Approximate subgroups

Massicot, Jean-Cyrille; Wagner, Frank Olaf

doi:10.5802/jep.17

Cited by 16 publications

(52 citation statements)

References 9 publications

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“…For instance, any large set with respect to the counting measure is an approximate subgroup. This can be deduced from the following version of Ruzsa's Covering Lemma (see, for instance, [, Fact 5]). Since its proof is short, we include if for completeness.…”

Section: Stabilizers and Product‐free Setsmentioning

confidence: 99%

On compactifications and product‐free sets

Palacín

2019

J. London Math. Soc.

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A subset of a group is said to be product free if it does not contain three elements satisfying the equation xy=z. We give a negative answer to a question of Babai and Sós on the existence of large product‐free sets in finite groups by model theoretic means. This question was originally answered by Gowers. Furthermore, we give a natural and sufficient model theoretic condition for a group to have a large product‐free subset, as well as a model theoretic account of a result of Nikolov and Pyber on triple products.

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Section: Stabilizers and Product‐free Setsmentioning

confidence: 99%

On compactifications and product‐free sets

Palacín

2019

J. London Math. Soc.

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“…In the definable context described above, this conjecture specializes to the theorem (easily deduced in [24] from [29]) saying that each definably amenable group G(M) satisfies G 00 M = G 000 M . On the other hand, in the topological context, Conjecture 0.1 specializes to Conjecture 0.2 from [24] which predicts that whenever G(M) is an amenable topological group, then G 00 top = G 000 top .…”

Section: Introductionmentioning

confidence: 97%

“…Similarly to [24], the proof is based on the Massicot-Wagner argument from [29], but here we use means on certain lattices instead of measures on Boolean algebras. Moreover, in Sect.…”

Section: Introductionmentioning

confidence: 99%

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Amenability, connected components, and definable actions

Hrushovski

Krupiński

Pillay

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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“…On the other hand, it seems hopeless to aim at classifying all infinite approximate subgroups. Some results in this direction for particular classes of infinite approximate subgroups can be found in [5,10,14]. Inspired by Yves Meyer's results on quasi-crystals [15], Michael Björklund and Tobias Hartnick have defined a class of infinite approximate subgroups called approximate lattices in [2].…”

Section: Introductionmentioning

confidence: 99%