The infinitesimal subgroup of interpretable groups in some dp-minimal valued fields

2025/02/12 by Yatir Halevi, Halevi, Yatir, Assaf Hasson +3
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2502.08163

openalex publication_date 2025/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We continue our local analysis of groups interpretable in various dp-minimal valued fields, as introduced in [8]. We associate with every infinite group G interpretable in those fields an infinite type-definable infinitesimal subgroup ν(G), generated by the four infinitesimal subgroups ν_D(G) associated with the distinguished sorts K, k, Γ and K/O. To show that ν(G) is type-definable, we show that the resulting subgroups ν_D(G) commute with each other as D ranges over the four distinguished sorts. We then study the basic properties of ν(G). Among others, we show that ν(G₁× G₂)=ν(G₁)× ν(G₂) and that if G₁≤ G is a definable subgroup then ν(G₁) is relatively definable in ν(G). We also discuss possible connections between dp-rk(ν(G)) and elimination of imaginaries.

Related

• On groups interpretable in various valued fields
• Semisimple groups interpretable in various valued fields
• Definably semisimple groups interpretable in p-adically closed fields
• On groups and fields interpretable in NTP₂ fields
• Interpretable Fields in Various Valued Fields