We prove that if
L
L
is a finite simple group of Lie type and
A
A
a set of generators of
L
L
, then either
A
A
grows, i.e.,
|
A
3
|
>
|
A
|
1
+
ε
|A^3| > |A|^{1+\varepsilon }
where
ε
\varepsilon
depends only on the Lie rank of
L
L
, or
A
3
=
L
A^3=L
. This implies that for simple groups of Lie type of bounded rank a well-known conjecture of Babai holds, i.e., the diameter of any Cayley graph is polylogarithmic. We also obtain new families of expanders.
A generalization of our proof yields the following. Let
A
A
be a finite subset of
S
L
(
n
,
F
)
SL(n,\mathbb {F})
,
F
\mathbb {F}
an arbitrary field, satisfying
|
A
3
|
≤
K
|
A
|
\big |A^3\big |\le \mathcal {K}|A|
. Then
A
A
can be covered by
K
m
\mathcal {K}^m
, i.e., polynomially many, cosets of a virtually soluble subgroup of
S
L
(
n
,
F
)
SL(n,\mathbb {F})
which is normalized by
A
A
, where
m
m
depends on
n
n
.