Dirichlet-Poincar\'e profiles of graphs and groups
Abstract
We define Poincar\'e profiles of Dirichlet type for graphs of bounded degree, in analogy with the Poincar\'e profiles (of Neumann type) defined in [HMT19]. The obvious first definition yields nothing of interest, but an alternative definition yields a spectrum of profiles which are quasi-isometry invariants and monotone with respect to subgroup inclusion. Moreover, in the extremal cases p=1 and p=∞, they detect the F lner function and the growth function respectively.
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