In this last post of this series, we want to complete the discussion of Oh–Benoist–Miquel theorem by giving a sketch of its proof in the cases not covered in previous posts.

More precisely, let us remind that Oh–Benoist–Miquel theorem (answering a conjecture of Margulis) asserts that:

Theorem 1Let be a semisimple algebraic Lie group of real rank . Denote by a horospherical subgroup of . If is a discrete Zariski-dense and irreducible subgroup such that is cocompact, then is commensurable to an arithmetic lattice .

Moreover, we remind that the proof of this result was worked out in the previous two posts of this series for and . Furthermore, we observed *en passant* that these arguments can be generalized without too much effort to yield a proof of Theorem 1 *when*

- is commutative;
- is reflexive (i.e., is conjugate to an opposite horospherical subgroup );
- is not compact (where , , and ).

Today, we will divide our discussion below into five sections discussing prototypical examples covering all possible remaining cases for and .

Remark 1The fact that Theorem 1 holds for the examples in Sections 1, 2 and 3 below is originally due to Oh. Similarly, the example in Section 4 was originally treated by Selberg. Finally, the original proof of Theorem 1 for the example in Section 5 is due to Benoist–Oh. Nevertheless, expect for Section 3, the arguments discussed below are some particular examples illustrating the general strategy of Benoist–Miquel and, hence, they provide some proofs which are different from the original ones.

**1. is not reflexive**

The prototype example of this case is and .

The corresponding parabolic subgroup is the stabilizer of the line :

Equivalently, is the stabilizer of the flag . Therefore, is *not* reflexive because its opposite is the stabilizer of a *plane*.

Since is Zariski-dense in , we can find such that is a basis of . Hence, there is no loss in generality in assuming that and . In this setting,

Also, we know that and are compact. Moreover, is a discrete and Zariski dense subgroup of the semi-direct product

of and .

In this context, a key fact is the following result of Auslander (compare with Proposition 4.17 in Benoist–Miquel paper):

Theorem 2 (Auslander)Let be an algebraic subgroup obtained from a semi-direct product of semisimple and solvable, and denote by the natural projection. If is discrete and Zariski dense, then is alsodiscreteand Zariski dense in .

The information about the discreteness of the projection in the previous statement is extremely precious for our purposes. Indeed, Auslander theorem implies that the projections and are discrete. Using these facts, one checks that

By repeating this argument with and in the place of , one can “fill all non-diagonal entries”, that is, one essentially gets that contains finite-index subgroups of

so that Raghunathan–Venkataramana–Oh theorem (stated in the previous post of this series) guarantees that is commensurable to .

This completes our sketch of proof of Theorem 1 for our prototype of non-reflexive subgroup above.

**2. is Heisenberg and is not compact**

A *Heisenberg* horospherical subgroup is a -step nilpotent whose associated parabolic group acts by similarities (of some Euclidean norm) on the center of the Lie algebra of .

A prototypical example of Heisenberg and non-compact is and

As it turns out, any Heisenberg is reflexive. Thus, we have that is opposite to for some adequate choice .

In particular, it is tempting to mimmic the arguments from the second and third posts of this series, namely, one introduces the lattices

so that the arithmeticity of follows from the closedness of the -orbit of in when is not compact; moreover, the closedness of is basically a consequence of the closedness and discreteness of in for an appropriate choice of polynomial function .

In the case of *commutative*, we took , where , and was the natural projection with respect to the decomposition .

As it turns out, the case of Heisenberg can be dealt with by slightly modifying the construction in the previous paragraph. More precisely, one considers a natural graduation

and one sets , , , and is the natural projection . In our prototypical example, the polynomial function is very explicit:

This completes our sketch of proof of Theorem 1 when is Heisenberg and is not compact.

**3. is not commutative and is not Heisenberg**

Our prototype of non-commutative and non-Heisenberg is the subgroup

of .

In this context, we will explore some well-known results from the theory of lattices in nilpotent groups to reduce our task to the case of commutative and reflexive.

More concretely, the properties of nilpotent groups together with our hypothesis that is a lattice in allow to conclude that is a lattice in

and, consequently, the centralizer of in is a lattice in the centralizer

of in . Therefore, we reduced matters to the case of commutative and reflexive which was discussed in the previous two posts of this series.

In particular, our sketch of proof of Theorem 1 when is non-commutative and non-Heisenberg is complete.

**4. commutative and is compact**

The basic example of commutative and compact is and

In this setting, we consider

the common Levi subgroup of and the parabolic subgroup normalizing an opposite of , and the “unimodular Levi subgroup”

The discussion in the second post of this series ensures that the -orbit of is closed in .

We affirm that is *compact*. Indeed, this fact can be proved via Mahler’s compactness criterion: more concretely, recall from the second post of this series that the proof of the closedness of produced a polynomial on which is -invariant and whose values on form a closed and discrete subset of ; in our prototypical example, a direct computation shows that

in particular, ; therefore, the -invariance of together with the closedness and discreteness of imply that

since is *irreducible*, , and, *a fortiori*, there are no arbitrarily short non-trivial vectors in the closed family of lattices ; hence, we can apply Mahler’s compactness criterion to complete the proof of our affirmation.

At this point, we observe that is not compact (because ), so that the compactness of means that the stabilizer of this orbit is infinite. Consequently, is infinite, and a quick inspection of the previous post reveals that this is *precisely* the information needed to apply Margulis’ construction of -forms and Raghunathan–Venkataramana theorem in order to derive the arithmeticity of . This completes our sketch of proof of Theorem 1 when is commutative and is compact (and the reader is invited to consult Section 4.6 of Benoist–Miquel paper for more details).

**5. is Heisenberg and compact**

Closing this series of post, let us discuss the remaining case of Heisenberg and compact. A concrete example of this situation is and

In this context, and an unimodular Levi subgroup is

Once again, let us recall that we know that is closed, where

We affirm that there is no loss of generality in assuming that and for all . Indeed, if this is not the case (say for some ), then we are back to the setting of Section 1 above (of the horospherical subgroup ).

Here, we can derive the arithmeticity of along the same lines in Section 4 above (where it sufficed to study an appropriate polynomial to employ Mahler’s compactness criterion). More precisely, one uses the fact that and for all to prove that is compact, so that is infinite and, thus, by Margulis’ construction of -forms and Raghunathan–Venkataramana theorem, is arithmetic.

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