Definable topological dynamics for trigonalizable algebraic groups over Qp

Yao, Ningyuan

doi:10.1002/malq.201900009

Cited by 8 publications

(8 citation statements)

References 16 publications

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“…As a consequence, using a recent result in [31], we could prove the following conjecture raised in [13] Conjecture 1. [13] Let G be a df g group definable in a NIP structure.…”

Section: Introductionmentioning

confidence: 83%

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Definable $f$-Generic Groups over $p$-Adic Numbers

Pillay¹,

Yao²

2019

Preprint

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The aim of this paper is to develop the theory for 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 p is eventually isomorphic to a finite index subgroup of a trigonalizable algebraic group over Q p . 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,[2]). We will also show that every open definable f -generic subgroup of a definable f -generic group has finite index, and every f -generic type of a definable f -generic group is almost periodic, which gives a positive answer on the problem raised in [13] of whether f -generic types coincide with almost periodic types in the p-adic case.

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“…As a consequence, using a recent result in [31], we could prove the following conjecture raised in [13] Conjecture 1. [13] Let G be a df g group definable in a NIP structure.…”

Section: Introductionmentioning

confidence: 83%

“…Fact 2.30. [31] Let H ⊆ Q n p be a trigonalizable algebraic group over Q p , N an elementary extension of M = (Q p , +, ×, 0). Then every f -generic type in S H (N ) is almost periodic.…”

Section: Df G Groups and F -Generic Typesmentioning

confidence: 99%

Definable $f$-Generic Groups over $p$-Adic Numbers

Pillay¹,

Yao²

2019

Preprint

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“…For simplity, We assume that B = A⋊H where A = T an and H = S ⋊B u . By [34], H has dfg and H 00 = H 0 = S 0 ⋊ B u . Let C = KA, it is easy to see that G = CH is a compact-dfg decomposition.…”

Section: Minimal Subflows and Ellis Groups Of Groups Admitting Iwasaw...mentioning

confidence: 99%

“…By [34], any algebraic group trigonalizable over Q p has a global definable f -generic (dfg) type, and by [22] any definably compact group over Q p has a global finitly satisfiable generic (fsg) type. So the above decomposition is a kind of "fsg-dfg" decomposition in the model-theoritic view when B is split (trigonalizable) over Q p .…”

Section: Introductionmentioning

confidence: 99%

Definably Topological Dynamics of $p$-Adic Algebraic Groups

Bao¹,

Yao²

2021

Preprint

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We study the p-adic algebraic groups G from the definable topologicaldynamical point of view. We consider the case that M is an arbitrary p-adic closed field and G an algebraic group over Q p admitting an Iwasawa decompostion G = KB, where K is open and definably compact over Q p , and B is a borel subgroup of G over Q p . Our main result is an explicit description of the minimal subflow and Ellis Group of the universal definable G(M )-flow S G (M ext ). We prove that the Ellis group of S G (M ext ) is isomorphic to the Ellis group of S B (M ext ), which is B/B 0 .As applications, we conclude that the Ellis groups corresponding to GL(n, M ) and SL(n, M ) are isomorphic to ( Ẑ × Z * p ) n and ( Ẑ × Z * p ) n−1 respectively, generalizing the main result of Penazzi, Pillay, and Yao in [23].

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“…As a corollary, the orbit of p consists of precisely one ideal subgroup. The paper [16] dealt with the question whether f-generic global types coincide with almost periodic types in the setup of definably amenable linear groups definable the field Q p of p-adic numbers. The results are done in the context of previous analyses by Pillay and Yao of groups definable in Q p , particularly dfg groups; and the work to classify groups of dimension 1.…”

Section: Definable F -Genericsmentioning

confidence: 99%

Topological dynamics and NIP fields

Jagiella¹

2020

Preprint

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We study definable topological dynamics of some algebraic group actions over an arbitrary NIP field K. We show that the Ellis group of the universal definable flow of SL 2 pKq is non-trivial if the multiplicative group of K is not type-definably connected, providing a way to find multiple counterexamples to the Ellis group conjecture, particularly in the case of dp-minimal fields. We also study some structure theory of algebraic groups over K with definable f-generics.

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Definable topological dynamics for trigonalizable algebraic groups over Qp

Cited by 8 publications

References 16 publications

Definable $f$-Generic Groups over $p$-Adic Numbers

Definable $f$-Generic Groups over $p$-Adic Numbers

Definably Topological Dynamics of $p$-Adic Algebraic Groups

Topological dynamics and NIP fields

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