A geometric approach to on-diagonal heat kernel lower bounds on groups. (English) Zbl 1137.58307
Summary: 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.
MSC:
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
| 60G50 | Sums of independent random variables; random walks |
| 22E30 | Analysis on real and complex Lie groups |
| 20E22 | Extensions, wreath products, and other compositions of groups |
Keywords:
heat kernels on manifolds; random walks on graphs; Følner sets; first eigenvalue for the Dirichlet problem; Lie groups; finitely generated groupsReferences:
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