Onshuus–Pillay conjecture on definable groups in P-minimal structures

Let $M$ be a highly saturated $P$-minimal structure over a $p$-adic field $K$, and let $G$ be an $n$-dimensional definable group. Recall that $G^{00}$ is the smallest type-definable subgroup of $G$ of small index, and that $H$ is compactly dominated by $H/H^{00}$ when the quotient map satisfies the compact-domination condition with respect to normalized Haar measure. Onshuus–Pillay conjecture. There is an $n$-dimensional definable open subgroup $H\subseteq G$ such that

$$H/H^{00}$$

is isomorphic to an $n$-dimensional Lie group over $K$, and $H$ is compactly dominated by $H/H^{00}$. This conjecture concerns the $P$-minimal analogue of Pillay's conjectures on o-minimal groups and combines a Lie-group conclusion with compact domination. The paper states that its main theorem technically resolves the conjecture, although the resulting subgroup may be much smaller than $G$ and the quotient may have a structure determined only by the dimension.