2019/01/08 by Isaac Goldbring, Goldbring, Isaac
Computer Science · Mathematics · #Advanced Operator Algebra Research #Computability, Logic, AI Algorithms #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #math.GR #math.LO
paper · pdf · doi:10.48550/arxiv.1901.02445
arxiv created 2019/01/08 · openalex publication_date 2019/01/08 · arxiv updated 2019/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that there is n∈ \mathbf N, a finite system Σ( x, y) of equations and inequations having a solution in some group, where x has length n, and ε>0 such that: for any group G and any a∈ Gn, if the system Σ( a, y) has a solution in G, then there is no ( a,ε)-Fø lner set in G. The proof uses ideas from model-theoretic forcing together with the observation that no amenable group can be existentially closed. Along the way, we also observe that no existentially closed group can be exact, have the Haagerup property, or have property (T). Finally, we show that, for n large enough and for ε small enough, the existence of (F,ε)-Fø lner sets, where F has size at most n, cannot be expressed in a first-order way uniformly in all groups.