On the length of non-solutions to equations with constants in some linear groups

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