The aim of this paper is to develop the theory for \emph{definable $f$-generic} groups in the $p$... more The aim of this paper is to develop the theory for \emph{definable $f$-generic} groups in the $p$-adic field within the framework of definable topological dynamics, here the "definable f-generic" means a definable group admits a global f-generic type which is definable over a small submodel. This "definable $f$-generic" is a dual concept to "finitely satisfiable generic", and a useful tool to describe the analogue of torsion free o-minimal groups in the $p$-adic context. In this paper we will show that every definable $f$-generic group definable in $\Q$ is eventually isomorphic to a finite index subgroup of a trigonalizable algebraic group over $\Q$. This is analogous to the $o$-minimal context, where every connected torsion free group definable in $\R$ is isomorphic to a trigonalizable algebraic group (Lemma 3.4, \cite{COS}). We will also show that every open definable $f$-generic subgroup of a definable $f$-generic group has finite index, and every $f...
In [16], Peterzil and Steinhorn proved that if a group G definable in an o-minimal structure is n... more In [16], 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 1. 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 1 which is not definably compact. In a future paper we will generalize this to non-abelian G.
The aim of this paper is to develop the theory for \emph{definable $f$-generic} groups in the $p$... more The aim of this paper is to develop the theory for \emph{definable $f$-generic} groups in the $p$-adic field within the framework of definable topological dynamics, here the "definable f-generic" means a definable group admits a global f-generic type which is definable over a small submodel. This "definable $f$-generic" is a dual concept to "finitely satisfiable generic", and a useful tool to describe the analogue of torsion free o-minimal groups in the $p$-adic context. In this paper we will show that every definable $f$-generic group definable in $\Q$ is eventually isomorphic to a finite index subgroup of a trigonalizable algebraic group over $\Q$. This is analogous to the $o$-minimal context, where every connected torsion free group definable in $\R$ is isomorphic to a trigonalizable algebraic group (Lemma 3.4, \cite{COS}). We will also show that every open definable $f$-generic subgroup of a definable $f$-generic group has finite index, and every $f...
In [16], Peterzil and Steinhorn proved that if a group G definable in an o-minimal structure is n... more In [16], 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 1. 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 1 which is not definably compact. In a future paper we will generalize this to non-abelian G.
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Papers by Ningyuan Yao