Transience of algebraic varieties in linear groups - applications to generic Zariski density

Richard Aoun[1]

  • [1] Université Paris Sud 11 Laboratoire de Mathématiques Bâtiment 425 91405 Orsay (France) Département de Mathématiques Faculté des Sciences de l’Université Saint-Joseph Campus des Sciences et Technologies B.P. 11-514 Riad El Solh Beyrouth 1107 205 (Liban)

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 5, page 2049-2080
  • ISSN: 0373-0956

Abstract

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We study the transience of algebraic varieties in linear groups. In particular, we show that a “non elementary” random walk in S L 2 ( ℝ ) escapes exponentially fast from every proper algebraic subvariety. We also treat the case where the random walk takes place in the real points of a semisimple split algebraic group and show such a result for a wide family of random walks.As an application, we prove that generic subgroups (in some sense) of linear groups are Zariski dense.

How to cite

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Aoun, Richard. "Transience of algebraic varieties in linear groups - applications to generic Zariski density." Annales de l’institut Fourier 63.5 (2013): 2049-2080. <http://eudml.org/doc/275633>.

@article{Aoun2013,
abstract = {We study the transience of algebraic varieties in linear groups. In particular, we show that a “non elementary” random walk in $SL_2(\{\mathbb\{R\}\})$ escapes exponentially fast from every proper algebraic subvariety. We also treat the case where the random walk takes place in the real points of a semisimple split algebraic group and show such a result for a wide family of random walks.As an application, we prove that generic subgroups (in some sense) of linear groups are Zariski dense.},
affiliation = {Université Paris Sud 11 Laboratoire de Mathématiques Bâtiment 425 91405 Orsay (France) Département de Mathématiques Faculté des Sciences de l’Université Saint-Joseph Campus des Sciences et Technologies B.P. 11-514 Riad El Solh Beyrouth 1107 205 (Liban)},
author = {Aoun, Richard},
journal = {Annales de l’institut Fourier},
keywords = {transience; algebraic varieties; Zariski density; random matrix products; random walks; probability of return; linear groups},
language = {eng},
number = {5},
pages = {2049-2080},
publisher = {Association des Annales de l’institut Fourier},
title = {Transience of algebraic varieties in linear groups - applications to generic Zariski density},
url = {http://eudml.org/doc/275633},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Aoun, Richard
TI - Transience of algebraic varieties in linear groups - applications to generic Zariski density
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 5
SP - 2049
EP - 2080
AB - We study the transience of algebraic varieties in linear groups. In particular, we show that a “non elementary” random walk in $SL_2({\mathbb{R}})$ escapes exponentially fast from every proper algebraic subvariety. We also treat the case where the random walk takes place in the real points of a semisimple split algebraic group and show such a result for a wide family of random walks.As an application, we prove that generic subgroups (in some sense) of linear groups are Zariski dense.
LA - eng
KW - transience; algebraic varieties; Zariski density; random matrix products; random walks; probability of return; linear groups
UR - http://eudml.org/doc/275633
ER -

References

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