As it was announced in the end of the first post of this series, we will discuss today the first half of the proof of the following result:

Theorem 1Let , , and . Suppose that is a discrete and Zariski dense subgroup of such that is cocompact. Then, is commensurable to some -form of .

Remark 1This statement is originally due to Hee Oh, but the proof below is a particular case of Benoist–Miquel’s arguments. In particular, our subsequent discussions can be generalized to obtain the statement of Theorem 1 of the previous post in full generality.

Remark 2Theorem 1 is not true without the higher rank assumption (i.e., ): indeed, has infinite index in .

Our task is to construct a -form satisfying the conclusions of Theorem 1. This is not very easy because it must cover all possible cases of -forms such as:

Example 1

- ;
- where are integers in a division algebra over ;
- “”, i.e., where is Galois conjugation (and is the transpose of ).

Before trying to construct adequate -forms, let us make some preliminary reductions.

We denote by the *parabolic subgroup* normalizing of in : more concretely,

where .

Next, we consider . In the literature, is called an opposite horospherical subgroup to .

Since is Zariski dense in , there exists such that (i.e., ). By taking a basis of such that , we have that

In particular, is cocompact in (thanks to the assumptions of Theorem 1).

Remark 3This is the one of the few places in Benoist–Miquel argument where the Zariski denseness of is used.

Remark 4In general, the argument above works when isreflexive, that is, is conjugated to an opposite horospherical subgroup .

We denote by an opposite parabolic subgroup, and

the common Levi subgroup of and . In particular, we have decompositions (in semi-direct products)

Let and be the Lie algebras of and . Note that and (resp.) are lattices in and (resp.). In other terms,

where is the space of lattices in .

Remark 5Note that for all in the context of the example and .

Observe that is the intersection of the normalizers of and . Therefore, acts on the spaces of lattices and via the adjoint map (i.e., by conjugation).

As it turns out, the key step towards the proof of Theorem 1 consists in showing that the -orbits of and are closed. In other terms, the proof of Theorem 1 can be divided into two parts:

- closedness of the -orbits of and ;
- construction of the -form based on the closedness of the -orbits above.

In the remainder of this post, we shall establish the closedness of relevant -orbits. Then, the next post of this series will be dedicated to obtain an adequate -form (i.e., arithmeticity) from this closedness property.

Remark 6Hee Oh’s original argument used Ratner’s theory for the semi-simple part of to derive the desired closedness property. The drawback of this strategy is the fact that it doesn’t allow to treat some cases (such as ), and, for this reason, Benoist and Miquel are forced to proceed along the lines below.

**1. Closedness of the -orbit of **

Consider Bruhat’s decomposition (where is the Lie algebra of ) and the corresponding projection .

Given , set , i.e.,

and consider the Zariski open set

Our first step towards the closedness of the -orbit of is to exploit the discreteness of and the commutativity of in order to get that the actions of the matrices , , on the vectors of the lattice do not produce arbitrarily short vectors:

*Proof:* Let with , , , and such that .

Our task is to show that for all sufficiently large.

For this sake, note that a direct calculation reveals that for all and . In particular, . Now, we use the cocompactness of to write with and (modulo taking subsequences).

By definition, . Since is *commutative*, and hence

as . Because is discrete, it follows that for all sufficiently large. Therefore, for all sufficiently large. This completes the proof of the proposition.

Remark 7The fact that iscommutativeplays a key role in the proof of this proposition.

Next, we shall combine this proposition with Mahler’s compactness criterion to study the set of determinants of the matrices for .

Proposition 3Let for . Then, the set is closed and discrete in .

*Proof:* Given such that , we want to show that for all sufficiently large.

By contradiction, let us assume that this is not the case. In particular, there is a subsequence with , i.e., , and also for all . Note that, by definition, is the covolume of .

By Proposition 2, doesn’t have small vectors. Since these lattices also have bounded covolumes (because ), we can invoke Mahler’s compactness criterion to extract a subsequence such that . In this setting, Proposition 2says that we must have for all sufficiently large, so that for all sufficiently large, a contradiction.

Now, we shall modify to obtain a polynomial function on . For this sake, we recall the element introduced in (1) (conjugating and ). In this context,

is a polynomial function of . Moreover, an immediate consequence of Proposition 3 is:

The polynomial is relevant to our discussion because it is intimately connected to the action of :

- on one hand, a straighforward calculation reveals that is proportional to for all (i.e., for some [explicit] ): see Lemma 3.12 of Benoist–Miquel paper;
- on the other hand, some purely algebraic considerations show that is the virtual stabilizer of the proportionality class of , i.e., is a finite-index subgroup of : see Proposition 3.13 of Benoist–Miquel paper.

At this stage, we are ready to establish the closedness of the -orbit of :

Theorem 5The -orbit of is closed in .

*Proof:* Given such that , we write with converging to the identity element .

Our task is reduce to show that for all sufficiently large. For this sake, it suffices to find stabilizing the proportionality class of the polynomial (thanks to Remark 8).

In this direction, we take . By definition, . Also, by Remark 8, we know that for some constant depending on the covolume of . In particular, .

As it turns out, since the lattices converge to , one can check that the quantities converge to some related to the covolume of . Moreover, because . Hence, we can apply Corollary 4 to deduce that and

for all sufficiently large depending on , say .

At this point, we observe that the degrees of the polynomials are *uniformly bounded* and is Zariski-dense in . Thus, we can choose such that

for all and .

In other terms, stabilizes the proportionality class of for all . It follows from Remark 8 that for all sufficiently large. This completes the argument.

**2. Closedness of the -orbit of **

The proof of the fact that the -orbit of is closed in follows the same ideas from the previous section: one introduces the polynomial for , one shows that is closed and discrete in , and one exploits this information to get the desired conclusion.

In particular, our discussion of the first half of the proof of Theorem 1 is complete. Next time, we will see how this information can be used to derive the arithmeticity of . We end this post with the following remark:

Remark 9Roughly speaking, we covered Section 3 of Benoist–Miquel article (and the reader is invited to consult it for more details about all results mentioned above).Finally, a closer inspection of the arguments shows that the statements are true in greater generality provided is reflexive and commutative (cf. Remarks 4 and 7).

## Leave a Reply