Zilber's Trichotomy in Hausdorff Geometric Structures
Benjamin Castle, Assaf Hasson, Jinhe Ye
TL;DR
This work develops a novel axiomatic framework of Hausdorff Geometric Structures (HGS) to advance Zilber’s Restricted Trichotomy for relics of algebraically closed fields, with $ACVF$ as the primary testbed. The strategy combines three pillars: (i) an axiomatic setup for taming topology via open maps and differentiability, (ii) a reduction showing that non-locally modular strongly minimal relics interpret a one-dimensional group, and (iii) a slope-based group-construction mechanism that translates geometric configurations into definable groups. Under ramification purity and TIMI (tangent intersections are multiple intersections), the theory proves that a non-locally modular strongly minimal $K$-relic has $ ext{dim}(M)=1$ and interprets a strongly minimal group, yielding the trichotomy for relics of $ACVF$ and related structures; the imaginary-sorts case reduces to a technical conjecture in characteristic $(0,0)$ that the authors verify. The framework not only encapsulates ACVF but also applies to o-minimal expansions and éz-field settings, providing a coherent path to reconstructing algebraic structures from definable geometry and offering new tools for further model-theoretic applications and Diophantine problems. The results illuminate how geometric tameness, transversality, and definable slopes interact to produce a robust trichotomy principle across several tame theories. $TIMI$, ramification purity, and open-mapping properties play central roles in linking local geometric data to global model-theoretic consequences, including group interpretation and potential field-interpretation in favorable contexts.
Abstract
We give a new axiomatic treatment of the Zilber trichotomy, and use it to complete the proof of the trichotomy for relics of algebraically closed fields, i.e., reducts of the ACF-induced structure on ACF-definable sets. More precisely, we introduce a class of geometric structures equipped with a Hausdorff topology, called \textit{Hausdorff geometric structures}. Natural examples include the complex field; algebraically closed valued fields; o-minimal expansions of real closed fields; and characteristic zero Henselian fields (in particular $p$-adically closed fields). We then study the Zilber trichotomy for relics of Hausdorff geometric structures, showing that under additional assumptions, every non-locally modular strongly minimal relic on a real sort interprets a one-dimensional group. Combined with recent results, this allows us to prove the trichotomy for strongly minimal relics on the real sorts of algebraically closed valued fields. Finally, we make progress on the imaginary sorts, reducing the trichotomy for \textit{all} ACVF relics (in all sorts) to a conjectural technical condition that we prove in characteristic $(0,0)$.