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

Abstract A mixed equation in a group G is given by a non-trivial element w ( x ) of the free product $$G *\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> , and a solution is some $$g\in G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> 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 $$\log (n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>log</mml:mo> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , 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.