Definable Equivariant Retractions in Non-Archimedean Geometry

TL;DR

This paper constructs definable, equivariant deformation retractions in non-Archimedean geometry, connecting algebraic groups, their stable completions, and Berkovich analytifications, revealing new structural insights.

Contribution

It introduces new definable deformation retractions for stable completions and Berkovich spaces, extending geometric understanding in non-Archimedean settings.

Findings

Constructed $G$-equivariant deformation retraction of $\uhat{G}$ onto generic type

Built $S$-equivariant retraction of $$ onto a definable group internal to the value group

Established that in certain cases, the retraction descends to the Berkovich analytification and onto the skeleton

Abstract

For $G$ an algebraic group definable over a model of $\operatorname{ACVF}$, or more generally a definable subgroup of an algebraic group, we study the stable completion $\widehat{G}$ of $G$, as introduced by Loeser and the second author. For $G$ connected and stably dominated, assuming $G$ commutative or that the valued field is of equicharacteristic 0, we construct a pro-definable $G$-equivariant strong deformation retraction of $\widehat{G}$ onto the generic type of $G$. For $G=S$ a semiabelian variety, we construct a pro-definable $S$-equivariant strong deformation retraction of $\widehat{S}$ onto a definable group which is internal to the value group. We show that, in case $S$ is defined over a complete valued field $K$ with value group a subgroup of $\mathbb{R}$, this map descends to an $S(K)$-equivariant strong deformation retraction of the Berkovich analytification $S^{\mathrm{an}}$ of $S$ onto a piecewise linear group, namely onto the skeleton of $S^{\mathrm{an}}$. This yields a construction of such a retraction without resorting to an analytic (non-algebraic) uniformization of $S$. Furthermore, we prove a general result on abelian groups definable in an NIP theory: any such group $G$ is a directed union of $\infty$-definable subgroups which all stabilize a generically stable Keisler measure on $G$.