Let X be a connected graph with uniformly bounded degree. We show that if there is a radius r such that, by removing from X any ball of radius r, we get at least three unbounded connected components, then X satisfies a strong isoperimetric inequality. In particular, the non-reduced -cohomology of X coincides with the reduced -cohomology of X and is of uncountable dimension. (Those facts are well known when X is the Cayley graph of a finitely generated group with infinitely many ends.)
@article{bwmeta1.element.bwnjournal-article-cmv78z2p307bwm,
author = {Christophe Pittet},
title = {On the isoperimetry of graphs with many ends},
journal = {Colloquium Mathematicae},
volume = {78},
year = {1998},
pages = {307-318},
zbl = {0923.05028},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv78z2p307bwm}
}
Pittet, Christophe. On the isoperimetry of graphs with many ends. Colloquium Mathematicae, Tome 78 (1998) pp. 307-318. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv78z2p307bwm/
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