On the isoperimetry of graphs with many ends

Christophe Pittet

Colloquium Mathematicae (1998)

  • Volume: 78, Issue: 2, page 307-318
  • ISSN: 0010-1354

Abstract

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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 l 2 -cohomology of X coincides with the reduced l 2 -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.)

How to cite

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Pittet, Christophe. "On the isoperimetry of graphs with many ends." Colloquium Mathematicae 78.2 (1998): 307-318. <http://eudml.org/doc/210617>.

@article{Pittet1998,
abstract = {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 $l^2$-cohomology of X coincides with the reduced $l^2$-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.)},
author = {Pittet, Christophe},
journal = {Colloquium Mathematicae},
keywords = {cohomology; connected graph; radius; isoperimetric inequality; Cayley graph; ends},
language = {eng},
number = {2},
pages = {307-318},
title = {On the isoperimetry of graphs with many ends},
url = {http://eudml.org/doc/210617},
volume = {78},
year = {1998},
}

TY - JOUR
AU - Pittet, Christophe
TI - On the isoperimetry of graphs with many ends
JO - Colloquium Mathematicae
PY - 1998
VL - 78
IS - 2
SP - 307
EP - 318
AB - 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 $l^2$-cohomology of X coincides with the reduced $l^2$-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.)
LA - eng
KW - cohomology; connected graph; radius; isoperimetric inequality; Cayley graph; ends
UR - http://eudml.org/doc/210617
ER -

References

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  12. [St] J. Stallings, Group Theory and Three-Dimensional Manifolds, Yale Math. Monographs 4, Yale Univ. Press, 1971. Zbl0241.57001
  13. [SW] P. Soardi and W. Woess, Amenability, unimodularity, and the spectral radius of random walks on infinite graphs, Math. Z. 205 (1990), 471-486. Zbl0693.43001

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