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

2025/04/21 by Bradford, Henry, Sisto, Alessandro · 6 citations
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2504.15456

Abstract

A mixed equation in a group G is given by a non-trivial element w (x) of the free product G ∗ ℤ, and a solution is some 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), 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.

Citations

• Mixed identities in linear groups -- effective version
• Structural properties of reduced C^*-algebras associated with higher-rank lattices
• Strict comparison in reduced group C^*-algebras
• On the length of non-solutions to equations with constants in some linear groups
• Some examples of separable convex-cocompact subgroups
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• Groups acting acylindrically on hyperbolic spaces
• Random subgroups of acylindrically hyperbolic groups and hyperbolic embeddings
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• Random walks on weakly hyperbolic groups
• Acylindrically hyperbolic groups
• Acylindrically hyperbolic groups
• Exponential decay in the mapping class group
• Conjugacy classes of solutions to equations and inequations over hyperbolic groups

Cited by

• Boundary dynamics, triple transitivity, and mixed identities in weakly hyperbolic groups
• Mixed identities in linear groups -- effective version

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