On the length of nonsolutions to equations with constants in some linear groups

Abstract

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 $O\left(\log \right(n\left)\right)$. 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 ${PSL}_d\left(Z\right)$ for all $d⩾2$, 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 nonsolution in $\Gamma$ is of word-length strictly greater than logarithmic.