On the edge of the stable range

Research output: Contribution to journalArticle

Abstract

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 stablility 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
Number of pages43
JournalMathematische Annalen
DOIs
Publication statusAccepted/In press - 6 Jan 2020

Fingerprint

General Linear Group
Unstable
Range of data
Principal ideal domain
Braid Group
Homology Groups
Stability Theorem
Free Group
Symmetric group
Automorphism Group
Lower bound
Integer
Arbitrary
Coefficient
Theorem
Family
Ambiguity

Keywords

  • Homological stability
  • general linear groups
  • automorphism groups of free groups

Cite this

On the edge of the stable range. / Hepworth, Richard Antony.

In: Mathematische Annalen, 06.01.2020.

Research output: Contribution to journalArticle

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