Random groups, random graphs and eigenvalues of p-Laplacians

Cornelia Druţu*, John M. Mackay

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)
202 Downloads (Pure)

Abstract

We prove that a random group in the triangular density model has, for densitylarger than 1/3, fixed point properties for actions on Lp-spaces (affineisometric, and more generally (2−2ϵ)1/2p-uniformly Lipschitz) withp varying in an interval increasing with the set of generators. In the samemodel, we establish a double inequality between the maximal p for whichLp-fixed point properties hold and the conformal dimension of the boundary.

In the Gromov density model, we prove that for every p0∈[2,∞)for a sufficiently large number of generators and for any density larger than1/3, a random group satisfies the fixed point property for affine actions onLp-spaces that are (2−2ϵ)1/2p-uniformly Lipschitz, and this forevery p∈[2,p0].

To accomplish these goals we find new bounds on the first eigenvalue of thep-Laplacian on random graphs, using methods adapted from Kahn and Szemeredi'sapproach to the 2-Laplacian. These in turn lead to fixed point properties usingarguments of Bourdon and Gromov, which extend to Lp-spaces previous resultsfor Kazhdan's Property (T) established by Zuk and Ballmann-Swiatkowski.
Original languageEnglish
Pages (from-to)188-254
Number of pages67
JournalAdvances in Mathematics
Volume341
Early online date30 Oct 2018
DOIs
Publication statusPublished - 7 Jan 2019

Keywords

  • Random groups
  • Property (T)
  • Fixed point properties
  • Conformal dimension
  • Expanders
  • Boundary of hyperbolic groups

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