Definable topological dynamics for trigonalizable algebraic groups over \(\mathbb{Q}_P\). (English) Zbl 1521.03092
Summary: We study the flow \((G(\mathbb{Q}_p),S_G(\mathbb{Q}_p))\) of trigonalizable algebraic group acting on its type space, focusing on the problem raised in [A. Pillay and the author, Adv. Math. 290, 483–502 (2016; Zbl 1386.03042)] of whether weakly generic types coincide with almost periodic types if the group has global definable f-generic types, equivalently whether the union of minimal subflows of a suitable type space is closed. We shall give a description of f-generic types of trigonalizable algebraic groups, and prove that every f-generic type is almost periodic.
MSC:
| 03C60 | Model-theoretic algebra |
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 03C45 | Classification theory, stability, and related concepts in model theory |
| 03C98 | Applications of model theory |
Citations:
Zbl 1386.03042References:
| [1] | J.Auslander, Minimal Flows and their Extensions, Mathematical Studies Vol. 153 (North Holland, 1988). · Zbl 0654.54027 |
| [2] | L.Belair, Panorama of p‐adic model theory, Ann. Sci. Math. Québec36(1), 43-75 (2012). · Zbl 1362.11107 |
| [3] | A.Chernikov and P.Simon, Definably amenable NIP groups, J. Amer. Math. Soc.31, 609-641 (2018). · Zbl 1522.03112 |
| [4] | E.Hrushovski, Y.Peterzil, and A.Pillay, Groups, measures, and the NIP, J. Amer. Math. Soc.21, 563-596 (2008). · Zbl 1134.03024 |
| [5] | E.Hrushovski and A.Pillay, Groups definable in local fields and pseudo‐finite fields, Isr. J. Math.85(1‐3), 203-262 (1994). · Zbl 0804.03024 |
| [6] | G.Jagiella, Definable topological dynamics and real Lie groups, Math. Log. Q.61(1‐2), 45-55 (2015). · Zbl 1341.03049 |
| [7] | A.MacIntyre, On definable subsets of p‐adic fields, J. Symb. Log.41, 605-610 (1976). · Zbl 0362.02046 |
| [8] | D.Marker, Model Theory: An Introduction, Graduate Texts in Mathematics Vol. 217 (Springer, 2002). · Zbl 1003.03034 |
| [9] | J. S.Milne, Algebraic Groups, The Theory of Group Schemes of Finite Type over a Field, Cambridge Studies in Advanced Mathematics Vol. 170 (Cambridge University Press, 2017). · Zbl 1390.14004 |
| [10] | L.Newelski, Topological dynamics of definable group actions, J. Symb. Log.74, 50-72 (2009). · Zbl 1173.03031 |
| [11] | L.Newelski and M.Petrykowski, Weak generic types and coverings of groups I, Fundam. Math.191, 201-225 (2006). · Zbl 1111.03036 |
| [12] | A.Onshuus and A.Pillay, Definable groups and compact p‐adic Lie groups, J. London Math. Soc.78(1), 233-247 (2008). · Zbl 1153.03015 |
| [13] | D.Penazzi, A.Pillay, and N.Yao, Some model theory and topological dynamics of p‐adic algebraic groups, Fundam. Math., to appear. · Zbl 1477.03139 |
| [14] | A.Pillay, On fields definable in \(\mathbb{Q}_p\), Arch. Math. Log.29, 1-7 (1989). · Zbl 0687.03016 |
| [15] | A.Pillay, Topological dynamics and definable groups, J. Symb. Log.78, 657-666 (2013). · Zbl 1278.03071 |
| [16] | A.Pillay and N.Yao, On groups over \(\mathbb{Q}_p\) with definable f‐generic types, preprint. |
| [17] | A.Pillay and N.Yao, On minimal flows, definably amenable groups, and o‐minimality, Adv. Math.290, 483-502 (2016). · Zbl 1386.03042 |
| [18] | B.Poizat, A Course in Model Theory, An Introduction to Contemporary Mathematical Logic, Universitext (Springer, 2000). · Zbl 0951.03002 |
| [19] | N.Yao, On f‐generic types in Presburger arithmetic, Stud. Log. (SYSU)12(3), 57-78 (2019). |
| [20] | N.Yao and D.Long, Topological dynamics for groups definable in real closed field, Ann. Pure Appl. Log.166(3), 261-273 (2015). · Zbl 1331.03029 |
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