On the edge of the stable range

Richard Hepworth* (Corresponding Author)

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)
9 Downloads (Pure)


We prove a general homological stability theorem for certain families of groups equipped with product maps, followed by two theorems of a new kind that give information about the last two homology groups outside the stable range. (These last two unstable groups are the ‘edge’ in our title.) Applying our results to automorphism groups of free groups yields a new proof of homological stability with an improved stable range, a description of the last unstable group up to a single ambiguity, and a lower bound on the rank of the penultimate unstable group. We give similar applications to the general linear groups of the integers and of the field of order 2, this time recovering the known stability range. The results can also be applied to general linear groups of arbitrary principal ideal domains, symmetric groups, and braid groups. Our methods require us to use field coefficients throughout.
Original languageEnglish
Pages (from-to)pages123–181
Number of pages59
JournalMathematische Annalen
Issue number1-2
Early online date5 Feb 2020
Publication statusPublished - 1 Jun 2020

Bibliographical note

Open access via the Springer Compact Agreement

Acknowledgements: My thanks to Rachael Boyd, Anssi Lahtinen, Martin Palmer, Oscar Randal-Williams, David Sprehn and Nathalie Wahl for useful discussions, and especially to Nathalie Wahl for explaining the main steps in the proof of Proposition 13.1.


  • Primary 20J06
  • Secondary 20F28
  • 57M07
  • 55R40


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