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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2018, Volume 26, Issue 2, Pages 190–246 (Mi adm680)

RESEARCH ARTICLE

Connectedness of spheres in Cayley graphs

Jérémie Brieussela, Antoine Gournayb

a Université de Montpellier, Institut Montpelliérain Alexander Grothendieck, 34095 Montpellier, France
b Institut für Geometrie, Fachrichtung Mathematik, TU Dresden, 01062 Dresden, Germany

Abstract: We introduce the notion of connection thickness of spheres in a Cayley graph, related to dead-ends and their retreat depth. It was well-known that connection thickness is bounded for finitely presented one-ended groups. We compute that for natural generating sets of lamplighter groups on a line or on a tree, connection thickness is linear or logarithmic respectively. We show that it depends strongly on the generating set. We give an example where the metric induced at the (finite) thickness of connection gives diameter of order $n^2$ to the sphere of radius $n$. We also discuss the rarity of dead-ends and the relationships of connection thickness with cut sets in percolation theory and with almost-convexity. Finally, we present a list of open questions about spheres in Cayley graphs.

Keywords: connectedness, sphere, connection thickness, Cayley graphs, dead-ends, wreath products.

 Funding Agency Grant Number European Union's Seventh Framework Programme 277728 Supported by the ERC-StG 277728 “GeomAnGroup”.

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MSC: 20F65
Revised: 12.11.2018
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Citation: Jérémie Brieussel, Antoine Gournay, “Connectedness of spheres in Cayley graphs”, Algebra Discrete Math., 26:2 (2018), 190–246

Citation in format AMSBIB
\Bibitem{BriGou18} \by J\'er\'emie~Brieussel, Antoine~Gournay \paper Connectedness of spheres in Cayley graphs \jour Algebra Discrete Math. \yr 2018 \vol 26 \issue 2 \pages 190--246 \mathnet{http://mi.mathnet.ru/adm680}