Finite dimensional amenable groups

2025/08/27 by Anna Erschler, Ivan Mitrofanov, Erschler, Anna +1
Computer Science · Mathematics · #Cellular Automata and Applications #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2508.20296

openalex publication_date 2025/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that an amenable group of finite Assouad-Nagata dimension satisfies the property H_FD of Shalom. Such infinite groups are known to admit a virtual homomorphism onto ℤ, and thus our result implies that an amenable group of finite AN-dimension cannot be a simple group. We can also conclude that an amenable group of finite AN-dimension cannot be a torsion group. Our proof is based on new estimates of diameters of Følner couples. We prove that any amenable group of finite AN-dimension admits Følner couples inside balls of linear diameter and more generally estimate the radius of the balls containing Følner couples in groups of finite asymptotic dimension. This result strengthens the result of Nowak about diameters of Følner sets.

Citations

• Small-ball estimates for random walks on groups
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Related

• Elementary amenable groups of cohomological dimension 3
• Dimension of elementary amenable groups
• Homological Dimension of Solvable Groups
• A groupoid approach to Lück's amenability conjecture
• Compactifications of pseudofinite and pseudo-amenable groups