Non-solutions to mixed equations in acylindrically hyperbolic groups coming from random walks
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917205339275264 |
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| author | Bradford, Henry Sisto, Alessandro |
| author_facet | Bradford, Henry Sisto, Alessandro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-solutions to mixed equations in acylindrically hyperbolic groups coming from random walks Bradford, Henry Sisto, Alessandro Group Theory 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. |
| title | Non-solutions to mixed equations in acylindrically hyperbolic groups coming from random walks |
| topic | Group Theory |
| url | https://arxiv.org/abs/2504.15456 |