Group stability and Property (T)

DOI10.1016/J.JFA.2019.108298arXiv1809.00632MaRDI QIDQ6306170

Publication date: 3 September 2018

Abstract: In recent years, there has been a considerable amount of interest in the stability of a finitely-generated group

G a m m a

with respect to a sequence of groups

l e f t {G_{n} i g h t}_{n = 1}^{i n f t y}

, equipped with bi-invariant metrics

l e f t {d_{n} i g h t}_{n = 1}^{i n f t y}

. We consider the case

G_{n} = o p e r a t o r n a m e U l e f t (n i g h t)

(resp.

G_{n} = o p e r a t o r n a m e S y m l e f t (n i g h t)

), equipped with the normalized Hilbert-Schmidt metric

d_{n}^{o p e r a t o r n a m e H S}

(resp. the normalized Hamming metric

d_{n}^{o p e r a t o r n a m e H a m m i n g}

). Our main result is that if

G a m m a

is infinite, hyperlinear (resp. sofic) and has Property

o p e r a t o r n a m e (T)

, then it is not stable with respect to

l e f t (o p e r a t o r n a m e U l e f t (n i g h t), d_{n}^{o p e r a t o r n a m e H S} i g h t)

(resp.

l e f t (o p e r a t o r n a m e S y m l e f t (n i g h t), d_{n}^{o p e r a t o r n a m e H a m m i n g} i g h t)

). This answers a question of Hadwin and Shulman regarding the stability of

o p e r a t o r n a m e {S L}_{3} l e f t (m a t h b b Z i g h t)

. We also deduce that the mapping class group

o p e r a t o r n a m e M C G l e f t (g i g h t)

,

g g e q 3

, and

o p e r a t o r n a m e A u t l e f t (m a t h b b F_{n} i g h t)

,

n g e q 3

, are not stable with respect to

l e f t (o p e r a t o r n a m e S y m l e f t (n i g h t), d_{n}^{o p e r a t o r n a m e H a m m i n g} i g h t)

. Our main result exhibits a difference between stability with respect to the normalized Hilbert-Schmidt metric on

o p e r a t o r n a m e U l e f t (n i g h t)

and the (unnormalized)

p

-Schatten metrics, since many groups with Property

o p e r a t o r n a m e (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

l e f t (o p e r a t o r n a m e U l e f t (n i g h t), d_{n}^{o p e r a t o r n a m e H S} i g h t)

and

l e f t (o p e r a t o r n a m e S y m l e f t (n i g h t), d_{n}^{o p e r a t o r n a m e H a m m i n g} i g h t)

.

Mathematics Subject Classification ID

Discrete subgroups of Lie groups (22E40) Asymptotic properties of groups (20F69) Symmetric groups (20B30)

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