Abstract
We show that for any finite‐rank–free group
Γ
$\Gamma$
, any word‐equation in one variable of length
n
$n$
with constants in
Γ
$\Gamma$
fails to be satisfied by some element of
Γ
$\Gamma$
of word‐length
O
(
log
(
n
)
)
$O(\log (n))$
. By a result of the first author, this logarithmic bound cannot be improved upon for any finitely generated group
Γ
$\Gamma$
. Beyond free groups, our method (and the logarithmic bound) applies to a class of groups including
PSL
d
(
Z
)
$\operatorname{PSL}_d(\mathbb {Z})$
for all
d
⩾
2
$d \geqslant 2$
, and the fundamental groups of all closed hyperbolic surfaces and 3‐manifolds. Finally, using a construction of Nekrashevych, we exhibit a finitely generated group
Γ
$\Gamma$
and a sequence of word‐equations with constants in
Γ
$\Gamma$
for which every nonsolution in
Γ
$\Gamma$
is of word‐length strictly greater than logarithmic.