated by $trx-2$ for all $x\in F_n$ is $Aut{F}_n$-invariant. We denote by ${\mathcal{ℰ}}_n\left(1\right)$ the subgroup of the automorphism group $Aut{F}_n$ of $F_n$ consisting of all automorphisms which act on $J∕J^2$ trivially. The group ${\mathcal{ℰ}}_n\left(1\right)$ is regarded as a Fricke character analogue of the IA-automorphism group of $F_n$ and the Torelli subgroup of the mapping class group of a surface. In our previous work, we constructed a homomorphism ${\eta }_1$ from ${\mathcal{ℰ}}_n\left(1\right)$ into ${Hom}_{\mathbb{Q}}\left(J∕J^2,J^2∕J^3\right)$ as a Fricke character analogue of the first Johnson homomorphisms of the mapping class group and $Aut{F}_n$.

In this paper, according to Morita’s work for the extension of the first Johnson homomorphism of the mapping class group, we extend ${\eta }_1$ to $Aut{F}_n$ as a crossed homomorphism. We see that the obtained crossed homomorphism $\eta$ is not null cohomologous. We also compute the images of Nielsen’s generators of $Aut{F}_n$ by $\eta$.

Keywords

ring of Fricke characters, automorphism group of free groups, IA-automorphism group, Andreadakis–Johnson filtration, Johnson homomorphisms

Mathematical Subject Classification 2010

Primary: 20F28

Secondary: 20J06

Milestones

Received: 21 July 2014

Revised: 24 October 2014

Accepted: 7 November 2014

Published: 15 May 2015

Correction: 16 July 2018