Vol. 312, No. 2, 2021

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

Takao Satoh

Vol. 312 (2021), No. 2, 477–504

We consider certain descending filtrations of the ring of the SL(m, )- 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(2, )-representation algebras.

$\mathrm{SL}(m,\mathbb{Q})$-representation varieties, automorphism groups of free groups, Johnson homomorphisms
Mathematical Subject Classification
Primary: 20F28
Secondary: 13A15, 20G05
Received: 23 April 2018
Revised: 23 December 2019
Accepted: 16 April 2021
Published: 31 August 2021
Takao Satoh
Tokyo University of Science