Harmonic functions with finite $p$-energy on lamplighter graphs are constant
| Autor: | Gournay, Antoine |
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| Rok vydání: | 2015 |
| Předmět: | |
| Zdroj: | Comptes Rendus Mathematique Volume 354, Issue 8, August 2016, Pages 762-765 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.crma.2014.05.005 |
| Popis: | The aim of this note is to show that lamplighter graphs where the space graph is infinite and at most two-ended and the lamp graph is at most two-ended do not admit harmonic functions with gradients in $\ell^p$ (\ie finite $p$-energy) for any $p\in [1,\infty[$ except constants (and, equivalently, that their reduced $\ell^p$ cohomology is trivial in degree one). Using similar arguments, it is also shown that many direct products of graphs (including all direct products of Cayley graphs) do not admit non-constant harmonic function with gradient in $\ell^p$. The proof relies on a theorem of Thomassen on spanning lines in squares of graphs. Comment: 6 pages |
| Databáze: | arXiv |
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