Dirichlet-Poincar\'{e} profiles of graphs and groups

David Hume

Dirichlet-Poincar\'{e} profiles of graphs and groups

Group Theory 2019-10-16 v1

Authors: David Hume

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 $p=\infty$ , they detect the F\o lner function and the growth function respectively.

Keywords

graph polynomials graph theory dirichlet problem

Cite

@article{arxiv.1910.06835,
  title  = {Dirichlet-Poincar\'{e} profiles of graphs and groups},
  author = {David Hume},
  journal= {arXiv preprint arXiv:1910.06835},
  year   = {2019}
}