Infinite approximate subgroups of soluble Lie groups

We study infinite approximate subgroups of soluble Lie groups. We show that approximate subgroups are close, in a sense to be defined, to genuine connected subgroups. Building upon 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.

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]. 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 [12]. 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 3 below). Along the way, we show a structure theorem for all approximate subgroups in soluble algebraic groups (Theorem 1).
Our main result is concerned with classifying certain soluble approximate subgroups up to compact commensurability: Theorem 1 Let Λ ⊂ GL n (R) be an approximate subgroup generating a soluble subgroup. Then Λ is compactly commensurable to a Zariski-closed soluble subgroup H of GL n (R) normal in the Zariski-closure of Λ ∞ .
Theorem 1 is a non-commutative generalisation of a theorem due to Jean-Pierre Schreiber [19,Proposition 2], which was recently given a new proof and an extension to discrete approximate subgroups of the Heisenberg group 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 (R) (see [8,Theorem 1.6] and Proposition 2 below). Another interesting corollary to this result is that approximate lattices (see [2,Definition 4.9]) in soluble algebraic groups are uniform (see Theorem 5).
An approximate subgroup Λ ⊂ G in a locally compact group is a uniform approximate lattice if Λ 2 is discrete and Λ is 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 ⊂ 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 (G, H , Γ ) consists of two locally compact groups G and H , and a uniform lattice Γ in G × H such that Γ ∩ ({e G } × H ) = {e G×H } and Γ projects densely to H . Given a cut-and-project scheme (G, H , Γ ) 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 (G × W 0 ) ∩ 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 [15] and extended by Michael Björklund and Tobias Hartnick [2].
In a similar fashion, for Λ ⊂ G an approximate subgroup, we say that a group homomorphism f : Λ ∞ → H with H a locally compact group is a good model In this situation, we say that Λ has a good model.
is a uniform approximate lattice constructed from a cut-and-project scheme |Γ is a good model of Λ, where p G and p H are the natural projections on G and H respectively. Conversely, if Λ ⊂ G is a uniform approximate lattice and has a good model f , the map is a cutand-project 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 [13].
In [15] 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 : ]. We answer this question in the soluble Lie case. This generalises, using completely different methods, a previous article by the author that dealt with uniform approximate lattices in nilpotent Lie groups [12].

Theorem 3 Let Λ ⊂ G be a uniform approximate lattice in a connected soluble Lie group. Then Λ is a Meyer subset.
Let us now give a brief overview of the proof strategy for Theorems 1 and 3. 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 ∈ Λ is relatively compact. Then we are able to show that Λ is close to the centre of G (R). The crux of the proof relies on the following fact that is specific to algebraic group homomorphisms: if φ is an algebraic group homomorphism and S is a set that has relatively compact image by φ, then S is contained in ker(φ)K for some compact subset K . This is because φ has (Hausdorff) closed image and is a homeomorphism onto its image (as a consequence of e.g. [25,Propositions 3.16]). 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 finite-index centre.
In order to prove Theorem 3, we first show that, although Λ ∞ is a priori only a soluble group, Λ is commensurable to a uniform approximate lattice Λ that generates a polycyclic group. Using Auslander's embedding theorem on polycyclic groups we embed Λ ∞ as a lattice in some soluble algebraic group. Then Λ is a Meyer subset according to Theorem 1.

Technical results about commensurability
In this section, we recall two technical results that will turn out to be particularly helpful. These are well-known results in the theory of finite approximate subgroups (see for instance [22]), but their proofs do not use the finiteness assumption. We will often use Lemma 1 with Ξ a subgroup. In this situation (Λ k ∩ Ξ) k≥2 is a family of pairwise commensurable approximate subgroups. When considering commensurable approximate subgroups we can obtain further information. In particular, we can see that when Λ and Ξ are genuine subgroups, commensurability as defined here is equivalent to commensurability of subgroups.

Lemma 2
Let Λ ⊂ G be an approximate subgroup and X ⊂ G be a subset such that Λ ⊂ F X for some finite set F. Then Λ 2 ∩ X −1 X is an approximate subgroup commensurable to Λ.
Proof Let F be a finite set such that Λ ⊂ F X and assume that it is minimal for this property with respect to inclusion. For all f ∈ F we can choose x f ∈ Λ ∩ f X. Then for any Now Λ 2 ∩ X −1 X is a symmetric subset containing e and But Λ 4 and Λ 2 ∩ X −1 X are commensurable according to the first part of the proof. We will extend Theorem 4 to soluble real algebraic groups. In the proof of the following proposition we rely on the theory of algebraic groups. See [25, §2 and 3] for a survey of the specific tools we will use and [20] for a general introduction to linear algebraic groups. Note that Proposition 1 below is in fact the main result of this paper. We will call upon Proposition 1 rather than Theorem 1 in the rest of the paper.
The subgroup H is the connected component of the identity of the group of R-points of some normal algebraic subgroup H of G. Moreover, there exists a unique minimal such H i.e. contained in every other normal algebraic subgroup L whose group of R-points is compactly commensurable to Λ.
Proof We start with the uniqueness part. It will be an easy consequence of the following: take two normal algebraic subgroups [25, 3.16]). In particular, if Λ is compactly commensurable to H 1 (R) and H 2 (R), then Λ is compactly commensurable to H 1 (R) ∩H 2 (R). Now, by the descending chain condition for Zariski-closed subsets of G(R) there is a Zariski-closed normal subgroup compactly commensurable to Λ that does not contain any Zariski-closed normal proper subgroup compactly commensurable to Λ. So by the discussion it must be contained in every other Zariski-closed normal subgroup compactly commensurable to Λ.
Let us now move on to proving existence. As . By Lemma 1 we know that Λ is an approximate subgroup. According now to the induction hypothesis there is a closed connected subgroup and Λ are commensurable. Therefore, H 1 and λH 1 λ −1 are compactly commensurable. We thus have Now H 1 is connected, so its normaliser N (H 1 ) is equal to the stabiliser of its Lie algebra. But then N (H 1 ) is Zariski-closed and contains Λ which is Zariski dense. So N (H 1 ) = G(R) and H 1 is normal. Since 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 (e.g. [25,Lemma 3.20]). Therefore, the natural map G(R)/H 1 (R) → (G/H 1 )(R) is an embedding and its image contains the connected component of the identity in (G/H 1 )(R). Moreover, the Zariski-closure of the image of Λ ∞ contains the image of G(R).
Claim The approximate subgroupΛ is compactly commensurable to a Zariski-closed subgroup of the centre Z (G) ofG.
Let us first show how Proposition 1 follows from this claim. There is a Zariskiclosed subgroup V ⊂ Z (G) such thatΛ is compactly commensurable to V . So we can find a compact subset According to the first inclusion we have Λ ⊂ K 1 p −1 (V ). On the other hand, there is Finally, we know that H(R) has finitely many connected components [23], so denoting by H the connected component of the identity in H(R) we indeed find a subgroup satisfying the conclusions of Proposition 1. Now let us move to the proof of the claim. 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 where log is the logarithm map from [G,G] to its Lie algebra. In addition, But by definition of H 1 we know thatΛ ∩ [G,G] is relatively compact. And thanks to Lemma 1 and sinceΛ = p(Λ) 2 the subsetΛ r ∩ [G,G] is an approximate subgroup commensurable toΛ ∩ [G,G] for all r ∈ N. The right-hand side of the above inclusion is therefore a relatively compact subset. Hence, the family of linear operators (Ad(λ) |Lie([G,G]) ) λ∈Λ is uniformly bounded. Since is an algebraic group homomorphism, there is a compact set K ⊂G such that In the following we will denote ker(ρ) by Z . Note that since [G,G] is connected, Z is in fact the centraliser of [G,G]. Now for any g ∈G define the map So θ g is an algebraic group homomorphism.
In particular So θ γ ( f (Λ)) is a relatively compact set. Set now where Υ is any finite family {γ 1 , . . . , γ n } of elements ofΛ. We readily see that We know that Z is an algebraic subgroup and θ Υ is an algebraic group morphism. Moreover, since θ Υ ( f (Λ)) is relatively compact as a subset ofG n and θ Υ (Z ) is closed, it is relatively compact as a subset of θ Υ (Z ). Thus, there is a compact set K 4 such that f (Λ) ⊂ K 4 ker(θ Υ ). Setting K Υ := K 3 K 4 , we getΛ ⊂ K Υ ker(θ Υ ). SinceΛ ∞ is Zariski-dense, the centraliser of elements are Zariski-closed and Zariski-closed subsets satisfy the descending chain condition, we can find Υ ⊂Λ ∞ finite such that the centraliser of Υ is equal to the centre Z (G) ofG. Hence, But Z (G) is a Zariski-closed subgroup, so it has a finite number of connected components [23]. Besides, the connected component of the identity is isomorphic to R k × T l for some k, l ∈ N. There is therefore a central subgroup W ⊂G and a compact subset Relying on the centrality of W we compute And we find We have thus proved that the subset g(Λ) is such that g(Λ) + g(Λ) is compactly commensurable to g(Λ). We then have that g(Λ) is compactly commensurable to an approximate subgroup (see Lemma 4 below). By Schreiber's theorem (Theorem 4) there is therefore a closed connected subgroup V 1 compactly commensurable to g(Λ) -hence toΛ. But V 1 ⊂ Z (G) is compactly commensurable to its Zariski-closure V 2 (see [24,Cororollary 4.1]). So V 2 is compactly commensurable toΛ.
We prove now the two technical lemmas used in the proof of Proposition 1. Proof Let K ⊂ N be a compact subset such that H 2 ⊂ H 1 K . We proceed by induction on the length of the upper central series. If N R n for some n ∈ N, the result is obvious. Otherwise let Z be the centre of N , by induction hypothesis the projections of H 1 and H 2 to N /Z are equal. So given g ∈ H 2 , there is z ∈ Z such that gz ∈ H 1 . Moreover for all n ∈ N there are h n ∈ H 1 and k n ∈ K such that g n = h n k n . Hence, ∀n ∈ N, z n k n = z n h −1 n g n = h −1 n (gz) n ∈ H 1 .

So,
where log is the logarithm map from N to its Lie alebra. But log(H 1 ) is closed (since it is a vector subspace of the Lie algebra) so we find z ∈ H 1 and g ∈ H 1 .

Lemma 4 Let A be a locally compact abelian group and Λ ⊂ V a symmetric subset
such that Λ + Λ is compactly commensurable to Λ. Then Λ is compactly commensurable to an approximate subgroup.
Proof Let U ⊂ V be a symmetric compact neighbourhood of 0, then Λ + U is a symmetric set compactly commensurable to Λ. Moreover, let K ⊂ V be a compact subset such that Λ + Λ ⊂ Λ + K and F ⊂ V be a finite subset such that K +U +U ⊂ U + F. Then we have,

Proof of Theorem 1
Let G be the Zariski-closure of Λ ∞ . Then G is the group of Rpoints of a soluble algebraic group. LetG denote the group of R-points of its Zariskiconnected component of the identity. The subgroup Λ ∞ ∩G is Zariski-dense inG so we can find a finite subset X ⊂ Λ ∞ ∩G such that XG = G. There is therefore an integer n ≥ 1 such that X ⊂ Λ n ∩G; a classical argument-found e.g. in [11]about generation in finite index subgroups now shows thatG ∩ Λ ∞ is generated by Λ 2n+1 ∩G. We show how this theorem is in fact a consequence of Theorem 1.
Proof Without loss of generality we can assume that Γ is Zariski-dense. Let G 0 be the Zariski-connected component of the identity of G. The subgroup Γ ∩ G 0 (R) has finite index in Γ and is Zariski-dense in G 0 (R). Then applying Proposition 1 to Γ ∩ G 0 (R) we get a closed connected normal subgroup H ⊂ G 0 (R) such that Γ is compactly commensurable to H . So the image of Γ ∩ G 0 (R) in G 0 (R)/H via p : G 0 (R) → G 0 (R)/H is contained in a compact subgroup K . Let K 0 be the connected component of the identity of K in the Euclidean topology, then set Γ = Γ ∩ p −1 (K 0 ). The subgroupΓ has finite index in Γ ∩ G 0 (R), hence in Γ , and is co-compact in the closed connected subgroup p −1 (K 0 ).
We also get a generalisation of the well-known fact that closed soluble subgroups of GL n (R) are compactly generated. Note that in Proposition 3 below we do not assume the approximate subgroup Λ considered to be closed. Specialised to (not necessarily closed) soluble subgroups of GL n (R) our conclusion is that any soluble subgroup is generated by a subset that is relatively compact in GL n (R).

Proposition 3 Let G be the R-points of a soluble real algebraic group and Λ ⊂ G an approximate subgroup. Then there is a compact subset K such that
Proof Let H be the Zariski-closure of Λ ∞ . Proceeding as in the proof of Theorem 1 we find an integer k ≥ 2 such that Λ := Λ k ∩ H 0 generates a Zariski-dense subgroup of the Zariski-connected component of the identity H 0 of H . Note moreover that Λ is an approximate subgroup commensurable to Λ (Lemma 2). Applying Proposition 1 to Λ we find a connected subgroup H ≤ G such that G and Λ are compactly commensurable. Since Λ and Λ are commensurable, there is a compact symmetric subset K ⊂ G such that Λ ⊂ K H and H ⊂ K Λ. 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 ∈ Λ choose h ∈ H such that λh −1 ∈ K . Since V generates H , we can find a sequence (h i ) 0≤i≤r of elements of H such that h 0 = e, h r = h and h i+1 h −1 i ∈ V . In addition, we can find a sequence (λ i ) 0≤i≤n of elements of Λ such that λ 0 = e, λ r = λ and for all 0

Corollary 1 If Λ is discrete, then Λ ∞ is finitely generated.
Finally, we give a partial generalisation of a theorem from [2,Theorem 4.25], where it was proved that weak approximate lattices in nilpotent locally compact groups are uniform. Our result is concerned with the more restrictive class of approximate lattices in soluble real algebraic groups. 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 5 Let Λ ⊂ G be a strong approximate lattice in the group of R-points of a connected soluble real algebraic group. Then Λ is relatively dense.
Proof Indeed, according to [2,Remark 4.14.(1) & Theorem 4.18] any approximate lattice is bi-syndetic in an amenable group i.e. there is K 1 ⊂ 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

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 3. 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 then φ is a group homomorphism and φ(Λ) ⊂ 1 m n Z is a discrete approximate subgroup bounded away from 0 so φ(Λ) is finite. As a consequence, is an approximate subgroup commensurable to Λ by Lemma 2.
Consider the diagonal embedding

Now, we can deduce
Proposition 5 Let Λ ⊂ GL n (R) be an approximate subgroup and suppose there are This result is not needed in the sequel, however it gives a good insight into the remaining part of the proof of Theorem 3. Indeed, a similar argument will be used to prove Proposition 7.
Proof According to Corollary 1 any approximate subgroup commensurable to Λ 1 generates a subgroup with finite rank. Take Λ 0 commensurable to Λ 1 generating a group of minimal rank. By Lemma 2 the approximate subgroup Λ 2 0 ∩ Λ 2 1 is commensurable to Λ 1 . But rank((Λ 2 0 ∩ Λ 2 1 ) ∞ ) ≤ rank(Λ ∞ 0 ), so there is equality of ranks. We will therefore assume that Λ 0 ⊂ Λ 2 1 . We also know that for all λ ∈ Λ the approximate group λ(Λ 0 ) is commensurable to λ(Λ 1 ) which in turn is commensurable to Λ 2 . So Λ 0 and λ(Λ 0 ) are commensurable. Hence, Λ 0 is commensurable to and Λ gives an approximate subgroup Ξ commensurable to Λ that consists of automorphisms of Λ ∞ 0 . Rewording the last statement using φ we find Remark 1 From the proof of Proposition 5, we have that for any discrete approximate lattice Λ ⊂ R n the subgroup {g ∈ GL n (R)|g(Λ) is commensurable to Λ} is isomorphic to a subgroup of GL m (Q) 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 3. Our first step towards this goal is to prove a version under an additional assumption.

Proposition 6 Let Λ ⊂ G be a uniform approximate lattice in a connected soluble Lie group. If Λ ∞ is polycyclic then Λ is a Meyer subset.
Proof According to a theorem of Auslander (see [1] or the proof of [17,Theorem 4.28]), Λ ∞ admits an embedding as a Zariski-dense lattice in R the group of R-points of a soluble algebraic group. In the following we will consider Λ ∞ 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 ∞ such that the Zariski closure of Γ is Zariski-connected. Furthermore, proceeding as in the proof of Theorem 1 we can find n ∈ N such that Λ n ∩ Γ is an approximate subgroup commensurable to Λ (Lemma 2) and generates Γ . Now according to Proposition 1 there is a closed connected normal subgroup N R such that Λ is compactly commensurable to N . Let p : R → R/N denote the natural projection. We know that p(Λ) is relatively compact, so we can choose a symmetric compact neighbourhood W 0 of p(Λ). Now p −1 (W 0 ) is an approximate subgroup compactly commensurable to N . But Λ ⊂ Λ ∞ ∩ p −1 (W 0 ) so both subsets are compactly commensurable to the subgroup N . So Λ and Λ ∞ ∩ p −1 (W 0 ) are compactly commensurable i.e. there is a compact subset K ⊂ R such that Hence, Λ is a Meyer subset.

Proposition 7 Let Λ ⊂ G be a uniform approximate lattice in a connected soluble Lie group. Then there is a uniform approximate lattice Λ commensurable to Λ such that (Λ ) ∞ 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 → G be a universal cover, then p −1 (Λ) is a uniform approximate lattice inG. Suppose p −1 (Λ) is commensurable to an approximate subgroup Λ such that Λ generates a polycyclic group. Then p(Λ ) is commensurable to Λ and p(Λ ) generates a polycyclic group as well.
From now on G is supposed simply connected. Let N denote the nilpotent radical of G, k ∈ N and Ξ ⊂ Λ k ∩ N be an approximate subgroup. First of all, let us show that Ξ ∞ is finitely generated. Since G is simply connected, G does not contain any nontrivial 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 [17,Theorem 4.1]) and Ξ is a discrete approximate subgroup. Hence, Ξ ∞ is finitely generated by Corollary 1.
The rest of the proof will rely on the following lemma that links finitely generated subgroups of connected simply connected nilpotent Lie groups to finite dimensional Q-Lie algebras.

Lemma 5 [17, Chapter IV]
Let Γ ⊂ N be a finitely generated group in a connected simply connected nilpotent Lie group. Then Γ is torsion-free nilpotent, Q log(Γ ) is a finite dimensional Q Lie algebra and dim Q (Q log(Γ )) = rank(Γ ).
Here 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 , and log denotes the logarithm map from N to its Lie algebra. Lemma 5 is a consequence of [17, Theorems 2.18, 2.12, 2.10 and 2.11].
The group Ξ ∞ is finitely generated, torsion-free and nilpotent so it has finite rank. Among all approximate subgroups Ξ commensurable to Λ 2 ∩N -note that Λ 2 ∩N is an approximate subgroup by Lemma 1 -such that there is k ∈ N satisfying Ξ ⊂ Λ k ∩ N , choose one with minimal rank. Let Ξ denote this approximate subgroup and let k be such that Ξ ⊂ Λ k ∩ N .
Finally, Λ is a uniform approximate lattice in G as it is commensurable to Λ. So it generates a finitely generated subgroup (as above see the proof of Proposition 3 or [2, Theorem 1.13]). Moreover, H ∩Λ ∞ ⊂ Λ ∞ is a finitely generated torsion-free nilpotent normal subgroup such that Λ ∞ /(H ∩Λ ∞ ) is abelian and finitely generated. Hence, Λ ∞ is polycyclic.

Proof of Theorem 3
Let Λ ⊂ G be a uniform approximate lattice in a connected soluble Lie group. According to Proposition 7 Λ is commensurable to a uniform approximate lattice Λ with Λ polycyclic. By Proposition 6 the uniform approximate lattice Λ is therefore a Meyer set. So Λ is a Meyer set.