STABLE RATIONAL HOMOLOGY OF THE IA-AUTOMORPHISM GROUPS OF FREE GROUPS

Abstract The rational homology of the IA-automorphism group IA n $\operatorname {IA}_n$ upper I upper A Subscript n of the free group F n $F_n$ upper F Subscript n is still mysterious. We study the quotient of the rational homology of IA n $\operatorname {IA}_n$ upper I upper A Subscript n that is obtained as the image of the map induced by the abelianization map, which we call the Albanese homology of IA n $\operatorname {IA}_n$ upper I upper A Subscript n . We obtain a representation-stable GL ( n , Q ) $\operatorname {GL}(n,\mathbb {Q})$ upper G upper L left parenthesis n comma double struck upper Q right parenthesis -subquotient of the Albanese homology of IA n $\operatorname {IA}_n$ upper I upper A Subscript n , which conjecturally coincides with the entire Albanese homology of IA n $\operatorname {IA}_n$ upper I upper A Subscript n . In particular, we obtain a lower bound of the dimension of the Albanese homology of IA n $\operatorname {IA}_n$