For a nite group
G
, a
character ratio
is a complex number of the form

(
x
)

(1)
,
where
x
2
G
and

is an irreducible character of
G
. Upper bounds for absolute
values of character ratios, particularly for simple groups, have long been of interest,
for various reasons; these include applications to covering numbers, mixing times
of random walks, and the study of word maps. In this article we shall survey some
results on character ratios for nite groups of Lie type, and their applications.
Character ratios for alternating and symmetric groups have been studied in great
depth also { see for example [
32, 33
] { culminating in the de nitive results and
applications 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 of
G.