Chabauty Limits of Subgroups of $SL(n, \mathbb{Q}_p)$

TL;DR

This paper investigates the limits of certain subgroups of $SL(n, Q_p)$ in the Chabauty topology, providing geometric proofs, classifications for small n, and demonstrating infinite limits for larger n.

Contribution

It offers a new geometric proof for the Chabauty compactification of parahoric subgroups and classifies limits of conjugates of diagonal subgroups in $SL(n, Q_p)$.

Findings

Classified Chabauty limits for $n \\leq 4$

Proved infinitely many nonconjugate limits for $n \\geq 7$

Constructed explicit homeomorphism between Lie algebra and group limits

Abstract

We study the Chabauty compactification of two families of closed subgroups of $SL(n,\mathbb{Q}_p)$. The first family is the set of all parahoric subgroups of $SL(n,\mathbb{Q}_p)$. Although the Chabauty compactification of parahoric subgroups is well studied, we give a different and more geometric proof using various Levi decompositions of $SL(n,\mathbb{Q}_p)$. Let $C$ be the subgroup of diagonal matrices in $SL(n, \mathbb{Q}_p)$. The second family is the set of all $SL(n,\mathbb{Q}_p)$-conjugates of $C$. We give a classification of the Chabauty limits of conjugates of $C$ using the action of $SL(n,\mathbb{Q}_p)$ on its associated Bruhat--Tits building and compute all of the limits for $n\leq 4$ (up to conjugacy). In contrast, for $n\geq 7$ we prove there are infinitely many $SL(n,\mathbb{Q}_p)$-nonconjugate Chabauty limits of conjugates of $C$. Along the way we construct an explicit homeomorphism between the Chabauty compactification in $\mathfrak{sl}(n, \mathbb{Q}_p)$ of $SL(n,\mathbb{Q}_p)$-conjugates of the $p$-adic Lie algebra of $C$ and the Chabauty compactification of $SL(n,\mathbb{Q}_p)$-conjugates of $C$.