Jean-Cyrille Massicot & Frank O. Wagner
Approximate subgroups Tome 2 (2015), p. 55-63.
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APPROXIMATE SUBGROUPS
by Jean-Cyrille Massicot & Frank O. Wagner
Abstract. — Given a definably amenable approximate subgroupAof a (local) group in some first-order structure, there is a type-definable subgroupHnormalized byAand contained inA4 such that every definable superset ofHhas positive measure.
Résumé(Sous-groupes approximatifs). — Étant donné un sous-groupe approximatif Adéfi- nissablement moyennable d’un groupe (local) dans une structure du premier ordre, il y a un sous-groupe H type-définissable normalisé parA et contenu dansA4 tel que tout ensemble définissable contenantHest de mesure positive.
Contents
Introduction. . . 55
1. A type-definable version of Sanders’ Theorem . . . 58
2. Normality. . . 62
References. . . 63
Introduction
LetG be a group and K >0 an integer, a subsetA ⊆G closed under inverse is a K-approximate subgroup if there is a finite subsetE ⊆G with|E|6K such that A2={ab:a, b∈A} ⊆EA. ThenAn ⊆En−1A.
Following work of Hrushovski [6] and many others, Breuillard, Green and Tao [1]
have classified finite approximate subgroups of local groups (see [2] for an excellent survey). In particular, they show that there is an approximate subgroupA∗⊆A4and an actualA∗-invariant subgroupH∗⊆A∗ such that
– finitely many left translates ofA∗ coverA, and – hA∗i/H∗ is nilpotent.
Mathematical subject classification (2010). — 11B30, 20N99, 03C98, 20A15.
Keywords. — Approximate subgroup, definability, definable amenability.
Partially supported by ValCoMo (ANR-13-BS01-0006).
56 J.-C. Massicot & F. O. Wagner
The result and its proof are inspired not only by Gleason’s and Yamabe’s solution of Hilbert’s5th problem [4, 10] and its extension to the local context by Goldbring [5], but also by Gromov’s Theorem on groups with polynomial growth, and is indeed a way to generalize this theorem. The three articles [6, 1, 2] provide some applications to geometric group theory.
The proof proceeds by considering a non-principal ultraproduct of a sequence of finite counterexamples (An : n < ω) with |An| → ∞, giving rise to a pseudofinite counterexample A. Then an A-invariant subgroup H ⊆A4 is constructed such that hAi/H is locally compact. From Yamabe’s theorem on the approximation of locally compact groups by Lie groups, it follows that there are suitableA∗ andH∗such that hA∗i/H∗ is a real Lie group; using pseudofiniteness, the final result is obtained.
The construction of the locally compact quotient hAi/H and the Lie model hA∗i/H∗was first shown by Hrushovski [6] by model-theoretic means inspired by and reminiscent of stability theory. Breuillard, Green and Tao use instead a (subsequent) theorem of Sanders [9] from finite combinatorics, constructing successively the traces of the definable supersets of H on the various finite approximate groups An. Using the ultraproduct construction, the pseudofinite counting measure and the translation- invariant ideal of measure zero sets, Hrushovski’s theorem allows to recover Sanders’
result at least qualitatively.
Definability. — The topology ofhAi/H was constructed analytically in [1], but has a natural model-theoretic interpretation already given in [6]. Recall that a subset of the ultraproduct isdefinableif it is the set of realizations of some first-order formula (usually involving quantifiers); it is type-definableif it is given as the intersection of a countable (say) family of definable sets. For instance, the centralizer of a group element g is defined by the formulaxg =gx, and ifG is a group defined by a for- mulaϕ(x), its centreZ(G)is defined by the formulaϕ(x)∧ ∀y(ϕ(y)→xy=yx). On the other hand, the group generated by an elementg, or the centre of a type-definable group, are in general not even type-definable.
If H is a type-definable normal subgroup of hAi, it has bounded index if any de- finable superset of H covers any definable subset of hAiin finitely many translates.
We can then endow the quotienthAi/H with thelogic topologywhose proper closed subsets are precisely those subsets whose preimage inhAiis type-definable; this will turn it into a locally compact topological group.
Of course, (type-)definability strongly depends on the language: if weexpand the structure, for instance by adding predicates for certain subsets, there will be more de- finable sets. While the groupH constructed by Hrushovski is naturally type-definable in the structure given, it only becomes so in Sanders’ Theorem (either in the ultra- product or in a suitable version using a bi-invariant measure instead of cardinality) after such an expansion of language. Hrushosvki, on the other hand, assumes the ex- istence of a bi-invariant S1 ideal (which should be thought of as the ideal of sets of measure zero) which in addition is automorphism invariant; in order for an ideal to be- come automorphism invariant, one would generally also have to expand the language.
J.É.P. — M., 2015, tome 2
Such an expansion does not matter much if the structure to start with is arbitrary, but should be avoided if the initial structure has particular model-theoretic properties one wants to preserve.
For example, Eleftheriou and Peterzil [3] construct H type-definably without expanding the language in the case whenAis definable in ano-minimal expansion of an ordered group (such as the field of real numbers with exponentiation), provided that hAiis abelian. Pillay [8], generalizing an argument in [7], generalizes this result if A is definable in a theory without the independence property, and is definably amenable (see below). In fact, in this setting there is a unique minimal choice forH, namely the unique minimal type-definable subgroup of bounded index,hAi00. Definable amenability. — We shall callAdefinably amenableifhAicarries a finitely additive left-invariant measureµon its definable subsets such thatµ(A) = 1. We can now state the main theorem of our paper.
Theorem1. — In any group G, a definably amenable approximate subgroup A gives rise to a type-definable subgroupH ⊆A4, such that finitely many left translates of any definable superset of H cover A. Hence a definably amenable approximate subgroup allows a real Lie model without expanding the language.
Our proof follows the ideas of Sanders, except that we use the measure not to define the subgroup we obtain, but only to show that the formulas we construct in the original language have the necessary properties. We conjecture that even without the definable amenability assumption a suitable Lie model exists.
The classification of approximate subgroups of real Lie groups is still an open problem. Since in a real Lie group any compact neighbourhood of the identity is an approximate subgroup, in particular no nilpotency (or even solubility) result can hold in general. We hope that under additional model-theoretic assumptions on the original structure, a partial classification might be easier to achieve.
We will end this introduction with two useful remarks. The first one concerns essentially the only (but crucial) use of model theory in this paper. The second one is an easy generalization which played a key role in the conclusion of [1], and thus seems worth noticing.
We shall assume that all structures under consideration are ω+-saturated, which means that any countable intersection of definable sets is non-empty as soon as all finite subintersections are. All non-principal ultraproducts areω+-saturated; the com- pactness theorem of model theory implies that we can replace any structure M by a superstructure M∗ satisfying the same first-order sentences with parameters inM (an elementary extension) which in addition isω+-saturated.
As in [1], the results in this paper remain true ifGis only alocal group, i.e. a set closed under inverse and endowed with a multiplication such that the product of up to 100 elements is well-defined and fully associative. For this, one can check throughout the proofs that one never needs to multiply more than 100 elements ofA.
58 J.-C. Massicot & F. O. Wagner
1. A type-definable version of Sanders’ Theorem
Definition2. — A subsetAof a (local) groupGissymmetricif1∈A, anda−1∈A for alla∈A.
IfK < ω, a symmetric subset AofGis aK-approximate subgroupif A2={aa0:a, a0∈A}
is contained in K left cosets ofA. An approximate subgroupis a symmetric subset which is aK-approximate subgroup for someK < ω.
From a model-theoretic point of view, a definable approximate subgroupAis just a symmetric generic set in hAi, i.e. a definable symmetric subset of hAi such that every definable subset of hAiis covered by finitely many left translates ofA.
Definition3. — A definable approximate subgroupAisdefinably amenableif there is a left translation-invariant finitely additive measureµon the definable subsets ofhAi withµ(A) = 1.
Note that by ω+-saturation, for any definable subset X of hAi there is n < ω with X ⊆ An. So if A is a definably amenable approximate subgroup, then µ(X)6µ(An)<∞.
Remark4. — Iflimn→∞µ(An)<∞, then there is n < ωwithAn=hAi.
Proof. — Suppose not. Then for every n < ω there is an ∈ An+1rAn. But then (a3kA : k 6 n) is a sequence of disjoint left translates of A inside A3n+2, whence µ(A3n+2)>(n+ 1)µ(A) =n+ 1, a contradiction.
For the remainder of the paper we fix a K-approximate subgroup A of a (local) group G, and consider the structure whose domain isG, with a predicate forA, and with group multiplication (which is a partial map in caseGis only local). We assume that An ⊆ G for all n < ω (in fact n 6 100 would be enough). Definability and type-definability will be with respect to this structure.
We assume that Ais definably amenable with limn→∞µ(An) =∞. We also fix a setE of size KwithA2⊆EA.
Fact 5 (Ruzsa’s covering lemma). — Let X, Y ⊆ G be definable such that µ(XY)6Kµ(Y). ThenX ⊆ZY Y−1 for some finiteZ ⊆X with |Z|6K.
Proof. — If X = ∅ there is nothing to show. Otherwise, consider a finite subset Z ⊆X such thatzY ∩z0Y =∅for allz6=z0 in Z. By left invariance,
|Z|µ(Y) =µ(ZY)6µ(XY)6Kµ(Y),
so |Z|6K, and there is a maximal suchZ. But then for anyx∈X there isz ∈Z
withzY ∩xY 6=∅by maximality, whence x∈zY Y−1.
J.É.P. — M., 2015, tome 2
Definition6. — A definable subset B⊂ hAiiswide in A ifA is covered by finitely many translates ofB.
Two approximate subgroups are said to be equivalent if each one is contained in finitely many translates of the other.
We will sometimes make explicit the finite constants and say thatBisL-wide inA, or thatAandA∗ areL-equivalent.
Lemma7. — LetB ⊂ hAi be definable.
(1) Ifµ(B)>0, thenBB−1 is wide in Aand symmetric.
(2) If B is wide in A and symmetric, then B is also an approximate subgroup equivalent to A.
Proof
(1) Clearly, BB−1 is symmetric. Since AB is a definable subset of hAi, we have µ(AB)<∞and there isL < ωwithµ(AB)6Lµ(B). By Fact 5, at mostLtranslates ofBB−1are needed to coverA.
(2) There is n < ω such that B2 ⊆ A2n ⊆ E2n−1A. Suppose Y is finite with A⊆Y B. Then
B2⊆E2n−1A⊆E2n−1Y B.
ThusB is an approximate subgroup; being wide inA, it must be equivalent toA.
Definition8. — A type-definable subgroupHof a (local) groupGhasboundedindex if there is some cardinalκsuch that in any elementary extension the index|G:H|is bounded byκ.
Remark9. — By ω+-saturationH has bounded index in Gif and only if for every definable subset X of G and every definable superset Y of H, finitely many left translates ofY coverX.
Lemma10. — If A and A∗ are equivalent, there exists an approximate subgroup in which both are wide, and another one which is wide in both. In particular,hAi ∩ hA∗i will have bounded index in both hAiandhA∗i.
Proof. — SupposeA∗2⊆E∗A∗, and put B=AA∗A, a symmetric set containing A andA∗. IfA⊆XA∗ andA∗⊆X∗A, then
B =AA∗A⊆XA∗A∗A⊆XE∗A∗A⊆XE∗X∗AA⊆XE∗X∗EA⊆XE∗X∗EXA∗, soA andA∗ are wide inB. Moreover
B2⊆XE∗X∗EAB=XE∗X∗EAAA∗A⊆XE∗X∗E2AA∗A=XE∗X∗E2B, so B is also an approximate subgroup. AshAiand hA∗ihave bounded index in hBi, the intersectionhAi ∩ hA∗ihas bounded index in hBi, and thus inhAiand inhAi∗.
Now note that hAi ∩ hA∗i =S
n<ω(An∩A∗n). By ω+-saturation there is n < ω such that finitely many translates of An∩A∗n coverB. SoAn∩A∗n is wide in B,
whence inAand inA∗.
60 J.-C. Massicot & F. O. Wagner
We now turn to the main result. We shall need the following Lemma due to Sanders.
Lemma11. — Let f : ]0,1]→[1, K] and ε >0. Then there exists n < ω depending only onK, εandt >1/(2K)2n−1 such that
f(t2/2K)>(1−ε)f(t).
Proof. — Define a sequence(tn) byt0 = 1 andtk+1 =t2k/2K, sotn = 1/(2K)2n−1. Forn < ω suppose that for alli < nwe havef(ti+1)<(1−ε)f(ti). Then
f(tn)<(1−ε)nf(t0)6(1−ε)nK.
Butf(tn)>1, so ifn < ωis such that(1−ε)nK <1, there must be somei < nwith
f(ti+1)>(1−ε)f(ti).
Theorem 12. — Let A be a K-approximate subgroup. For any m < ω there is a definableL-wide approximate subgroupS withSm⊆A4, whereLdepends only onK andm.
Proof. — Let us show first that if B ⊆ A is definable withµ(B)> tµ(A) for some 0< t61ands=t/2K, then Ais covered byN =b1/sctranslates of
X ={g∈A2:µ(gB∩B)>stµ(A)}
by elements ofA. So suppose not. Then inductively we find a sequence(gi :i6N) of elements ofA such thatµ(giB∩gjB)< stµ(A)for alli < j6N, since forj6N the setS
i<jgiX cannot coverA. But then Kµ(A)>µ(A2)>µ
S
i6N
giB
>(N+ 1)µ(B)− X
i<j6N
µ(giB∩gjB)
>(N+ 1)tµ(A)−N(N+ 1)
2 stµ(A) = (1−Ns
2)(N+ 1)tµ(A)
>(1−1 s s 2)1
stµ(A) =1 2
2K
t tµ(A) =Kµ(A), a contradiction.
However, asµis not supposed to be definable,X need not be definable either. We shall hence look for definable sets with similar properties. To this end, consider the following conditionsPnt(X)on definable subsets ofA, forn < ω and0< t61:
– P0t(B)ifB6=∅.
– Pn+1t (B)ifPnt(B), andAis covered byb2K/tctranslates of
Xn+1t (B) ={g∈A2:Pnt2/2K(gB∩B)andPnt2/2K(g−1B∩B)}.
Clearly, if (Bx)x is a family of uniformly definable subsets of A, then Pnt(Bx) is definable by a formula θnt(x) for alln < ω and 0 < t 61. As Xn+1t (B)⊆ A2, the translating elements for the covering ofAmust come fromA3, so thePnt are definable even in a local group (where we can only quantify over finite powers ofA).
J.É.P. — M., 2015, tome 2
For0 < t61 we consider the family Bt of definable subsetsB ofA withPnt(B) for all n < ω. The first paragraph implies inductively that for definable B ⊆ A, if µ(B) > tµ(A) then Pnt(B) holds, whence B ∈ Bt. In particular, A ∈ Bt so Bt is non-empty. Note thatPnt impliesPnt0 fort>t0, so Bt⊆Bt0.
Define a functionf : ]0,1]→Rby
f(t) = infnµ(BA)
µ(A) :B∈Bt
o .
Fixε >0. Since16f(t)6Kfor all0< t61, by Lemma 11 there ist >0depending only onKandεsuch that
f(t2/2K)>(1−ε)f(t).
ChooseB∈Btwithµ(BA)/µ(A)6(1 +ε)f(t). Put
Xn=Xnt(B) ={g∈A2:Pnt2/2K(gB∩B)andPnt2/2K(g−1B∩B)}
andX=T
n<ωXn. ThenXnis symmetric,Xn+1⊆Xnandb2K/tctranslates ofXn cover A, for alln < ω. By ω+-saturation, b2K/tc translates of X cover A, so X is nonempty. Moreover, forg∈X we havegB∩B∈Bt2/2K, whence
µ(gBA∩BA)>µ((gB∩B)A)>f(t2/2K)µ(A)
>(1−ε)f(t)µ(A)> 1−ε
1 +εµ(BA).
Hence forg∈X,
µ(gBA4BA)6 4ε
1 +εµ(BA)<4ε µ(BA).
It follows that forg1, . . . , gm∈X, µ(g1· · ·gmBA4BA)
6µ((BA4g1BA)∪g1(BA4g2BA)∪ · · · ∪g1· · ·gm−1(BA4gmBA)) 6µ(BA4g1BA) +µ(BA4g2BA) +· · ·+µ(BA4gmBA)
<4m ε µ(BA).
In particular, if ε 6 1/4m, then g1· · ·gmBA∩BA 6= ∅, whence Xm ⊆ A4. By ω+-saturation there isn < ωsuch thatXnm⊆A4. Note thatS:=Xn isb2K/tc-wide in A, and thus an approximate subgroup equivalent toAby Lemma 7.
Corollary13. — There is a type-definable subgroupH ⊆A4such that every definable superset ofH contained inhAiis wide in A.
Proof. — Put S0=Aand apply inductively Theorem 12 form= 8with Si instead ofA, in order to obtain a sequence of approximate subgroups(Si:i < ω)with Si+1 wide in Si (whence in A) and Si+18 ⊆ S4i. Then H = T
i<ωSi4 is a type-definable subgroup ofA4. Any definable superset ofH must contain someSi4byω+-saturation,
and hence be wide inA.
62 J.-C. Massicot & F. O. Wagner
2. Normality
Since we want to consider the quotienthAi/H, we shall look for a stronger version of Theorem 12 whereH will be normal.
Lemma14. — Let X1, . . . , Xn be definable subsets of A with Niµ(Xi) > µ(A) for someNi< ω. Then there is a definableD⊆Asuch that
D−1D⊆(X1X1−1)2∩ · · · ∩(XnXn−1)2 and Kn−1N1· · ·Nnµ(D)>µ(A).
Proof. — Since µ(AX2) 6K µ(A) 6KN2µ(X2), by Fact 5 there are g1, . . . , gKN2 such that
A⊆
KN2
S
i=1
giX2X2−1. Then there is anisuch that
KN1N2µ(X1∩giX2X2−1)>µ(A).
We setD0=X1∩giX2X2−1 and note thatD−10 D0⊆X1−1X1∩(X2X2−1)2.
Then we can iterate the construction, replacing X1 byD0 andX2 byX3. Induc- tively we obtain a suitableD withKn−1N1· · ·Nnµ(D)>µ(A)such that
D−1D⊆X1−1X1∩(X2X2−1)2∩ · · · ∩(XnXn−1)2.
Notice thatD−1D isKn−1N1· · ·Nn-wide in Aby Fact 5.
Theorem15. — Let Abe aK-approximate subgroup, andRa definableN-wide sym- metric subset withR4⊆A4. Then there exists a definableL-wide symmetric subsetS with (S8)A⊆R4, whereLdepends only on K andN.
Proof. — IfA⊆XR, then
R2⊆A4⊆E3A⊆E3XR,
so R is a K3N-approximate subgroup. Theorem 12 yields the existence of some T ⊆ R4 equivalent to R with T48 ⊆ R4. Then T is wide in A and there exists n < ω depending only onK andN and some elementsai ofAsuch that
A⊆
n
S
i=1
aiT.
Consider the measureµon definable subsets ofhAidefined by µ(X) := 1
n
n
X
i=1
µ(Xai).
Clearlyµis left translation invariant, we have µ(A) = 1
n
n
X
i=1
µ(Aai)6n
nµ(A2)6Kµ(A), and
µ(aiT a−1i )> 1
nµ(T)> 1
n2µ(A)> 1 Kn2µ(A).
J.É.P. — M., 2015, tome 2
Since all theaiT a−1i are subsets ofA6 and
K6n2µ(aiT a−1i )>K5µ(A)>µ(A6),
Lemma 14 applied to theK6-approximate subgroupA6yields a subsetD⊆A6 with (K6)n−1(K6n2)nµ(D)>µ(A6)
such that fori= 1,2, . . . , nwe have
S :=D−1D⊆aiT4a−1i .
ThenSis symmetric, wide in AandSai⊆T4fori= 1, . . . , n. SinceA⊆SaiT, this
means thatSA⊆T6, so (S8)A⊆T48⊆R4.
Corollary16. — There is a type-definable normal subgroupHofhAicontained inA4 such that every definable superset ofH contained inhAiis wide in A.
Proof. — As Corollary 13, using Theorem 15 instead of Theorem 12.
References
[1] E. Breuillard, B. Green&T. Tao– “The structure of approximate groups”,Publ. Math. Inst.
Hautes Études Sci.116(2012), p. 115–221.
[2] L. van den Dries – “Approximate groups [after Hrushovski, and Breuillard, Green, Tao]”, in Séminaire Bourbaki (2013/14), Astérisque, Société Mathématique de France, Exp. no1077, to appear.
[3] P. E. Eleftheriou&Y. Peterzil– “Definable quotients of locally definable groups”,Selecta Math.
(N.S.)18(2012), no. 4, p. 885–903.
[4] A. M. Gleason– “Groups without small subgroups”,Ann. of Math. (2)56(1952), p. 193–212.
[5] I. Goldbring– “Hilbert’s fifth problem for local groups”,Ann. of Math. (2)172(2010), no. 2, p. 1269–1314.
[6] E. Hrushovski – “Stable group theory and approximate subgroups”,J. Amer. Math. Soc.25 (2012), no. 1, p. 189–243.
[7] E. Hrushovski&A. Pillay– “On NIP and invariant measures”,J. Eur. Math. Soc. (JEMS)13 (2011), no. 4, p. 1005–1061.
[8] A. Pillay– Private communication, 2014.
[9] T. Sanders– “On a nonabelian Balog-Szemerédi-type lemma”,J. Aust. Math. Soc.89(2010), no. 1, p. 127–132.
[10] H. Yamabe– “A generalization of a theorem of Gleason”,Ann. of Math. (2)58(1953), p. 351–365.
Manuscript received September 17, 2014 accepted April 1, 2015
Jean-Cyrille Massicot, École normale supérieure de Rennes, Campus de Ker lann Avenue Robert Schuman, 35170 Bruz, France
E-mail :[email protected]
Frank O. Wagner, Université Lyon 1, CNRS, Institut Camille Jordan UMR 5208 21 avenue Claude Bernard, 69622 Villeurbanne cedex, France
E-mail :[email protected]
Url :http://math.univ-lyon1.fr/homes-www/wagner/
