This is an announcement for the paper “On the bi-Lipschitz geometry of lamplighter graphs” by Florent P. Baudierhttps://arxiv.org/search/math?searchtype=author&query=Baudier%2C+F+P, Pavlos Motakishttps://arxiv.org/search/math?searchtype=author&query=Motakis%2C+P, Thomas Schlumprechthttps://arxiv.org/search/math?searchtype=author&query=Schlumprecht%2C+T, András Zsákhttps://arxiv.org/search/math?searchtype=author&query=Zs%C3%A1k%2C+A.
Abstract: In this article we start a systematic study of the bi-Lipschitz geometry of lamplighter graphs. We prove that lamplighter graphs over trees bi-Lipschitzly embed into Hamming cubes with distortion at most $6$. It follows that lamplighter graphs over countable trees bi-Lipschitzly embed into $\ell_1$. We study the metric behaviour of the operation of taking the lamplighter graph over the vertex-coalescence of two graphs. Based on this analysis, we provide metric characterizations of superreflexivity in terms of lamplighter graphs over star graphs or rose graphs. Finally, we show that the presence of a clique in a graph implies the presence of a Hamming cube in the lamplighter graph over it.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1902.07098