Strong Banach property (T) for simple algebraic groups of higher rank

Liao, Benben

doi:10.1142/s1793525314500010

Cited by 33 publications

(42 citation statements)

References 11 publications

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“…If S denotes a finite symmetric generating set of Γ, and if Y i = Γ/Γ i is the corresponding graph with natural metric denoted by In the non-Archimedean setting, a strong analogue of Theorem A is known. Indeed, for a non-Archimedean local field F , Lafforgue proved that SL(3, F ) has property (T strong Banach ) (see [Laf08] and [Laf09]), and Liao proved that Sp(2, F ) has it as well, implying that any connected almost F -simple algebraic group with F -split rank at least 2 has property (T strong Banach ) (see [Lia13]). Analogues of the corollaries above follow as well.…”

Section: Strong E4mentioning

confidence: 99%

Strong property (T) for higher-rank simple Lie groups: Figure 1.

Laat

Salle

2015

Proc. London Math. Soc.

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Abstract. We prove that connected higher rank simple Lie groups have Lafforgue's strong property (T) with respect to a certain class of Banach spaces E 10 containing many classical superreflexive spaces and some non-reflexive spaces as well. This generalizes the result of Lafforgue asserting that SL(3, R) has strong property (T) with respect to Hilbert spaces and the more recent result of the second named author asserting that SL(3, R) has strong property (T) with respect to a certain larger class of Banach spaces. For the generalization to higher rank groups, it is sufficient to prove strong property (T) for Sp(2, R) and its universal covering group. As consequences of our main result, it follows that for X ∈ E 10 , connected higher rank simple Lie groups and their lattices have property (F X ) of Bader, Furman, Gelander and Monod, and that the expanders contructed from a lattice in a connected higher rank simple Lie group do not admit a coarse embedding into X.

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Section: Strong E4mentioning

confidence: 99%

Strong property (T) for higher-rank simple Lie groups: Figure 1.

Laat

Salle

2015

Proc. London Math. Soc.

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“…We refer to Definition 2.2 for the definitions of type and cotype. Banach Strong (T) with respect to spaces of nontrivial type was recently extended to all higher rank algebraic groups over non-Archimedean local fields in [18]. This proves the conjecture in [1] in the non-Archimedean case, since every superreflexive Banach space has nontrivial type by [24].…”

Section: In the Norm Topology Of B(x) To A Projection On The π(G)-invmentioning

confidence: 71%

“…Lafforgue proved that hyperbolic groups do not satisfy Strong Banach (T) with respect to Hilbert spaces (T Strong Hilb ), but for a local field F , SL(3, F ) has (T Strong Hilb ). In the case when F is non-Archimedean, it was proved in [13] and [14] (see also [17]) that SL(3, F ) satisfies Strong Banach property (T) with respect to the class of all Banach spaces of nontrivial (Rademacher) type, which is essentially the largest class of Banach spaces for which strong (T) could hold (for a non compact group). We refer to Definition 2.2 for the definitions of type and cotype.…”

Section: In the Norm Topology Of B(x) To A Projection On The π(G)-invmentioning

confidence: 99%

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Towards strong Banach property (T) for SL(3,ℝ)

Salle

2015

Isr. J. Math.

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Abstract. We prove that SL(3, R) has Strong Banach property (T) in Lafforgue's sense with respect to the Banach spaces that are θ > 0 interpolation spaces (for the complex interpolation method) between an arbitrary Banach space and a Banach space with sufficiently good type and cotype. As a consequence, every action of SL(3, R) or its lattices by affine isometries on such a Banach space X has a fixed point, and the expanders contructed from SL(3, Z) do not admit a coarse embedding into X. We also prove a quantitative decay of matrix coefficients (Howe-Moore property) for representations with small exponential growth of SL(3, R) on X.This class of Banach spaces contains many superreflexive spaces and some nonreflexive spaces as well. We see no obstruction for this class to be equal to all spaces with nontrivial type.

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“…In particular, this implies that these expanders are superexpanders, since the class of Banach spaces with nontrivial type is known to be larger than the class of uniformly convex spaces. More examples of groups with strong property (T) or other Banach space versions of property (T) relative to various classes of Banach spaces were provided in [11][12][13]16,22,25]. Other examples of superexpanders, obtained by means of a combinatorial construction, were provided in the groundbreaking work of Mendel and Naor [20].…”

Section: Introductionmentioning

confidence: 99%

Superexpanders from group actions on compact manifolds

Laat

Vigolo

2018

Geom Dedicata

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It is known that the expanders arising as increasing sequences of level sets of warped cones, as introduced by the second-named author, do not coarsely embed into a Banach space as soon as the corresponding warped cone does not coarsely embed into this Banach space. Combining this with non-embeddability results for warped cones by Nowak and Sawicki, which relate the non-embeddability of a warped cone to a spectral gap property of the underlying action, we provide new examples of expanders that do not coarsely embed into any Banach space with nontrivial type. Moreover, we prove that these expanders are not coarsely equivalent to a Lafforgue expander. In particular, we provide infinitely many coarsely distinct superexpanders that are not Lafforgue expanders. In addition, we prove a quasi-isometric rigidity result for warped cones.

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Strong Banach property (T) for simple algebraic groups of higher rank

Cited by 33 publications

References 11 publications

Strong property (T) for higher-rank simple Lie groups: Figure 1.

Strong property (T) for higher-rank simple Lie groups: Figure 1.

Towards strong Banach property (T) for SL(3,ℝ)

Superexpanders from group actions on compact manifolds

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