Vol. 14, No. 1, 2021

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Banach space actions and $L^2$-spectral gap

Tim de Laat and Mikael de la Salle

Vol. 14 (2021), No. 1, 45–76

Żuk proved that if a finitely generated group admits a Cayley graph such that the Laplacian on the links of this Cayley graph has a spectral gap > 1 2, then the group has property (T), or equivalently, every affine isometric action of the group on a Hilbert space has a fixed point. We prove that the same holds for affine isometric actions of the group on a uniformly curved Banach space (for example an Lp-space with 1 < p < or an interpolation space between a Hilbert space and an arbitrary Banach space) as soon as the Laplacian on the links has a two-sided spectral gap > 1 𝜀.

This criterion applies to random groups in the triangular density model for densities > 1 3. In this way, we are able to generalize recent results of Druţu and Mackay to affine isometric actions of random groups on uniformly curved Banach spaces. Also, in the setting of actions on Lp-spaces, our results are quantitatively stronger, even in the case p = 2. This naturally leads to new estimates on the conformal dimension of the boundary of random groups in the triangular model.

Additionally, we obtain results on the eigenvalues of the p-Laplacian on graphs, and on the spectrum and degree distribution of Erdős–Rényi graphs.

fixed-point properties, affine group actions on Banach spaces, random groups, Erdős–Rényi graphs
Mathematical Subject Classification 2010
Primary: 05C80, 20F65, 20P05, 46B20
Secondary: 20F67, 46B70
Received: 31 May 2018
Revised: 11 August 2019
Accepted: 25 October 2019
Published: 19 February 2021
Tim de Laat
Mathematisches Institut
Westfälische Wilhelms-Universität Münster
Mikael de la Salle