Homomorphisms to ℝ constructed from random walks

Anna Erschler[1]; Anders Karlsson[2]

  • [1] Université Paris Sud Laboratoire de Mathématiques d’Orsay Bâtiment 425 91405 Orsay Cedex (France)
  • [2] Université de Genève Section de mathématiques 2-4 rue du Lièvre Case postale 64 1211 Genève 4 (Suisse)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 6, page 2095-2113
  • ISSN: 0373-0956

Abstract

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We give a construction of homomorphisms from a group into the reals using random walks on the group. The construction is an alternative to an earlier construction that works in more general situations. Applications include an estimate on the drift of random walks on groups of subexponential growth admitting no nontrivial homomorphism to the integers and inequalities between the asymptotic drift and the asymptotic entropy. Some of the entropy estimates obtained have applications independent of the homomorphism construction, for example a Liouville-type theorem for slowly growing harmonic functions on groups of subexponential growth and on some groups of exponential growth.

How to cite

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Erschler, Anna, and Karlsson, Anders. "Homomorphisms to $\mathbb{R}$ constructed from random walks." Annales de l’institut Fourier 60.6 (2010): 2095-2113. <http://eudml.org/doc/116326>.

@article{Erschler2010,
abstract = {We give a construction of homomorphisms from a group into the reals using random walks on the group. The construction is an alternative to an earlier construction that works in more general situations. Applications include an estimate on the drift of random walks on groups of subexponential growth admitting no nontrivial homomorphism to the integers and inequalities between the asymptotic drift and the asymptotic entropy. Some of the entropy estimates obtained have applications independent of the homomorphism construction, for example a Liouville-type theorem for slowly growing harmonic functions on groups of subexponential growth and on some groups of exponential growth.},
affiliation = {Université Paris Sud Laboratoire de Mathématiques d’Orsay Bâtiment 425 91405 Orsay Cedex (France); Université de Genève Section de mathématiques 2-4 rue du Lièvre Case postale 64 1211 Genève 4 (Suisse)},
author = {Erschler, Anna, Karlsson, Anders},
journal = {Annales de l’institut Fourier},
keywords = {Random walks on groups; Liouville type theorems; growth of harmonic functions; homomorphisms to $\mathbb\{R\}$; groups of intermediate growth; entropy; drift; Gaussian estimates; random walks on groups; homomorphisms to },
language = {eng},
number = {6},
pages = {2095-2113},
publisher = {Association des Annales de l’institut Fourier},
title = {Homomorphisms to $\mathbb\{R\}$ constructed from random walks},
url = {http://eudml.org/doc/116326},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Erschler, Anna
AU - Karlsson, Anders
TI - Homomorphisms to $\mathbb{R}$ constructed from random walks
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 6
SP - 2095
EP - 2113
AB - We give a construction of homomorphisms from a group into the reals using random walks on the group. The construction is an alternative to an earlier construction that works in more general situations. Applications include an estimate on the drift of random walks on groups of subexponential growth admitting no nontrivial homomorphism to the integers and inequalities between the asymptotic drift and the asymptotic entropy. Some of the entropy estimates obtained have applications independent of the homomorphism construction, for example a Liouville-type theorem for slowly growing harmonic functions on groups of subexponential growth and on some groups of exponential growth.
LA - eng
KW - Random walks on groups; Liouville type theorems; growth of harmonic functions; homomorphisms to $\mathbb{R}$; groups of intermediate growth; entropy; drift; Gaussian estimates; random walks on groups; homomorphisms to
UR - http://eudml.org/doc/116326
ER -

References

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