A geometric approach to on-diagonal heat kernel lower bounds on groups

Thierry Coulhon[1]; Alexander Grigor'yan[2]; Christophe Pittet[3]

  • [1] Université de Cergy-Pontoise, Département de Mathématiques, 2 avenue Adolphe Chauvin, 95032 Cergy Cedex (France)
  • [2] Imperial College, London DW7 2BZ (Grande-Bretagne)
  • [3] Université Paul Sabatier, Laboratoire Émile Picard, 118 route de Narbonne, 31062 Toulouse Cedex (France)

Annales de l’institut Fourier (2001)

  • Volume: 51, Issue: 6, page 1763-1827
  • ISSN: 0373-0956

Abstract

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We introduce a new method for obtaining heat kernel on-diagonal lower bounds on non- compact Lie groups and on infinite discrete groups. By using this method, we are able to recover the previously known results for unimodular amenable Lie groups as well as for certain classes of discrete groups including the polycyclic groups, and to give them a geometric interpretation. We also obtain new results for some discrete groups which admit the structure of a semi-direct product or of a wreath product. These include the two- generators groups of affine transformations of the real line 〈 x ↦ x + 1 , x ↦ λ x 〉 with λ algebraic, as well as lamplighter groups with nilpotent base.

How to cite

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Coulhon, Thierry, Grigor'yan, Alexander, and Pittet, Christophe. "A geometric approach to on-diagonal heat kernel lower bounds on groups." Annales de l’institut Fourier 51.6 (2001): 1763-1827. <http://eudml.org/doc/115967>.

@article{Coulhon2001,
abstract = {We introduce a new method for obtaining heat kernel on-diagonal lower bounds on non- compact Lie groups and on infinite discrete groups. By using this method, we are able to recover the previously known results for unimodular amenable Lie groups as well as for certain classes of discrete groups including the polycyclic groups, and to give them a geometric interpretation. We also obtain new results for some discrete groups which admit the structure of a semi-direct product or of a wreath product. These include the two- generators groups of affine transformations of the real line $\langle x\mapsto x+1,x\mapsto \lambda x\rangle $ with $\lambda $ algebraic, as well as lamplighter groups with nilpotent base.},
affiliation = {Université de Cergy-Pontoise, Département de Mathématiques, 2 avenue Adolphe Chauvin, 95032 Cergy Cedex (France); Imperial College, London DW7 2BZ (Grande-Bretagne); Université Paul Sabatier, Laboratoire Émile Picard, 118 route de Narbonne, 31062 Toulouse Cedex (France)},
author = {Coulhon, Thierry, Grigor'yan, Alexander, Pittet, Christophe},
journal = {Annales de l’institut Fourier},
keywords = {heat kernels on manifolds; random walks on graphs; Følner sets; first eigenvalue for the Dirichlet problem; Lie groups; finitely generated groups},
language = {eng},
number = {6},
pages = {1763-1827},
publisher = {Association des Annales de l'Institut Fourier},
title = {A geometric approach to on-diagonal heat kernel lower bounds on groups},
url = {http://eudml.org/doc/115967},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Coulhon, Thierry
AU - Grigor'yan, Alexander
AU - Pittet, Christophe
TI - A geometric approach to on-diagonal heat kernel lower bounds on groups
JO - Annales de l’institut Fourier
PY - 2001
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 6
SP - 1763
EP - 1827
AB - We introduce a new method for obtaining heat kernel on-diagonal lower bounds on non- compact Lie groups and on infinite discrete groups. By using this method, we are able to recover the previously known results for unimodular amenable Lie groups as well as for certain classes of discrete groups including the polycyclic groups, and to give them a geometric interpretation. We also obtain new results for some discrete groups which admit the structure of a semi-direct product or of a wreath product. These include the two- generators groups of affine transformations of the real line $\langle x\mapsto x+1,x\mapsto \lambda x\rangle $ with $\lambda $ algebraic, as well as lamplighter groups with nilpotent base.
LA - eng
KW - heat kernels on manifolds; random walks on graphs; Følner sets; first eigenvalue for the Dirichlet problem; Lie groups; finitely generated groups
UR - http://eudml.org/doc/115967
ER -

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