Planar lamplighter is not of negative type

Gioacchino Antonelli, Emanuele Caputo, Nicola Cavallucci, Luca Nalon, Assaf Naor, Pietro Wald

Abstract

The lamplighter group over the planar integer grid is proved to not be bi-Lipschitz equivalent to any metric space of negative type, so in particular it does not admit a bi-Lipschitz embedding into $L_1$. This shows the existence of finitely generated metabelian groups on which word metrics are never comparable up to constant factors to conditionally negative definite (CND) kernels, and that the property of admitting a word metric-comparable CND kernel is not preserved by wreath products.

Disclosure

“on Analysis and Geometry in Metric Spaces (Trento, June 2026), where he recalled Problem 1.2 because he suspected that the method of the breakthrough [GO26] should be relevant to it. G.A., E.C., N.C., L.N., P.W. asked (not involving A.N.) ChatGPT-5.5 Pro for assistance. It suggested an argument which indeed cleverly considers steps and objects that are in the spirit of [GO26]. After G.A., E.C., N.C., L.N., P.W. wrote the resulting proof, they sent it to A.N., who proceeded by build”
PDF page 5

Classification: Proof ideas or individual proof-step assistance
Multiplier: 8
Verified: 2026-08-23T05:20:04.129186Z

Structural counts

Pages 16 pdf

Theorems 3 source

Lemmas 5 source

Propositions 3 source

Corollaries 0 source

Definitions 0 source

Displayed equations 69 source

Bibliography entries 105 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file Paper.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.