Abstract A mixed equation in a group G is given by a non-trivial element w (x) of the free product G * Z G ∗ Z, and a solution is some g G g ∈ G such that w (g) is the identity. For G acylindrically hyperbolic with trivial finite radical (e. g. torsion-free), we show that any mixed equation of length n has a non-solution of length comparable to (n) log (n), which is the best possible bound. Similarly, we show that there is a common non-solution of length O (n) to all mixed equations of length n, again the best possible bound. In fact, in both cases, we show that a random walk of appropriate length yields a non-solution with positive probability.
Bradford et al. (Tue,) studied this question.