Let X be a connected graph with uniformly bounded degree. We show that if there is a radius r such that, by removing from X any ball of radius r, we get at least three unbounded connected components, then X satisfies a strong isoperimetric inequality. In particular, the non-reduced -cohomology of X coincides with the reduced -cohomology of X and is of uncountable dimension. (Those facts are well known when X is the Cayley graph of a finitely generated group with infinitely many ends.)
The isoperimetric inequality |∂Ω| / |Ω| = constant / log |Ω| for finite subsets Ω in a finitely generated group Γ with exponential growth is optimal if Γ is polycyclic.
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....
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