Let
$w\in {F}_{k}$ be a nontrivial word
and denote by
$w\left(G\right)\subseteq G$ the image of
the associated word map
$w:{G}^{k}\to G$.
Let
$G$ be one of the
finite groups
${S}_{n},{GL}_{n}\left(q\right),{Sp}_{2{n}^{\prime}}\left(q\right),{GO}_{2{n}^{\prime}}^{\pm}\left(q\right),{GO}_{2{n}^{\prime}+1}\left(q\right),{GU}_{n}\left(q\right)$
($q$ a prime
power,
$n\ge 2$,
${n}^{\prime}\ge 1$), or the
unitary group
${U}_{n}$
over
$\u2102$.
Let
${d}_{G}$
be the normalized Hamming metric resp. the normalized rank metric on
$G$ when
$G$
is a symmetric group resp. one of the other classical groups and write
$n\left(G\right)$ for the
degree of
$G$.
For
$\mathit{\epsilon}>0$, we prove that
there exists an integer
$N\left(\mathit{\epsilon},w\right)$
such that
$w\left(G\right)$
is
$\mathit{\epsilon}$dense in
$G$ with respect
to the metric
${d}_{G}$
if
$n\left(G\right)\ge N\left(\mathit{\epsilon},w\right)$.
This confirms metric versions of conjectures by Shalev and Larsen. Equivalently, we
prove that any nontrivial word map is surjective on a metric ultraproduct of groups
$G$ from above
such that
$n\left(G\right)\to \infty $
along the ultrafilter.
As a consequence of our methods, we also obtain an
alternative proof of the result of Hui, Larsen, and Shalev that
${w}_{1}\left({SU}_{n}\right){w}_{2}\left({SU}_{n}\right)={SU}_{n}$ for nontrivial
words
${w}_{1},{w}_{2}\in {F}_{k}$
and
$n$
sufficiently large.
