On the structure and arithmeticity of lattice envelopes
We announce results about the structure and arithmeticity of all possible lattice embeddings of a class of countable groups which encompasses all linear groups with simple Zariski closure, all groups with non-vanishing first -Betti number, word hyperbolic groups, and, more general, convergence groups.
Key words and phrases:Lattices, locally compact groups
2000 Mathematics Subject Classification:Primary 22D99; Secondary 20F65
Let be a countable group. We are concerned with the study of its lattice envelopes, i.e. the locally compact groups containing as a lattice. We aim at structural results that impose no restrictions on the ambient locally compact group and only abstract group-theoretic conditions on . We say that satisfies if every finite index subgroup of a quotient of by a finite normal subgroup
is not isomorphic to a product of two infinite groups, and
does not possess infinite amenable commensurated subgroups, and
satisfies: For a normal subgroup and a commensurated subgroup with there exists a finite index subgroup such that and commute.
Let be a lattice in a locally compact group . If has no infinite amenable commensurated subgroups, then the amenable radical of is compact.
The role of 3 is less transparent, but be aware of the obvious observation: If are both normal with , then and commute. There are lattices in , where is the universal cover of the -skeleton of the Bruhat-Tits building of (see [4, 6.C] and [3, Prop. 1.8]). They are built in such a way that , which is a free group of infinite rank, is a normal subgroup of . Let be the stabilizer of a vertex. Then is commensurated, but violate 3. Moreover, this group satisfies 1 and 2.
Linear groups with semi-simple Zariski closure satisfy the conditions 2 and 3. Groups with some positive -Betti number satisfy the condition 2. All the conditions are satisfied by all linear groups with simple Zariski closure, by all groups with positive first -Betti number, by all non-elementary word hyperbolic or, more generally, convergence groups.
For a concise formulation of our main result, we introduce the following notion of -arithmetic lattice embeddings up to tree extension: Let be a number field. Let be a connected, absolutely simple adjoint -group, and let be a set of (equivalence classes) of places of that contains every infinite place for which is isotropic and at least one finite place for which is isotropic. Let denote the -integers. The (diagonal) inclusion of (a finite index subgroup of) into is the prototype of an -arithmetic lattice. Let be a group obtained from by possibly replacing each factor with -rank by an intermediate closed subgroup where is a tree with a cofinite -action. The lattice embedding into is called an -arithmetic lattice embedding up to tree extension.
A typical example is embedded diagonally as a lattice into . The latter is a closed cocompact subgroup of , where is the Bruhat-Tits tree of , i.e. a -regular tree. So is an -arithmetic lattice embedding up to tree extension. We now state the main result :
Let be a finitely generated group satisfying , e.g one of the groups considered in Proposition 1.2. Then every embedding of as a lattice into a locally compact group is, up to passage to finite index subgroups and dividing out a normal compact subgroup of , isomorphic to one of the following cases:
an irreducible lattice in a center-free, semi-simple Lie group without compact factors;
an -arithmetic lattice embedding up to tree extension;
a lattice in a totally disconnected group with trivial amenable radical.
The same conclusion holds true if one replaces the assumption that is finitely generated by the assumption that is compactly generated.
Finite generation of implies compact generation of any locally compact group containing as a lattice. The examples above for show that condition 3 in Theorem 1.3 is indispensable. Since non-uniform lattices with a uniform upper bound on the order of finite subgroups do not exist in totally disconnected groups, our main theorem yields the following classification of non-uniform lattice embeddings.
Let be a group that satisfies and admits a uniform upper bound on the order of all finite subgroups. Then every non-uniform lattice embedding of into a compactly generated locally compact group is, up to passage to finite index subgroups and dividing out a normal compact subgroup of , either a lattice in a center-free, semi-simple Lie group without compact factors or an -arithmetic lattice embedding up to tree extension.
The following arithmeticity theorem  is at the core of the proof of Theorem 1.3. Actually, it is a more general version that is used in which we drop condition 1 (see the comment in Step 3 of Section 2).
In the proof of Theorem 1.3 we only need Theorem 1.5 in the case where , thus , is compactly generated which means that the set of primes is finite. Caprace-Monod [4, Theorem 5.20] show Theorem 1.5 for compactly generated and under the hypothesis that is the -points of a simple -group (where is a local field) but the latter hypothesis is too restrictive for our purposes. Morevover, our proof of Theorem 1.5 does not become much easier if we assume compact generation from the beginning. Regardless of its role in Theorem 1.3 we consider the following result as a first step in the classification of lattices in locally compact groups that are not necessarily compactly generated.
Let be a connected center-free semi-simple Lie group without compact factors, and let be a totally disconnected locally compact group without compact normal subgroups. Let be a lattice such that the projections of to both and are dense and the projection of to is injective and satisfies 1.
Then there exists a number field , a (possibly infinite) set of places of , and a connected adjoint, absolutely simple -group such that the following holds:
Let be the simply connected cover of in the algebraic sense. Let be the -integers of . The group embeds as a closed subgroup into the restricted (adelic) product . Under this embedding and under passing to a finite index subgroup, is contained in and the intersection of with the image of is commensurable to the image of .
The above theorem states essentially that all lattice inclusions satisfying some natural conditions could be constructed from arithmetic data. The following example describes, up to passing to finite index subgroups, all pairs of groups , and embeddings satisfying the condition of the theorem, for the special case . These are obtained by the choice of the set and the subgroups and described below. Similar classifications for other semi-simple groups could be achieved using Galois cohomology.
Fix a possibly infinite set of primes and consider the localization . Fix a closed subgroup in the compact group and a subgroup in the discrete group (note that for and ). The determinant homomorphism from to the corresponding idele group, which we naturally identify with , restricts to . The determinant is well defined on modulo squares, thus we obtain a map which restricts to . Let be the preimage of under the first map and be the preimage of under the second. Denote and embed in via . Thus embeds diagonally in and its image is a lattice, whose projections to both and are dense and the projection of to is injective while satisfies 1.
2. Sketch of proof of Theorem 1.3
Step 1: Using 2 to reduce to products.
Burger and Monod [2, Theorem 3.3.3] observed that one obtains as a consequence of the positive solution of Hilbert’s 5th problem: Every locally compact group has a finite index subgroup that modulo its amenable radical splits as a product of a connected center-free semi-simple real Lie group without compact factors and a totally disconnected group with trivial amenable radical. By Proposition 1.1 we conclude that the amenable radical of , thus that of any of its finite index subgroups, is compact. Therefore we may assume (up to passage to finite index subgroups and by dividing out a normal compact subgroup) that is of the form with and as above.
If we assume that not all -Betti numbers of vanish, we can reach the same conclusion without appealing to Proposition 1.1 but by using -Betti numbers of locally compact groups  instead. Since has a positive -Betti number in some degree, the same is true for [7, Theorem B], thus has a compact amenable radical [7, Theorem C].
Step 2: Separating according to discrete and dense projections to the connected factor
The connected Lie group factor splits as a product of simple Lie groups . The projection of to might not have dense image. It is easy to see that there is a maximal subset such that the projection of to has discrete image. Then and are lattices in and , respectively. So we obtain an extension of groups
which are lattices in the corresponding (split) extension of locally compact groups . The projection of to turns out to be dense.
Notice that finite generation of does not guarantee that is finitely generated. However is still compactly generated if is so.
Step 3: Distinguishing cases of the theorem
Let be a compact open subgroup. Let and . We prove that is totally disconnected if is finite and that if is infinite, but is finite. The latter step involves condition 1. Hence if or is finite, the proof is finished. In the remainder we discuss the case that is infinite. For simplicity let us first assume that the projection of to is dense; we return to this issue in the last step.
Consider which can also be regarded as a subgroup of . As such it is also normal by denseness of the projection . The assumptions of Theorem 1.5 apart from condition 1 are satisfied for the lattice embedding . From a more general version of Theorem 1.5 one concludes a posteriori that satisfies 1, so the conclusion of Theorem 1.5 holds true for . Because of compact generation of we can exclude the adelic case and conclude that is an -arithmetic lattice for a finite set of primes.
Step 4: Using 3 to identify group extensions
In this step we show that is finite, thus trivial (since has no compact normal subgroups). The proof involves the use of 3 for and and Margulis’ normal subgroup theorem. Hence is an -arithmetic lattice. As such it has a finite outer automorphism group which implies that, after passing to finite index subgroups, the extension (1) splits. By 1 is an -arithmetic lattice embedding.
Step 5: Quasi-isometric rigidity results
We have previously assumed that the projection is dense. If it is not we have to identify the difference between the closure of the image of the projection and . The subgroup is cocompact, thus is a quasi-isometry. By the argument before we know that is a product of algebraic groups over non-Archimedean fields and thus acts by isometries on a product of Bruhat-Tits buildings. By conjugating with we obtain a homomorphism of to the quasi-isometry group of . We finally appeal to the quasi-isometric rigidity results of Kleiner-Leeb  and Mosher-Sageev-White  to conclude that is an -arithmetic lattice up to tree extension.
3. Special cases
It is instructive to investigate the consequences of Theorem 1.3 for specific groups. Rather than just applying Theorem 1.3 we sketch a blend of ad hoc arguments and techniques of the proof of Theorem 1.3 to most easily classify all lattice embeddings in the following three cases.
3.1. is a free group.
Let be a non-commutative finitely generated free group. Let be a lattice embedding. We show that, up to finite index and dividing out a normal compact subgroup, is or embeds as a closed cocompact subgroup in the automorphism group of a tree.
As explained in the first step of Subsection 2 one can avoid the use of Proposition 1.1 by using the positivity of the first -Betti number of to conclude that has a compact amenable radical. Up to passage to a finite index subgroup and dividing out a compact amenable radical we may assume that is a product . By the Künneth formula can have positive first -Betti number only if one of the factors is compact. Since has trivial amenable radical, this implies that is either or . In the first case must be . In the second case is totally disconnected, and as a torsion-free lattice must be cocompact. By [8, Theorem 9] embeds as a closed cocompact subgroup of the automorphism group of a tree.
3.2. is a surface group.
Let be the fundamental group of a closed oriented surface of genus . Let be a lattice embedding. Similarly as for free groups, by using the positivity of the first -Betti number, we conclude that , up to passage to a finite index subgroup and dividing out a compact amenable radical, is either or a totally disconnected group with trivial amenable radical. In the latter case is cocompact.
We argue that the totally disconnected case cannot happen unless is discrete and so is the trivial lattice embedding: The inclusion is a quasi-isometry in that case. So we obtain a homomorphism . Each quasi-isometry induces a homeomorphism of the boundary , so we obtain a homomorphism . One can verify that is continuous [5, Theorem 3.5] and is compact, thus trivial by the triviality of the amenable radical. Let be a compact-open subgroup. Then is a compact subgroup, hence is either finite or isomorphic to [5, Lemma 3.6]. But it cannot be isomorphic to a connected group. Therefore is finite, which implies that is discrete.
3.3. , .
Recall that embeds as a non-uniform lattice in via ; we denote by and the injective projections. Let us verify 2: For any commensurated amenable subgroup , the connected component of the Zariski closure is amenable and normal in because replacing by a finite index subgroup does not change . Hence is trivial, and so and are finite.
Let be embedded as a lattice in some locally compact group . Using 2 as in the first step of Subsection 2 we replace by , where is a (possibly trivial) connected real Lie group, is totally disconnected, and both have trivial amenable radicals. Let denote the closure of ; then where is totally disconnected, and has finite covolume in .
Case corresponds to the trivial lattice embedding . Indeed, in this case is a lattice in a totally disconnected , and having bounded torsion, it is cocompact. This allows us to use the results on quasi-isometric rigidity  and obtain a homomorphism (hereafter stands for commensurability) that can be further shown to have an image commensurable to .
If is non-trivial, then it is a center-free, semi-simple Lie group without compact factors. By Borel’s density theorem the projection map has Zariski dense image, and Margulis’ superrigidity implies that and (this can also be shown by elementary means by conjugating unipotent matrices). Let denote the closure of ; we get a lattice embedding where is totally disconnected, and has finite covolume in . If is a compact open subgroup, is a lattice in , that projects to a lattice . We claim that . Indeed, it follows from Margulis’ superrigidity (recall that ) applied to that is contained in a maximal compact subgroup commensurated to , yielding .
Let be the closure in of . Then is the closure of , thus compact because of . Also is compact. The group is compact, hence closed and equals . Since is dense in and contains the open subgroup , we obtain that . Similarly, . This implies that is a graph of a continuous surjective homomorphism whose kernel is contained in , thus compact.
Finally, to recover the original group , one uses the fact that is cocompact in , so quasi-isometric rigidity  of the Bruhat-Tits building of gives .
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