Infinite Approximate Subgroups of Soluble Lie Groups

We study infinite approximate subgroups of soluble Lie groups. Generalising a theorem of Fried and Goldman we show that approximate subgroups are close, in a sense to be defined, to genuine connected subgroups. Building up on this result we prove a structure theorem for approximate lattices in soluble Lie groups. This extends to soluble Lie groups a theorem about quasi-crystals due to Yves Meyer.


Introduction
Approximate subgroups were defined by Terence Tao in [16] in order to give a non-commutative generalisation of results from additive combinatorics. On the one hand, finite approximate subgroups have been extensively studied in particular by Ehud Hrushovski [8] and by Emmanuel Breuillard, Ben Green and Terence Tao [4], leading to the structure theorem [4]. This asserts that finite approximate subgroups are commensurable to coset nilprogressions, which are a certain non-commutative generalisation of arithmetic progressions. 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 [8], [5] and [11]. Inspired by Yves Meyer's results on quasi-crystals ( [12]), Michael Björklund and Tobias Hartnick have defined a class of infinite approximate subgroups called approximate lattices in [2]. These approximate subgroups generalise lattices (discrete subgroups of Lie groups with finite co-volume) and share many properties with them. For instance, lattices and approximate lattices in nilpotent Lie groups have a very similar theory, see [9]. Whether similar results hold for other types of locally compact groups is the open question that motivates this article. Here, we address the case of soluble Lie groups (Theorem 2 below). Along the way, we show a structure theorem for all approximate subgroups in soluble algebraic groups (Theorem 1).
A subset Λ of a group G containing the identity is an approximate subgroup if it is symmetric, i.e. Λ " Λ´1, and if there exists a finite subset F Ă G such that Λ 2 Ă F Λ. Here, Λ 2 :" tλ 1 λ 2 |λ 1 , λ 2 P Λu, F Λ :" tf λ|f P F, λ P Λu and more generally Λ n :" tλ 1¨¨¨λn |λ 1 , . . . , λ n P Λu. Moreover, set Λ 8 :" Ť ně0 Λ n the subgroup generated by Λ. We will say that two subsets Λ, Ξ Ă G are commensurable if there is a finite set F such that Λ Ă F Ξ and Ξ Ă F Λ. If G is endowed with the structure of a topological group, we say that subsets Λ, Ξ Ă G are compactly commensurable if there is a compact subset K Ă G with Λ Ă KΞ and Ξ Ă KΛ. Commensurability and compact commensurability are equivalence relations. An approximate subgroup Λ Ă G in a locally compact group is a uniform approximate lattice if it is discrete and compactly commensurable to G. The approximate group condition arises naturally from the combination of discreteness and compact commensurability to the ambient group: if a subset Λ Ă G is symmetric, compactly commensurable to G and Λ 6 is discrete, then Λ is a uniform approximate lattice. See [2] for this and more on the general theory of approximate lattices.
Examples of uniform approximate lattices are given by cut-and-project schemes. A cut-and-project scheme pG, H, Γq is the datum of two locally compact groups G and H, and a uniform lattice Γ in GˆH such that Γ X G " teu and Γ projects densely into H. Given a cut-and-project scheme pG, H, Γq and a symmetric relatively compact neighbourhood W 0 of e H in H, one gets a uniform approximate lattice when considering the projection Λ of pGˆW 0 q X Γ to G. Any approximate subgroup of G which is commensurable to such a Λ is called a Meyer subset of G. This construction was first introduced by Yves Meyer in the abelian case [12] and extended by Michael Björklund and Tobias Hartnick [2].
In a similar fashion, for Λ Ă G a symmetric subset, we say that a group homomorphism f : Λ 8 Ñ H with H a locally compact group is a good model (for Λ) if: piq f pΛq is relatively compact, and piiq there is V a neighbourhood of the identity in H such that f´1pV q Ă Λ. In this situation, we say that Λ has a good model. In particular, note that if Λ has a good model, then Λ is an approximate subgroup.
If Λ :" pGˆW 0 q X Γ is a uniform approximate lattice constructed from a cutand-project scheme pG, H, Γq, then f " p H˝p p G Γ q´1 is a good model for Λ, where p G and p H are the natural projections on G and H respectively. Conversely, if Λ Ă G is a uniform approximate lattice and has a good model f , the map Λ 8 Ñ Gˆf pGq γ Þ Ñ pγ, f pγqq embeds Λ 8 in Gˆf pGq as a uniform lattice. Thus, pG, f pGq, Λ 8 q is a cut-andproject scheme. Therefore, both constructions are equivalent and we will use the latter as it is handier in our case. For further results on good models in groups see [10]. Now, we state our main results. The first theorem is concerned with general approximate subgroups in soluble algebraic groups. Theorem 1. Let Λ Ă GL n pRq be an approximate subgroup generating a soluble subgroup. Then Λ is compactly commensurable to a Zariski-closed soluble subgroup of GL n pRq that is normalised by and contained in the Zariski-closure of an approximate subgroup commensurable to Λ. Theorem 1 is a non-commutative generalisation of a theorem due to Jean-Pierre Schreiber [14,Proposition 2], which was recently given a new proof by Alexander Fish in [6, Theorem 2.2]. Theorem 1 also generalises a result of Fried and Goldman about the existence of syndetic hulls for virtually solvable subgroups of GL n pRq (see [7,Theorem 1.6] and Proposition 2 below). Another interesting corollary to this result is that strong approximate lattices (see [2,Definition 4.9]) in soluble algebraic groups are uniform (see Theorem 4).
In [12] Yves Meyer proved a structure theorem for what later came to be known as mathematical quasi-crystals. Quasi-crystals correspond to uniform approximate lattices in locally compact abelian groups. Rephrased with our terminology, Meyer's theorem becomes : Theorem (Theorem 3.2, [12]). Let Λ be a uniform approximate lattice in a locally compact abelian group G. Then Λ is a Meyer subset.
Motivated by this result the authors of [2] asked whether similar results would hold for other classes of locally compact groups [2,Problem 1.]. We answer this question in the soluble Lie case. This improves, using completely different methods, a previous article by the author that dealt with uniform approximate lattices in nilpotent Lie groups [9]. Let us now give a brief overview of the proof strategy for Theorems 1 and 2. Theorem 1 will be proved by induction on the derived length. We use induction to reduce the proof to the case where λ 1 λ 2 λ´1 1 λ´1 2 |λ 1 , λ 2 P Λ ( is relatively compact. Then we are able to show that Λ is close to the centre of G pRq. The crux of the proof relies on the following fact that is specific to algebraic group homomorphisms: if ϕ is a algebraic group homomorphism and S is a set that has relatively compact image by ϕ, then S is contained in kerpϕqK for some compact subset K. Applied to inner automorphisms, this yields a result reminiscent of a classical theorem of Schur, according to which a group with a finite set of commutators has a finiteindex centre. Finally, we conclude using ideas developed by Alexander Fish in his new proof of the abelian case [6].
In order to prove Theorem 2, we first show that, although Λ 8 is a priori only a soluble group, Λ is commensurable to a uniform approximate lattice Λ 1 that generates a polycyclic group. Using Auslander's embedding theorem on polycyclic groups we embed pΛ 1 q 8 as a lattice in some soluble algebraic group. Then Λ 1 is a Meyer subset according to Theorem 1.

Technical Results about Commensurability
In this section, we prove two technical results that will turn out to be particularly helpful. These are well-known results in the theory of finite approximate subgroups, but their proofs do not use the finiteness assumption.
The first result is about the intersection of commensurable approximate subgroups.
In particular, we can see that when Λ and Ξ are genuine subgroups, commensurability as defined here is equivalent to commensurability of subgroups.
In the same line of ideas, we have the following lemma about intersections of general approximate subgroups. Lemma 2. Let Λ, Ξ Ă G be approximate subgroups. Then Λ 2 X Ξ 2 is an approximate subgroup. Moreover, pΛ k X Ξ k q kě2 is a family of pairwise commensurable approximate subgroups. Therefore, So Λ 4 X Ξ 4 and Λ 2 X Ξ 2 are commensurable and by induction Λ 2 n X Ξ 2 n is commensurable to Λ 2 X Ξ 2 .
In particular, we will often use Lemma 2 with Ξ a subgroup. Then pΛ k X Ξq kě2 is a family of pairwise commensurable approximate subgroups.

Approximate subgroups in soluble linear groups
In this section we prove Theorem 1. Let us first consider approximate subgroups in vector spaces.
Theorem 3 (Proposition 2, [14]). Let V be a real vector space and Λ Ă V an approximate subgroup. There exists a vector subspace W Ă V compactly commensurable to Λ.
In our proof of Proposition 1 we will need a slightly different result that is an easy consequence of Theorem 3.

Lemma 3. Let V be a real vector space and
Proof. According to Theorem 3 we only need to show that Λ is compactly commensurable to an approximate subgroup.
Let U Ă V be a symmetric compact neighbourhood of 0, then Λ`U is a symmetric set compactly commensurable to Λ. Moreover, let K Ă V be a compact subset such that Λ`Λ Ă Λ`K and F Ă V be a finite subset such that K`U`U Ă U`F . Then we have, pΛ`U q`pΛ`U q Ă Λ`K`U`U Ă pΛ`U q`F. Now, we will extend Theorem 3 to soluble real algebraic groups. In the proof of the following proposition we rely on the theory of algebraic groups. See [15] for a general introduction to linear algebraic groups. Proof of Proposition 1. As Λ 8 is Zariski-dense, we know that Λ 8 X rGpRq, GpRqs is Zariski-dense in rGpRq, GpRqs. Moreover, rGpRq, GpRqs is a connected simply connected nilpotent Lie group so Λ 8 X rGpRq, GpRqs is co-compact by [13, Theorem 2.1]. As a consequence, there is k P N greater than 2 such that Λ 1 :" Λ k X rGpRq, GpRqs is an approximate subgroup with Λ 18 Zariski-dense in rGpRq, GpRqs.
According to the induction hypothesis there is a closed connected subgroup H 1 Ÿ rGpRq, GpRqs compactly commensurable to Λ 1 . In addition, for all λ P Λ, we have λ pΛ 1 q λ´1 Ă Λ k`2 X rGpRq, GpRqs. But, according to Lemma 2 approximate subgroups Λ k`2 X rGpRq, GpRqs and Λ 1 are commensurable. Therefore, H 1 and λH 1 λ´1 are compactly commensurable. Indeed, let K Ă rGpRq, GpRqs be a compact subset such that λH 1 λ´1 Ă H 1 K. We proceed by induction on the length of the upper central series. If rGpRq, GpRqs » R n for some n P N, the result is obvious. Otherwise, let Z be the centre of rGpRq, GpRqs, by induction hypothesis the projections of H 1 and λH 1 λ´1 to rGpRq, GpRqs{Z are equal. So choose g P λH 1 λ´1zH 1 , there is z P Z such that gz P H 1 , moreover for all n P N there are h n P H 1 and k n P K such that g n " h n k n . As a consequence, @n P N, z n k n " z n h´1 n g n " h´1 n pgzq n P H 1 .
Thus, @n P N, logpzq`l ogpk n q n P logpH 1 q, where log is the logarithm map from rGpRq, GpRqs to its Lie alebra. But logpH 1 q is closed (since log is a homeomorphism) so we get z P H 1 and g P H 1 , hence Claim 1.
Now, Λ is Zariski dense and H 1 is connected, so H 1 is normal. Moreover, as H 1 is connected in a unipotent subgroup, H 1 is the group of R-points of an algebraic subgroup H 1 normal in G.
Therefore, the natural map GpRq{H 1 pRq Ñ pG{H 1 qpRq is an embedding and its image contains the connected component of the identity in pG{H 1 qpRq. Moreover, the Zariski-closure of the image of Λ 8 contains the image of GpRq.
Claim 2. The approximate subgroupΛ is compactly commensurable to a closed connected subgroup of the centre ZpGq ofG.
Let us first show how Proposition 1 follows from this claim. There is some closed connected subgroup V Ă ZpGq such thatΛ is compactly commensurable to V . Hence, we can find a compact subset K 1 Ă GpRq such that Λ Ă ppK 1 qV and V Ă ppK 1 qΛ.
In particular, according to the first inclusion, Λ Ă K 1 p´1pV q. On the other hand, there is K 2 Ă GpRq compact such that H 1 Ă K 2 Λ, where H 1 is the subgroup defined above. Finally, So p´1pV q is the subgroup we are looking for. Now, let us move to the proof of Claim 2. We will use the fact that the set of commutators of elements ofΛ is relatively compact to show thatΛ is contained in a 'neighbourhood' of the centre.
The groupΛ 8 X rG,Gs is co-compact in rG,Gs. So there is k P N such that where log is the logarithm map from rG,Gs to its Lie algebra. In addition, ď λPΛ λ´Λ k X rG,Gs¯λ´1 ĂΛ k`2 X rG,Gs.
But the right-hand side is a relatively compact set. Hence, pAdpλq LieprG,Gsq q λPΛ is a uniformly bounded family of linear operators. Since ρ : pG{HqpRq Ñ GLpLieprG,Gsqq g Þ Ñ Adpgq LieprG,Gsq is an algebraic group homomorphism, there is a compact set K ĂG such that Λ Ă kerpρqK.
The Kernel Z :" kerpρq has finitely many connected components so there is a compact set K 3 such thatΛ Ă Z 0 K 3 where Z 0 is the connected component of the identity. Now, for any g PG define the map where rg, hs denotes ghg´1h´1. For h 1 , h 2 P Z we have So θ g is an algebraic group homomorphism. For all λ PΛ let f pλq denote an element of Z such that λf pλq´1 P K 3 . Now, for γ PΛ and λ PΛ we have γλγ´1λ´1 P γK 3 γ´1θ γ pf pλqqK´1 3 .
We know that Z is an algebraic subgroup and θ F is an algebraic group morphism. Moreover, since θ F pf pΛqq is relatively compact as a subset ofG n and θ F pZq is closed, it is relatively compact as a subset of θ F pZq. Thus, there is a compact set Moreover,Λ 8 is Zariski-dense so we can choose F such that it generates a Zariski dense subgroup. Hence,Λ where ZpGq is the centre ofG. But ZpGq has a finite number of connected components and the connected component of the identity is isomorphic to R kˆTl for some k, l P N. Therefore, there is a central subgroup W ĂG and a compact subset K 5 ĂG such thatΛ Ă K 5 W and W » R k Finally, choose a function g :Λ Ñ W such that for all λ PΛ, gpλ´1q " gpλq´1 and bpλq :" gpλqλ´1 P K 5 . There is a finite subset F Ă G such that for all λ 1 , λ 2 PΛ there is λ PΛ satisfying λλ´1 1 λ´1 2 P F . As a consequence, gpλqgpλ 1 q´1gpλ 2 q´2 " λλ´1 1 λ´1 2 bpλqbpλ 1 q´1bpλ 2 q´1 P F K 5 K´2 5 .
By Lemma 3 we obtain a closed connected subgroup V 1 compactly commensurable toΛ. As V 1 Ă ZpGq it is compactly commensurable to its Zariski-closure V 2 . The connected component of the identity of V 2 is the subgroup we are looking for.
Proof of Theorem 1. Let G be the Zariski-closure of Λ 8 . Then G is the group of R-points of a soluble algebraic group. LetG denote the group of R-points of its Zariski-connected component of the identity. Proposition 1 applied to Λ 2 XG yields Theorem 1.

Consequences of Theorem 1
In [7] Fried and Goldman proved that every soluble subgroup H of GL n pRq admit a syndetic hull i.e. a closed connected subgroup of GL n pRq containing H and such that H is co-compact in it. We show how this theorem is a consequence of Theorem 1.
Proposition 2 (Theorem 1.6,[7]). Let G be the R-points of a soluble real algebraic group and H a subgroup. Then there is B ă G such that B is a closed connected subgroup (for the Euclidean topology), H X B has finite index in H and H and B are compactly commensurable.
Proof. Without loss of generality we can assume that H is Zariski-dense. Then applying Proposition 1 to H we get a closed connected normal subgroup B Ÿ G such that H is compactly commensurable to B. So the image of H in G{B via p : G Ñ G{H is contained in a compact subgroup K. Let K 0 be its connected component of the identity in the Euclidean topology, then setH " H X p´1pK 0 q. The subgroupH has finite index in H and is co-compact in p´1pK 0 q.
We also get a generalisation of the well-known fact that closed soluble subgroups of GL n pRq are compactly generated.
Proposition 3. Let G be the R-points of a soluble real algebraic group and Λ Ă G an approximate subgroup. Then there is a compact subset K such that Λ 2 X K generates Λ 8 .
Proof. As a consequence of Proposition 1, there is a connected subgroup H ď G such that G and Λ are compactly commensurable. Let K Ă G be a compact symmetric subset such that Λ Ă KH and H Ă KΛ. Choose also V a compact neighbourhood of the identity in H. As H is connected for the Euclidean topology, V generates H. Now, for any λ P Λ choose h P H such that λh´1 P K. Since V generates H, we can find a sequence ph i q 0ďiďr of elements of H such that h 0 " e, h r " h and h i`1 h´1 i P V . In addition, we can find a sequence pλ i q 0ďiďn of elements of Λ such that λ 0 " e, λ r " λ and for all 0 ď i ď r, λ i h´1 i P K´1 " K. Thus, λ i`1 λ´1 i P KV´1K´1. Finally, Λ 8 is generated by Λ 2 X KV´1K´1.
In particular, when Λ is discrete, this implies that Λ 8 is finitely generated. This fact will be used in the proof of Theorem 2 below.
Finally, we generalise a theorem from [2,Theorem 4.25], who handled the nilpotent case. This result is concerned with strong approximate lattices. Strong approximate lattices are defined by measure-theoretic conditions on an associated dynamical system called the invariant hull. We refer the reader to [2, Section 4] for precise definitions.
Theorem 4. Let Λ Ă G be a strong approximate lattice in the group of R-points of a soluble real algebraic group. Then Λ is relatively dense.
Proof. Indeed, according to [2,Theorem 4.18] any strong approximate lattice is bi-syndetic i.e. there is K 1 Ă G compact such that G " K 1 ΛK 1 . Moreover, Λ is Zariski-dense according to [3]. Now let H and K 2 be given by Proposition 1 so that Since H is normal we have that G "

Uniform approximate lattices in abelian groups
We will investigate morphisms commensurating approximate subgroups in R n . This will turn out to be useful in the proof of Theorem 2. Our goal is to understand morphisms that commensurate a uniform approximate lattice. Let us start with a result concerning lattices.
Proof. We can assume that Γ 1 " Z n . Let m be the order of Γ 2 {Γ 1 and p 1 , . . . , p r the prime factors of m. Then any matrix in Λ has entries lying in 1 m Z. Set ϕ : GL n pRq Ñ RM Þ Ñ | detpM q| then ϕ is a group homomorphism and ϕpΛq Ă 1 m n Z is a discrete approximate subgroup bounded away from 0 so ϕpΛq is finite. As a consequence, is an approximate subgroup commensurable to Λ. Now,Λ is commensurable to the approximate subgroup Λ 2 X ι´1p r ź i"1 SL n pZ pi qq "Λ 2 X SL n pZq.

Now, we can deduce
Proposition 5. Let Λ Ă GL n pRq be an approximate subgroup and suppose there are Λ 1 Ă Λ 2 approximate lattices in R n such that λpΛ 1 q Ă Λ 2 for all λ P Λ.
Then there are Ξ Ă Λ 4 commensurable to Λ and an injective group homomorphism Ξ 8 Ñ SL m pZq for some m ě n. .
This result is not needed in the sequel, however it gives a good insight into the remaining part of the proof of Theorem 2. Indeed, a similar argument will be used to prove Proposition 7.
Proof. For any Ξ commensurable to Λ 1 the subgroup Ξ 8 has finite rank. Choose Ξ commensurable to Λ 1 with minimal rank, then Ξ 2 X Λ 2 1 is a uniform approximate lattice as well and rankpΞ 2 X Λ 2 1 q ď rankpΞ 2 q, so there is equality. Thus, we can assume that Ξ Ă Λ 2 1 . As a consequence, for all λ P Λ the approximate group λpΞq is commensurable to λpΛ 2 1 q which in turn is commensurable to Λ 2 2 . So Ξ and λpΞq are commensurable. Hence, Ξ is commensurable to Ξ 2 X λpΞ 2 q. By minimality of rankpΞ 8 q we get that rankppΞ 2 X λpΞ 2 qq 8 q " rankpΞ 8 q " rankpλpΞ 8 qq, Therefore, λ is an isomorphism of span Q pΞq and as span R pΞq " R n we get an injective morphism Λ 8 Ñ GL m pQq where m " rankpΞq.
Finally, for all λ P Λ we have λpΞ 8 q Ă Λ 8 2 X span Q pΞq. Since rank`Λ 8 2 X span Q pΞq˘" dim Q pspan Q pΞqq we get that Ξ 8 has finite index in Λ 8 2 X span Q pΞq. So Proposition 4 applied to Ξ 8 , Λ 8 2 X span Q pΞq and Λ gives the desired morphism. Remark 1. From the proof of Proposition 5, we have that for any discrete approximate lattice Λ Ă R n the subgroup tg P GL n pRq|gpΛq is commensurable to Λu is isomorphic to a subgroup of GL m pQq where m is the minimal rank of an approximate subgroup commensurable to Ξ.

Meyer's Theorem for soluble Lie groups
We will now turn to the proof of Theorem 2. As a first step, let us prove it with an additional assumption. Proposition 6. Let Λ Ă G be a uniform approximate lattice in a connected soluble Lie group. If Λ 8 is polycyclic it is a Meyer subset.
Proof. According to a theorem of Auslander (see [1] or the proof of [13,Theorem 4.28]), Λ 8 admits an embedding as a Zariski-dense lattice in R the group of Rpoints of a soluble algebraic group. In the following we will consider Λ 8 as a subgroup of R. Moreover, we can assume without loss of generality that R is Zariski-connected. Indeed, there is a finite index subgroup Γ of Λ 8 such that the Zariski closure of Γ is Zariski-connected. Furthermore, the approximate subgroup Λ 2 X Γ is commensurable to Λ according to Lemma 1. Now, according to Proposition 1 there is a closed connected normal subgroup N Ÿ R such that Λ is compactly commensurable to N . Let p : R Ñ R{N denote the natural projection. We know that ppΛq is relatively compact, so we can choose a compact neighbourhood W 0 of ppΛq. Now, Λ is compactly commensurable to Λ 8 X p´1pW 0 q, so there is a compact subset K Ă R such that Λ Ă KpΛ 8 X p´1pW 0 qq and Λ 8 X p´1pW 0 q Ă KΛ.
But Λ 8 is a discrete subgroup in R so K X Λ 8 is finite and Λ is commensurable to Λ 8 X p´1pW 0 q.
Finally, p pΛ 8 Xp´1pW0qq 8 is a good model for Λ 8 Xp´1pW 0 q . Hence, Λ is a Meyer subset.
Proposition 7. Let Λ Ă G be a uniform approximate lattice in a connected soluble Lie group. Then there is Λ 1 commensurable to Λ such that pΛ 1 q 8 is polycyclic.
Proof. Let us first show that we can assume G to be simply connected. Indeed, if G is not simply connected we proceed as follows. Let p :G Ñ G be a universal cover, then p´1pΛq is a uniform approximate lattice inG. Suppose p´1pΛq is commensurable to an approximate subgroup Λ 1 such that Λ 1 generates a polycyclic group. Then ppΛ 1 q is commensurable to Λ and ppΛ 1 q generates a polycyclic group as well.
From now on G is supposed simply connected. Let N denote the nilradical of G, k P N and Ξ Ă Λ k X N be an approximate subgroup.
First of all, let us show that Ξ 8 is finitely generated. Since G is simply connected, G does not contain any non-trivial compact subgroup. So N does not contain any non-trivial compact subgroup, and thus N is simply connected. Now, N is a connected simply connected nilpotent Lie group so it is the group of R-points of a unipotent algebraic group (see [13,Theorem 4.1]) and Ξ is a discrete approximate subgroup. Hence, Ξ 8 is finitely generated by Proposition 3.
The proof will rely on the following lemma that links finitely generated subgroups of connected simply connected nilpotent Lie group to finite dimensional Q Lie algebras.
Lemma 4. [13, Chapter IV] Let Γ Ă N be a finitely generated group in a connected simply connected nilpotent Lie group. Then Γ is torsion-free nilpotent, Q logpΓq is a finite dimensional Q Lie algebra and dim Q pQ logpΓqq " rankpΓq.
Where the rank of Γ is the dimension of its Malcev completion, i.e. the unique connected simply connected nilpotent Lie group that admits a lattice isomorphic to Γ. Lemma 4 is a consequence of [13, Theorems 2.18, 2.12, 2.10 and 2.11]. Now, Ξ 8 is a finitely generated torsion-free nilpotent group so it has finite rank. Among all approximate subgroups Ξ commensurable to Λ 2 X N such that there is k P N satisfying Ξ Ă Λ k X N , choose one with minimal rank. Let Ξ denote this approximate subgroup and let k be such that Ξ Ă Λ k X N . Now, for λ P Λ, Ξ and λΞλ´1 are contained in and commensurable to Λ k`2 X N , so Ξ 2 X λΞ 2 λ´1 is commensurable to Ξ. But`Ξ 2 X λΞ 2 λ´1˘8 Ă Ξ 8 so they have the same rank. As a consequence, it is a finite index subgroup, so the groups Ξ 8 and λΞ 8 λ´1 are commensurable.
Proof of Theorem 2. Let Λ Ă G be a uniform approximate lattice in a connected soluble Lie group. According to Proposition 7 Λ is commensurable to an approximate subgroup Λ 1 with Λ 1 polycyclic. Now, by Proposition 6 the approximate subgroup Λ 1 is a Meyer set, so Λ is a Meyer set as well.

Acknowledgements
I am deeply grateful to my supervisor, Emmanuel Breuillard, for his patient guidance and encouragements.