Generic differentiability and $P$-minimal groups

TL;DR

This paper establishes generic differentiability in P-minimal theories, leading to a classification of definable groups and fields, and confirming conjectures about the structure of P-minimal groups.

Contribution

It proves generic differentiability in P-minimal theories and applies this to classify definable groups and fields, confirming conjectures on their structure.

Findings

Existence of an open definable subgroup with compact domination

The quotient by H^{00} is a p-adic Lie group of expected dimension

Classification of interpretable fields in P-minimal theories

Abstract

We prove generic differentiability in $P$-minimal theories, strengthening an earlier result of Kuijpers and Leenknegt. Using this, we prove Onshuus and Pillay's $P$-minimal analogue of Pillay's conjectures on o-minimal groups. Specifically, let $G$ be an $n$-dimensional definable group in a highly saturated model $M$ of a $P$-minimal theory. Then there is an open definable subgroup $H \subseteq G$ such that $H$ is compactly dominated by $H/H^{00}$, and $H/H^{00}$ is a $p$-adic Lie group of the expected dimension. Additionally, the generic differentiability theorem immediately implies a classification of interpretable fields in $P$-minimal theories, by work of Halevi, Hasson, and Peterzil.