Non-solutions to mixed equations in acylindrically hyperbolic groups coming from random walks

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 log (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.