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

Coulhon, Thierry; Grigor'yan, Alexander; Pittet, Christophe

Annales de l'Institut Fourier, Tome 51 (2001) no. 6, pp. 1763-1827.

On introduit une nouvelle méthode pour minorer sur la diagonale les noyaux de la chaleur des groupes de Lie non-compacts et des groupes infinis de type fini. Cette méthode permet de retrouver les bornes inférieures optimales pour les groupes de Lie unimodulaires moyennables et pour certains groupes de type fini, parmi lesquels les groupes polycycliques. Elle permet aussi de donner une interprétation géométrique de ces résultats. On obtient des résultats nouveaux pour certains groupes discrets admettant une structure de produit semi-direct avec groupe quotient abélien ou nilpotent. Parmi ces groupes, on trouvera ceux des transformations affines de la droite réelle engendrés par la translation $\langle x\mapsto x+1\rangle$ et une homothétie $\langle x\mapsto \lambda x\rangle$ avec $\lambda$ algébrique. On trouvera aussi certains produits en couronnes, comme les groupes d’allumeurs de réverbères à base nilpotente.

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.

Classification : 58J35, 60G50, 22E30, 20E22
Keywords: heat kernels on manifolds, random walks on graphs, Følner sets, first eigenvalue for the Dirichlet problem, Lie groups, finitely generated groups
Mot clés : noyaux de la chaleur sur les varietés, marches aléatoires sur les graphes, ensembles de Følner, première valeur propre pour le problème de dirichlet, groupes de Lie, groupes finiment engendrés

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