Abstract
Let
S
⊂
GL
n
(
Z
)
S\subset\mathrm{GL}_{n}(\mathbb{Z})
be a finite symmetric set.
We show that if the Zariski closure of
Γ
=
⟨
S
⟩
\Gamma=\langle S\rangle
is a product of special linear groups or a special affine linear group, then the diameter of the Cayley graph
Cay
(
Γ
/
Γ
(
q
)
,
π
q
(
S
)
)
\operatorname{Cay}(\Gamma/\Gamma(q),\pi_{q}(S))
is
O
(
log
q
)
O(\log q)
, where 𝑞 is an arbitrary positive integer,
π
q
:
Γ
→
Γ
/
Γ
(
q
)
\pi_{q}\colon\Gamma\to\Gamma/\Gamma(q)
is the canonical projection induced by the reduction modulo 𝑞, and the implied constant depends only on 𝑆.