Under mild axioms on a graded E2-algebra over a positive-characteristic field, higher-order homological stability maps exist with slopes approaching 1, and the patterns are governed by a stability Hopf algebra.
Homological stability for general linear groups over Dedekind domains
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We prove a new kind of homological stability theorem for automorphism groups of finitely-generated projective modules over Dedekind domains, which takes into account all possible stabilisation maps between these, rather than only stabilisation by the free module of rank 1. We show the same kind of stability holds for Clausen and Jansen's reductive Borel--Serre spaces.
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A chromatic approach to homological stability
Under mild axioms on a graded E2-algebra over a positive-characteristic field, higher-order homological stability maps exist with slopes approaching 1, and the patterns are governed by a stability Hopf algebra.