Residually Dominated Groups in Henselian Valued Fields of Equicharacteristic Zero

TL;DR

This paper introduces residually dominated groups in henselian valued fields of equicharacteristic zero, establishing their properties, connections to stably dominated groups, and algebraic characterizations within model theory.

Contribution

It defines residually dominated groups, explores their functorial and universality properties, and generalizes previous work on stably dominated types in algebraically closed valued fields.

Findings

Existence of a finite-to-one homomorphism into stably dominated groups.

Residual domination characterized by homomorphisms into residue field groups.

Algebraic characterization of residually dominated types.

Abstract

We introduce \emph{residually dominated groups} in pure henselian valued fields of equicharacteristic zero, as an analogue of stably dominated groups introduced by Hrushovski and Rideau-Kikuchi. We show that when $G$ is a residually dominated group, there is a finite-to-one group homomorphism from its connected component into a connected stably dominated group, and we study the functoriality and universality properties of this map. Moreover, we prove that residual domination is witnessed by a group homomorphism into a definable group in the residue field. In our proofs, we use the results of Montenegro, Onshuus, and Simon on groups definable in $\mathrm{NTP}_2$-theories that extend the theory of fields. Along the way, we also provide an algebraic characterization of residually dominated types, generalizing the work by Ealy, Haskell and Simon for stably dominated types in algebraically closed valued fields, and we study their properties.