Non-solutions to mixed equations in acylindrically hyperbolic groups coming from random walks
Henry Bradford, Alessandro Sisto
TL;DR
This work studies mixed identities in finitely generated acylindrically hyperbolic groups with trivial finite radical by quantifying shortest non-solutions via MIF-growth and proving the existence of non-solutions produced by random walks. The authors establish that such groups are sharply MIF with $ \mathcal{M}_G^S(n)=O(\log n)$, and they further show that random walks yield simultaneous non-solutions to all mixed identities of length $n$ with length bounds $O(n)$, which is optimal. The main technical tool is a concatenation lemma in hyperbolic spaces together with Maher–Tiozzo linear progress results, enabling bi-infinite quasi-geodesic constructions that certify non-triviality of mixed words. They also connect these results to the selfless-ness framework, proving acylindrically hyperbolic groups with trivial finite radical are selfless with $f(n)=O(n^2)$ and outlining open questions on possible improvements. Together, these results provide sharp probabilistic and geometric insights into mixed identities and their non-solutions in a broad and important class of groups, with implications for operator-algebraic questions.
Abstract
A mixed equation in a group $G$ is given by a non-trivial element $w (x)$ of the free product $G \ast \mathbb{Z}$, and a solution is some $g\in G$ 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 $\log(n)$, 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.