On stable homology of congruence groups

Abstract

We show in this work that homology in degree d of a congruence group, in a very general framework, defines a weakly polynomial functor of degree at most 2d and we describe this functor modulo polynomial functors of smaller degree. Our main tool is a spectral sequence connecting homology of congruence-like groups (in a formal setting close to the one introduced with Vespa in 2010 for orthogonal groups) and functor homology. We prove and use in a crucial way properties of some tensor structures and derived Kan extensions on polynomial functors.Our results extend especially, with different methods, the work by Suslin on excision in integer algebraic K-theory and a recent preprint by Church-Miller-Nagpal-Reinhold.