Quantum isometry groups of dual of finitely generated discrete groups and quantum groups

Abstract

We study quantum isometry groups, denoted by Q(Γ, S), of spectral triples on C * r (Γ) for a finitely generated discrete group Γ coming from the word-length metric with respect to a symmetric generating set S. We first prove a few general results about Q(Γ, S) including:• For a group Γ with polynomial growth property, the dual of Q(Γ, S) has polynomial growth property provided the action of Q(Γ, S) on C * r (Γ) has full spectrum. • Q(Γ, S) ∼ = QISO(Γ, d) for any discrete abelian group Γ, where d is a suitable metric on the dual compact abelian groupΓ.We then carry out explicit computations of Q(Γ, S) for several classes of examples including free and direct product of cyclic groups, Baumslag-Solitar group, Coxeter groups etc. In particular, we have computed quantum isometry groups of all finitely generated abelian groups which do not have factors of the form Z k 2 or Z l 4 for some k, l in the direct product decomposition into cyclic subgroups.

Wreath products of finite groups by quantum groups

“…Note that the above quantum groups appeared as quantum isometry groups of the free products of finite cyclic groups in [4] and [14]. There is also a classical version of this result.…”

Example of a Group Whose Quantum Isometry Group Does Not Depend on the Generating Set

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In this article we have shown that the quantum isometry group of C * r (Z), denoted by Q(Z, S) as in [16], with respect to a symmetric generating set S does not depend on the generating set S. Moreover, we have proved that the result is no longer true if the group Z is replaced by Z × Z × · · · × Z n copies for n > 1.

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“…Consider the generating sets S 1 = {(1, 0), (0, 1), (−1, 0), (0, −1)} and S 2 = {(1, 1), (−1, −1)} respectively for Z n ×Z 4 . The underlying C * -algebra of Q(Z n ×Z 4 , S 1 ) is noncommutative by Theorem 4.10 of [16]. On the other hand, Q(Z n × Z 4 , S 2 ) is the doubling of C * (Z n × Z 4 ) corresponding to the automorphism given by a → a −1 ∀ a ∈ Z n × Z 4 from [8].…”

Quantum isometry group of dual of finitely generated discrete groups - II

“…There have been several articles already on computations and study of the quantum isometry groups of such spectral triples, e.g [11], [23], [16], [6], [4] and references therein. In [19] together with Goswami we also studied the quantum isometry groups of such spectral triples in a systematic and unified way. Here we compute Q(Γ, S) for more examples of groups including braid groups, Z 4 * Z 4 · · · * Z 4 n copies etc.…”

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As a continuation of the programme of [19], we carry out explicit computations of Q(Γ, S), the quantum isometry group of the canonical spectral triple on C * r (Γ) coming from the word length function corresponding to a finite generating set S, for several interesting examples of Γ not covered by the previous work [19]. These include the braid group of 3 generators, Z * n 4 etc.

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