TY - JOUR
T1 - Group stability and Property (T)
AU - Becker, Oren
AU - Lubotzky, Alexander
N1 - Publisher Copyright:
© 2019 Elsevier Inc.
PY - 2020/1/1
Y1 - 2020/1/1
N2 - 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).
AB - 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).
KW - Group theoretic stability
KW - Property (T)
UR - http://www.scopus.com/inward/record.url?scp=85070496569&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2019.108298
DO - 10.1016/j.jfa.2019.108298
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AN - SCOPUS:85070496569
SN - 0022-1236
VL - 278
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 1
M1 - 108298
ER -