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

We show that for any finite‐rank–free group , any word‐equation in one variable of length with constants in fails to be satisfied by some element of of word‐length . By a result of the first author, this logarithmic bound cannot be improved upon for any finitely generated group . Beyond free groups, our method (and the logarithmic bound) applies to a class of groups including for all , and the fundamental groups of all closed hyperbolic surfaces and 3‐manifolds. Finally, using a construction of Nekrashevych, we exhibit a finitely generated group and a sequence of word‐equations with constants in for which every nonsolution in is of word‐length strictly greater than logarithmic.