Residually Dominated Groups in Henselian Valued Fields of Equicharacteristic Zero

Abstract

We study the model theory of henselian valued fields of equicharacteristic zero by generalizing results from the complete theory of algebraically closed valued fields ($\ACVF$) in the literature. Haskell, Hrushovski and Macpherson introduced the notion of \emph{stable domination} which provides a tameness condition to study the model theory of $\ACVF$. It was later generalized to \emph{residual domination} in the work by Haskell, Ealy and Simon, and independently by Vicaria for pure henselian valued fields of equicharacteristic zero.

In this thesis, we study residual domination itself, giving characterizations that are similar to those for stably dominated types in $\ACVF$. We then introduce \emph{residually dominated groups}, which are the analogue of the \emph{stably dominated groups} introduced and studied extensively by Rideau-Kikuchi and Hrushovski. We show that the connected components of residually dominated groups are subgroups of stably dominated groups that are definable in the algebraic closure of the given henselian valued field. This allows us to extend results from $\ACVF$ to the henselian setting.