On groups and fields interpretable in $\mathrm{NTP_2}$ fields

TL;DR

The paper develops a cohesive model-theoretic framework to study interpretable fields and definably amenable groups in tame settings such as $NIP$ and $NTP_2$ fields. Central to the approach is an abstract stabilizer-based construction that yields definable group homomorphisms from generic data, together with a relative stable group configuration tool for handling imaginaries. Under algebraic boundedness and either $NIP$ or $NTP_2$ with bounded Galois group hypotheses, the authors obtain structure theorems showing that definably amenable groups map to algebraic groups with purely imaginary kernels, which in turn implies strong classification results for interpretable fields in DCVF and in finitely ramified henselian valued fields of characteristic $0$. The results unify and extend known classifications, providing explicit embeddings and rigidity phenomena for interpretable objects across differential, valued, and henselian contexts, with significant implications for understanding the model-theoretic anatomy of these theories. The methodology hinges on stabilizers, f-generics, and group configurations, offering a versatile toolkit for future investigations into interpretable algebraic structures in broad tame frameworks.

Abstract

This paper aims at developing model-theoretic tools to study interpretable fields and definably amenable groups, mainly in $\mathrm{NIP}$ or $\mathrm{NTP_2}$ settings. An abstract theorem constructing definable group homomorphisms from generic data is proved. It relies heavily on a stabilizer theorem of Montenegro, Onshuus and Simon. The main application is a structure theorem for definably amenable groups that are interpretable in algebraically bounded perfect $\mathrm{NTP_2}$ fields with bounded Galois group (under some mild assumption on the imaginaries involved), or in algebraically bounded theories of (differential) NIP fields. These imply a classification of the fields interpretable in differentially closed valued fields, and structure theorems for fields interpretable in finitely ramified henselian valued fields of characteristic $0$, or in NIP algebraically bounded differential fields.

Theorems & Definitions (165)

• Theorem 1.1: see Theorem A in HruPil-GpPFF
• Theorem 1.2: see Theorem C in HruPil-GpPFF
• Theorem 1.3: Theorem \ref{['theo_group_with_kernel_blind_to_S']}
• Theorem 1.4: Theorem \ref{['theo_general_morphism_with_imaginary_kernel']}
• Theorem 1.5: Corollary \ref{['coro_fields_are_either_pur_im_or_algebraic']}
• Theorem 1.6: Theorem \ref{['theo_groups_in_algebraically_bounded_differential_fields']}
• Theorem 1.7: Theorem \ref{['theo_fields_hen_0']}
• Definition 2.1
• Definition 2.2
• Definition 2.4