On minimal flows, definably amenable groups, and o-minimality

Pillay, Anand; Yao, Ningyuan

doi:10.1016/j.aim.2015.12.010

Cited by 16 publications

(29 citation statements)

References 15 publications

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“…(i). This follows by using the proof of Lemma 1.15 of [23]. More specifically it is proved there that in the NIP environment, a global definable f -generic type p is almost periodic by showing that in fact the orbit of p is closed.…”

Section: The Additive and Multiplicative Groupsmentioning

confidence: 93%

Some model theory and topological dynamics of $p$-adic algebraic groups

2019

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We initiate the study of p-adic algebraic groups G from the stability-theoretic and definable topological-dynamical points of view, that is, we consider invariants of the action of G on its space of types over Q p in the language of fields. We consider the additive and multiplicative groups of Q p and Z p , the group of upper triangular invertible 2 × 2 matrices, SL(2, Z p ), and, our main focus, SL(2, Q p ). In all cases we identify f -generic types (when they exist), minimal subflows, and idempotents. Among the main results is that the "Ellis group" of SL(2, Q p ) iŝ Z, yielding a counterexample to Newelski's conjecture with new features: G = G 00 = G 000 but the Ellis group is infinite. A final section deals with the action of SL(2, Q p ) on the type-space of the projective line over Q p .

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Section: The Additive and Multiplicative Groupsmentioning

confidence: 93%

Some model theory and topological dynamics of $p$-adic algebraic groups

2019

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“…This was proved to be the case in stable theories [12] and for definably compact groups definable in o-minimal structures [11]. This result was later extended to definably amenable groups definable in o-minimal structures [15] and finally to all definably amenable groups in NIP theories [5]. Partial results also hold in the general case.…”

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confidence: 91%

“…These results have been generalized in [16] to allow the calculation of the Ellis groups over larger models, establishing the isomorphism with groups calculated over the reals. A tangent case of definably amenable groups definable in o-minimal expansions of arbitrary real closed fields have been solved in [14]. The methods used to determine the Ellis groups have been progressively less specific to the o-minimal setting.…”

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confidence: 99%

The Ellis Group Conjecture and Variants of Definable Amenability

Jagiella¹

2018

J. symb. log.

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We consider definable topological dynamics for N IP groups admitting certain decompositions in terms of specific classes of definably amenable groups. For such a group, we find a description of the Ellis group of its universal definable flow. This description shows that the Ellis group is of bounded size. Under additional assumptions, it is shown to be independent of the model, proving a conjecture proposed by Newelski. Finally we apply the results to new classes of groups definable in o-minimal structures, generalizing all of the previous results for this setting.2010 Mathematics Subject Classification. 03C99, 54H20 (primary), and 03C64 (secondary).

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“…So Question 3.35 of [3] asked that wether weakly generic types coincide with almost periodic types for definably amenable groups in NIP environment. One of the main results of [17] saying that the answer in negative even for o-minimal case. On the other hand, [17] proved that any definable weakly generic type is almost periodic for definably amenable groups in NIP environment.…”

Section: Introductionmentioning

confidence: 99%

“…One of the main results of [17] saying that the answer in negative even for o-minimal case. On the other hand, [17] proved that any definable weakly generic type is almost periodic for definably amenable groups in NIP environment. Based on this positive result, [17] conjectured that Conjecture 1.…”

Section: Introductionmentioning

confidence: 99%

Definable topological dynamics for trigonalizable algebraic groups over Qp

Yao

2019

Mathematical Logic Qtrly

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We study the flow (G(Q p ), S G (Q p )) of trigonalizable algebraic group acting on its type space, focusing on the problem raised in [17] of whether weakly generic types coincide with almost periodic types if the group has global definable fgeneric types, equivalently whether the union of minimal subflows of a suitable type space is closed. We will give a description of of f -generic types of trigonalizable algebraic groups, and prove that every f -generic type is almost periodic.

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On minimal flows, definably amenable groups, and o-minimality

Cited by 16 publications

References 15 publications

Some model theory and topological dynamics of $p$-adic algebraic groups

Some model theory and topological dynamics of $p$-adic algebraic groups

The Ellis Group Conjecture and Variants of Definable Amenability

Definable topological dynamics for trigonalizable algebraic groups over Qp

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