Volume of spheres in doubling metric measured spaces and in groups of polynomial growth

Romain Tessera

Bulletin de la Société Mathématique de France (2007)

  • Volume: 135, Issue: 1, page 47-64
  • ISSN: 0037-9484

Abstract

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Let G be a compactly generated locally compact group and let U be a compact generating set. We prove that if G has polynomial growth, then ( U n ) n ∈ ℕ is a Følner sequence and we give a polynomial estimate of the rate of decay of μ ( U n + 1 ∖ U n ) μ ( U n ) . Our proof uses only two ingredients: the doubling property and a weak geodesic property that we call Property (M). As a matter of fact, the result remains true in a wide class of doubling metric measured spaces including manifolds and graphs. As an application, we obtain a L p -pointwise ergodic theorem ( 1 ≤ p < ∞ ) for the balls averages, which holds for any compactly generated locally compact group G of polynomial growth.

How to cite

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Tessera, Romain. "Volume of spheres in doubling metric measured spaces and in groups of polynomial growth." Bulletin de la Société Mathématique de France 135.1 (2007): 47-64. <http://eudml.org/doc/272467>.

@article{Tessera2007,
abstract = {Let $G$ be a compactly generated locally compact group and let $U$ be a compact generating set. We prove that if $G$ has polynomial growth, then $(U^n)_\{n\in \mathbb \{N\}\}$ is a Følner sequence and we give a polynomial estimate of the rate of decay of $ \frac\{\mu (U^\{n+1\}\setminus U^n)\}\{\mu (U^n)\}. $ Our proof uses only two ingredients: the doubling property and a weak geodesic property that we call Property (M). As a matter of fact, the result remains true in a wide class of doubling metric measured spaces including manifolds and graphs. As an application, we obtain a $L^p$-pointwise ergodic theorem ($1\le p&lt;\infty $) for the balls averages, which holds for any compactly generated locally compact group $G$ of polynomial growth.},
author = {Tessera, Romain},
journal = {Bulletin de la Société Mathématique de France},
keywords = {isoperimetry; spheres; locally compact groups; volume growth in groups; metric measure spaces; doubling property},
language = {eng},
number = {1},
pages = {47-64},
publisher = {Société mathématique de France},
title = {Volume of spheres in doubling metric measured spaces and in groups of polynomial growth},
url = {http://eudml.org/doc/272467},
volume = {135},
year = {2007},
}

TY - JOUR
AU - Tessera, Romain
TI - Volume of spheres in doubling metric measured spaces and in groups of polynomial growth
JO - Bulletin de la Société Mathématique de France
PY - 2007
PB - Société mathématique de France
VL - 135
IS - 1
SP - 47
EP - 64
AB - Let $G$ be a compactly generated locally compact group and let $U$ be a compact generating set. We prove that if $G$ has polynomial growth, then $(U^n)_{n\in \mathbb {N}}$ is a Følner sequence and we give a polynomial estimate of the rate of decay of $ \frac{\mu (U^{n+1}\setminus U^n)}{\mu (U^n)}. $ Our proof uses only two ingredients: the doubling property and a weak geodesic property that we call Property (M). As a matter of fact, the result remains true in a wide class of doubling metric measured spaces including manifolds and graphs. As an application, we obtain a $L^p$-pointwise ergodic theorem ($1\le p&lt;\infty $) for the balls averages, which holds for any compactly generated locally compact group $G$ of polynomial growth.
LA - eng
KW - isoperimetry; spheres; locally compact groups; volume growth in groups; metric measure spaces; doubling property
UR - http://eudml.org/doc/272467
ER -

References

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