Group stability and Property (T)

Oren Becker*, Alexander Lubotzky

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

18 Scopus citations

Abstract

In recent years, there has been a considerable amount of interest in the stability of a finitely-generated group Γ with respect to a sequence of groups {Gn}n=1 , equipped with bi-invariant metrics {dn}n=1 . We consider the case Gn=U(n) (resp. Gn=Sym(n)), equipped with the normalized Hilbert-Schmidt metric dn HS (resp. the normalized Hamming metric dn Hamming). Our main result is that if Γ is infinite, hyperlinear (resp. sofic) and has Property (T), then it is not stable with respect to (U(n),dn HS) (resp. (Sym(n),dn Hamming)). This answers a question of Hadwin and Shulman regarding the stability of SL3(Z). We also deduce that the mapping class group MCG(g), g≥3, and Aut(Fn), n≥3, are not stable with respect to (Sym(n),dn Hamming). Our main result exhibits a difference between stability with respect to the normalized Hilbert-Schmidt metric on U(n) and the (unnormalized) p-Schatten metrics, since many groups with Property (T) are stable with respect to the latter metrics, as shown by De Chiffre-Glebsky-Lubotzky-Thom and Lubotzky-Oppenheim. We suggest a more flexible notion of stability that may repair this deficiency of stability with respect to (U(n),dn HS) and (Sym(n),dn Hamming).

Original languageEnglish
Article number108298
JournalJournal of Functional Analysis
Volume278
Issue number1
DOIs
StatePublished - 1 Jan 2020

Bibliographical note

Publisher Copyright:
© 2019 Elsevier Inc.

Keywords

  • Group theoretic stability
  • Property (T)

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