On the Grossman representations of the automorphism groups of free groups

Abstract The goal of this paper is to show that, for any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> k\geq 2 , the subgroup of the automorphism group of a free group consisting of all automorphisms that act trivially on the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (k+1) -th derived quotient of the free group is not finitely generated in some stable range. In order to show this, we consider a slight generalization of results of Grossman in [Representations of the automorphism groups of free groups, J. Algebra 30 (1974), 388–399], and a sequence of descending filtrations of the IA-automorphism group of a free group.