On non-compact $p$-adic definable groups

Abstract

Peterzil and Steinhorn proved that if a group $G$ definable in an $o$-minimal structure is not definably compact, then $G$ contains a definable torsion-free subgroup of dimension one. We prove here a $p$-adic analogue of the Peterzil-Steinhorn theorem, in the special case of abelian groups. Let $G$ be an abelian group definable in a $p$-adically closed field $M$. If $G$ is not definably compact then there is a definable subgroup $H$ of dimension one which is not definably compact. In a future paper we will generalize this to non-abelian $G$.

28 pages; typos corrected, proofs cleaned up, numbering changed

References in corpus (2)

• Interpretable sets in dense o-minimal structures
• On Dimensions, Standard Part Maps, and $p$-Adically Closed Fields

Cited by in corpus (4)

• A note on $\mu$-stabilizers in ACVF
• Around definable types in $p$-adically closed fields
• Semisimple groups interpretable in various valued fields
• A note on fsg groups in p-adically closed fields