Amenability, definable groups, and automorphism groups

Krupiński, Krzysztof; Pillay, Anand

doi:10.1016/j.aim.2019.01.033

“…(But note that there is a model-theoretic account of the dynamics of definable groups with a definable topology. See [17] for example. And it might be worthwhile to prove and compare results.…”

Section: Topological Dynamicsmentioning

confidence: 99%

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

Penazzi

¹

,

Pillay

²

,

Yao

³

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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“…

We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version (Massicot and Wagner in J Ec Polytech Math 2:55-63, 2015) of the stabilizer theorem (Hrushovski in J Am Math Soc 25:189-243, 2012), and also some results about measures and measure-like functions (which we call means and pre-means).As an application we show that if G is an amenable topological group, then the Bohr compactification of G coincides with a certain "weak Bohr compactification" introduced in Krupiński and Pillay (Adv Math 345:1253-1299, 2019. In other words, the conclusion says that certain connected components of G coincide: G 00 top = G 000 top .

…”

mentioning

confidence: 76%

“…As an application we show that if G is an amenable topological group, then the Bohr compactification of G coincides with a certain "weak Bohr compactification" introduced in Krupiński and Pillay (Adv Math 345:1253-1299, 2019. In other words, the conclusion says that certain connected components of G coincide: G 00 top = G 000 top .…”

mentioning

confidence: 77%

“…We conclude that any group G definable in a sufficiently saturated structure is "weakly definably amenable" in the sense of Krupiński and Pillay (2019), namely any definable action of G on a compact space supports a G-invariant probability measure. This gives negative solutions to some questions and conjectures raised in Krupiński (J Symb Log 82:1080-1105, 2017) and Krupiński and Pillay (2019). Stability in continuous logic will play a role in some proofs in this part of the paper.…”

mentioning

confidence: 88%

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Amenability, connected components, and definable actions

Hrushovski

¹

,

Krupiński

²

,

Pillay

³

2021

Sel. Math. New Ser.

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We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version (Massicot and Wagner in J Ec Polytech Math 2:55–63, 2015) of the stabilizer theorem (Hrushovski in J Am Math Soc 25:189–243, 2012), and also some results about measures and measure-like functions (which we call means and pre-means). As an application we show that if G is an amenable topological group, then the Bohr compactification of G coincides with a certain “weak Bohr compactification” introduced in Krupiński and Pillay (Adv Math 345:1253–1299, 2019). In other words, the conclusion says that certain connected components of G coincide: $$G^{00}_{{{\,\mathrm{{top}}\,}}} = G^{000}_{{{\,\mathrm{{top}}\,}}}$$ G top 00 = G top 000 . We also prove wide generalizations of this result, implying in particular its extension to a “definable-topological” context, confirming the main conjectures from Krupiński and Pillay (2019). We also introduce $$\bigvee $$ ⋁ -definable group topologies on a given $$\emptyset $$ ∅ -definable group G (including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of G implies (under some assumption) that $${{\,\mathrm{{cl}}\,}}(G^{00}_M) = {{\,\mathrm{{cl}}\,}}(G^{000}_M)$$ cl ( G M 00 ) = cl ( G M 000 ) for any model M. Secondly, we study the relationship between (separate) definability of an action of a definable group on a compact space [in the sense of Gismatullin et al. (Ann Pure Appl Log 165:552–562, 2014)], weakly almost periodic (wap) actions of G [in the sense of Ellis and Nerurkar (Trans Am Math Soc 313:103–119, 1989)], and stability. We conclude that any group G definable in a sufficiently saturated structure is “weakly definably amenable” in the sense of Krupiński and Pillay (2019), namely any definable action of G on a compact space supports a G-invariant probability measure. This gives negative solutions to some questions and conjectures raised in Krupiński (J Symb Log 82:1080–1105, 2017) and Krupiński and Pillay (2019). Stability in continuous logic will play a role in some proofs in this part of the paper. Thirdly, we give an example of a $$\emptyset $$ ∅ -definable approximate subgroup X in a saturated extension of the group $${{\mathbb {F}}}_2 \times {{\mathbb {Z}}}$$ F 2 × Z in a suitable language (where $${{\mathbb {F}}}_2$$ F 2 is the free group in 2-generators) for which the $$\bigvee $$ ⋁ -definable group $$H:=\langle X \rangle $$ H : = ⟨ X ⟩ contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact (and in consequence also Lie) “model” exists for each approximate subgroup does not work in general (they proved in (Massicot and Wagner 2015) that it works for definably amenable approximate subgroups).

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“…Now, we give a few details on Galois groups in model theory. For more detailed expositions the reader is referred to [10,Subsection 1.3] or [9,Subsection 4.1]. If the reader is interested in yet more details and proofs, he or she may consult fundamental papers around this topic, e.g.…”

Section: Some Preliminariesmentioning

confidence: 99%

Boundedness and absoluteness of some dynamical invariants in model theory

Krupiński

¹

,

Newelski

²

,

Simon

³

2019

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Let C be a monster model of an arbitrary theory T , letᾱ be any (possibly infinite) tuple of bounded length of elements of C, and letc be an enumeration of all elements of C (so a tuple of unbounded length). By Sᾱ(C) we denote the compact space of all complete types over C extending tp(ᾱ/∅), and Sc(C) is defined analogously. Then Sᾱ(C) and Sc(C) are naturally Aut(C)flows (even Aut(C)-ambits). We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of C), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend (as groups equipped with the so-called τ -topology) on the choice of the monster model C; thus, we say that these Ellis groups are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows Sᾱ(C) and Sc(C). We give an example of a NIP theory in which the minimal left ideals are of unbounded size. Then we show that in each of these two cases, boundedness of a minimal left ideal (equivalently, of all the minimal left ideals) is an absolute property (i.e. it does not depend on the choice of C) and that whenever such an ideal is bounded, then in some sense its isomorphism type is also absolute.Under the assumption that T has NIP, we give characterizations (in various terms) of when a minimal left ideal of the Ellis semigroup of Sc(C) is bounded. Then we adapt the proof of [3, Theorem 5.7] to show that whenever such an ideal is bounded, a certain natural epimorphism (described in [10]) from the Ellis group of the flow Sc(C) to the Kim-Pillay Galois group Gal KP (T ) is an isomorphism (in particular, T is G-compact). We also obtain some variants of these results, formulate some questions, and explain differences (providing a few counter-examples) which occur when the flow Sc(C) is replaced by Sᾱ(C).2010 Mathematics Subject Classification. 03C45, 54H20, 37B05.

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Amenability, definable groups, and automorphism groups

Cited by 17 publications

References 28 publications

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

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

Amenability, connected components, and definable actions

Boundedness and absoluteness of some dynamical invariants in model theory

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