On the graded quotients of the $\mathrm{SL}_m$-representation algebras of groups

Takao Satoh

Vol. 312 (2021), No. 2, 477–504

DOI: 10.2140/pjm.2021.312.477

Abstract

We consider certain descending filtrations of the ring of the $SL\left(m,\mathbb{Q}\right)$- representation varieties of free groups and free abelian groups. By using them, we introduce analogs of the Johnson homomorphisms of the automorphism groups of free groups. We show that the first homomorphisms are extended to the automorphism groups of free groups as crossed homomorphisms. Furthermore we show that the extended crossed homomorphisms induce Kawazumi’s cocycles and Morita’s cocycles. This is a generalization of our previous results for the $SL\left(2,ℂ\right)$-representation algebras.

Keywords

$\mathrm{SL}(m,\mathbb{Q})$-representation varieties, automorphism groups of free groups, Johnson homomorphisms

Mathematical Subject Classification

Primary: 20F28

Secondary: 13A15, 20G05

Milestones

Received: 23 April 2018

Revised: 23 December 2019

Accepted: 16 April 2021

Published: 31 August 2021