#### Vol. 312, No. 2, 2021

 Recent Issues Vol. 314: 1  2 Vol. 313: 1  2 Vol. 312: 1  2 Vol. 311: 1  2 Vol. 310: 1  2 Vol. 309: 1  2 Vol. 308: 1  2 Vol. 307: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Contacts Submission Guidelines Submission Form Policies for Authors ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Special Issues Author Index To Appear Other MSP Journals
On the graded quotients of the $\mathrm{SL}_m$-representation algebras of groups

### Takao Satoh

Vol. 312 (2021), No. 2, 477–504
##### Abstract

We consider certain descending filtrations of the ring of the $SL\left(m,ℚ\right)$- 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