## Abstract

Kleiner’s theorem is the assertion that for a finitely generated group of polynomial growth, the spaces of polynomially growing harmonic functions are finite dimensional. Kleiner used this to provide a new proof for Gromov’s theorem: polynomial growth of a group is equivalent to it being virtually nilpotent. In this work we study the structure of finitely generated groups for which a

space of harmonic functions with fixed polynomial growth is finite dimensional. It is conjectured that such groups must be virtually nilpotent (the converse direction to Kleiner’s theorem). We prove that this is indeed the case for linear groups. For non-linear groups with this property, we describe the structure of the linearly growing harmonic functions, providing evidence that the converse direction to Kleiner’s theorem may indeed hold in general.

space of harmonic functions with fixed polynomial growth is finite dimensional. It is conjectured that such groups must be virtually nilpotent (the converse direction to Kleiner’s theorem). We prove that this is indeed the case for linear groups. For non-linear groups with this property, we describe the structure of the linearly growing harmonic functions, providing evidence that the converse direction to Kleiner’s theorem may indeed hold in general.

Original language | English |
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Publisher | arXiv:1408.6243 |

State | Published - 2014 |