The stable Albanese homology of the IA-automorphism groups of free groups

2024/04/24 by Katada, Mai
#20F28 #20J06 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2404.15901

Abstract

The IA-automorphism group IA_n of the free group F_n of rank n is a normal subgroup of the automorphism group Aut(F_n) of F_n. We study the Albanese homology of IA_n, which is the quotient of the rational homology of IA_n defined as the image of the map induced by the abelianization map of IA_n on homology. The Albanese homology of IA_n is an algebraic GL(n,ℚ)-representation. We determine the representation structure of the Albanese homology of IA_n for n greater than or equal to three times the homological degree. We also determine the structure of the stable Albanese homology of the analogue of IA_n to the outer automorphism group of F_n. Moreover, we identify the relation between the stable Albanese (co)homology of IA_n and the stable cohomology of Aut(F_n) with certain twisted coefficients.

Related

• Stable rational homology of the IA-automorphism groups of free groups
• The walled Brauer category and stable cohomology of IA_n
• On the stable cohomology of the IA-automorphism groups of free groups
• Stable homology of automorphism groups of free groups
• Cohomoloy of Aut(F_n)