Geodesic
Connectedness of spheres in Cayley graphs
Algebra and discrete mathematics, Tome 26 (2018) no. 2, pp. 190-246

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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. 
@article{ADM_2018_26_2_a2,
     author = {J\'er\'emie Brieussel and Antoine Gournay},
     title = {Connectedness of spheres in {Cayley} graphs},
     journal = {Algebra and discrete mathematics},
     pages = {190--246},
     publisher = {mathdoc},
     volume = {26},
     number = {2},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2018_26_2_a2/}
}
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Jérémie Brieussel; Antoine Gournay. Connectedness of spheres in Cayley graphs. Algebra and discrete mathematics, Tome 26 (2018) no. 2, pp. 190-246. http://geodesic.mathdoc.fr/item/ADM_2018_26_2_a2/