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

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.