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@article{Liebeck:2017:conm/694,
author = {Liebeck, MW},
doi = {conm/694},
journal = {Contemporary Mathematics},
title = {Character ratios for finite groups of Lie type, and applications},
url = {http://dx.doi.org/10.1090/conm/694},
volume = {694},
year = {2017}
}

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TY  - JOUR
AB  - For a  nite groupG, acharacter  ratiois a complex number of the form (x) (1),wherex2Gand is an irreducible character ofG.  Upper bounds for absolutevalues of character ratios, particularly for simple groups, have long been of interest,for various reasons; these include applications to covering numbers, mixing timesof random walks, and the study of word maps.  In this article we shall survey someresults  on  character  ratios  for   nite  groups  of  Lie  type,  and  their  applications.Character ratios for alternating and symmetric groups have been studied in greatdepth also { see for example [32, 33] { culminating in the de nitive results andapplications to be found in [20]; but we shall not discuss these here.It is not hard to see the connections between character ratios and group struc-ture.  Here are three well known, elementary results illustrating these connections.The  rst two go back to Frobenius.  Denote by Irr(G) the set of irreducible charac-ters ofG.
AU  - Liebeck,MW
DO  - conm/694
PY  - 2017///
SN  - 0271-4132
TI  - Character ratios for finite groups of Lie type, and applications
T2  - Contemporary Mathematics
UR  - http://dx.doi.org/10.1090/conm/694
UR  - http://hdl.handle.net/10044/1/43131
VL  - 694
ER  -

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