ON NON-COMPACT <i>p</i>-ADIC DEFINABLE GROUPS

Johnson, Will; Yao, Ningyuan

doi:10.1017/jsl.2021.93

Cited by 10 publications

(6 citation statements)

References 27 publications

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“…Proof. The equivalence of (1)-( 4) is Remark 5.12 and Lemma 5.13 in [JY22]. The equivalence of (4) and (5) follows by a similar argument to the proof of [JY22, Lemma 6.2], using Lemma 5.7(3) instead of [JY22, Lemma 2.25].…”

Section: Interpretable Groupsmentioning

confidence: 72%

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Abelian groups definable in $p$-adically closed fields

Johnson¹,

Yao²

2022

Preprint

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Recall that a group G has finitely satisfiable generics (fsg) or definable f -generics (dfg) if there is a global type p on G and a small model M such that every left translate of p is finitely satisfiable in M 0 or definable over M 0 , respectively. We show that any abelian group definable in a p-adically closed field is an extension of a definably compact fsg definable group by a dfg definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where G is an abelian group definable in the standard model Q p , we show that G 0 = G 00 , and that G is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in Q p .

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Section: Interpretable Groupsmentioning

confidence: 72%

“…Proof. The proofs of Lemmas 4.9, 4.10, 4.11 in [JY22] work here, after making a couple trivial changes. The interpretable group X has finite dp-rank because dp-rk(X) ≤ dp-rk( X) = dim( X) < ∞.…”

Section: Interpretable Groupsmentioning

confidence: 99%

Abelian groups definable in $p$-adically closed fields

Johnson¹,

Yao²

2022

Preprint

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“…The equivalence of (1) and (2) follows from [6, Proposition 2.10]. The equivalence of (1) and (3) is [6, Proposition 2.16]. The equivalence of (1) and (4) is [6, Proposition 2.24].…”

Section: Definable Compactness In Pcfmentioning

confidence: 99%

A note on fsg$\text{fsg}$ groups in p‐adically closed fields

Johnson

2023

Mathematical Logic Qtrly

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“…In future work with Yao [JY22a], Theorem 1.2 will be used to generalize some of the results of [JY22b] to interpretable groups. In the present paper, we apply Theorem 1.2 to the study of interpretable groups with fsg.…”

Section: Application To Fsg Groupsmentioning

confidence: 99%

“…. If M is a p-adically closed field, a definable set X ⊆ M n is definably compact iff X is closed and bounded[JY22b, Lemmas 2.4,2.5].2. Consequently, if M= Q p , then a definable set X ⊆ M n is definably compact iff it is compact.Work in a monster model M of pCF.…”

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confidence: 99%

Topologizing interpretable groups in $p$-adically closed fields

Johnson¹

2022

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We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of "admissible topologies" with topological tameness properties like generic continuity, similar to the topology on definable subsets of K n . We show every interpretable set has at least one admissible topology, and every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an interpretable group is definably compact if and only if it has finitely satisfiable generics (fsg), generalizing an earlier result on definable groups. As a consequence, we see that fsg is a definable property in definable families of interpretable groups, and that any fsg interpretable group defined over Q p is definably isomorphic to a definable group. the subspace topology. Then D is admissible. We will see below that D is not manifold dominated (Example 6.5). Closure propertiesRemark 4.11. Let X be a finite interpretable set with the discrete topology. Then X is strongly admissible. In fact, X is a definable manifold.In the category of topological spaces, the class of open maps is closed under composition and base change. Consequently, ifProposition 4.12. The following classes of interpretable topological spaces are closed under finite disjoint unions and finite products.1. The class of locally definable spaces. The class of locally Euclidean spaces.3. The class of definably dominated spaces. The class of manifold dominated spaces.5. The class of admissible spaces. The class of strongly admissible spaces.Proof. We focus on binary disjoint unions and binary products. (The 0-ary cases are handled by Remark 4.11.)Cases ( 2) and (1) are straightforward. For (3), let X 1 , X 2 be two definably dominated spaces. Let f i : Y i → X i be a map witnessing definable domination for i = 1, 2. Up to definable homeomorphism, Y 1 ⊔ Y 2 is a definable set, and sois definably dominated. The proof for (4) is similar. Then (1) and (3) give (5), while (2) and ( 4) give (6).Say that a class C of interpretable topological spaces is "closed under subspaces" if C contains every interpretable subspace of every X ∈ C. Proposition 4.13. The following classes of interpretable topological spaces are closed under subspaces:1. The class of locally definable spaces.2. The class of definably dominated spaces.

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ON NON-COMPACT p-ADIC DEFINABLE GROUPS

Cited by 10 publications

References 27 publications

Abelian groups definable in $p$-adically closed fields

Abelian groups definable in $p$-adically closed fields

A note on fsg$\text{fsg}$ groups in p‐adically closed fields

Topologizing interpretable groups in $p$-adically closed fields

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