Last week, Jon Chaika, Jing Tao and I co-organized the Summer School on Teichmüller Theory and its Connections to Geometry, Topology and Dynamics at Fields Institute.

This activity was part of the Thematic Program on Teichmüller Theory and its Connections to Geometry, Topology and Dynamics, and it consisted of four excellent minicourses by Yves Benoist, Hee Oh, Giulio Tiozzo and Alex Wright.

These minicourses were fully recorded and the corresponding videos will be available at Fields Institute video archive in the near future.

Meanwhile, I decided to transcript my notes of Benoist’s minicourse in a series of four posts (corresponding to the four lectures delivered by him).

Today, we shall begin this series by discussing the statement of the main result of Benoist’s minicourse, namely:

Theorem 1 (Oh, Benoist–Miquel)Let be a semisimple algebraic Lie group of real rank . Suppose that is a horospherical subgroup of , and assume that is a Zariski dense and irreducible subgroup of such that is cocompact. Then, there exists an arithmetic subgroup such that and are commensurable.

The basic reference for the proof of this theorem (conjectured by Margulis) is the original article by Benoist and Miquel. This theorem completes the discussion in Hee Oh’s thesis where she dealt with many families of examples of semisimple Lie groups (as Hee Oh kindly pointed out to me, the reader can find more details about her contributions to Theorem 1 in these articles here).

Remark 1I came across Benoist–Miquel theorem during my attempts to understand a question by Sarnak about the nature of Kontsevich–Zorich monodromies. In particular, I’m thankful to Yves Benoist for explaining in his minicourse the proof of a result that Pascal Hubert and I used as a black box in our recent preprint here.

Below the fold, the reader will find my notes of the first lecture of Benoist’s minicourse (whose goal was simply to discuss several keywords in the statement of Theorem 1).

**1. Examples**

Let be a Lie group and consider a discrete subgroup.

Definition 2

- is a
latticewhen , i.e., there exists such that and where is a right-invariant Haar measure on .- is
cocompactif is compact (i.e., the subset can be chosen compact).

Example 1is (discrete and) cocompact in : indeed, for .

In general, is cocompact is lattice. However, the converse is not true:

Example 2is a lattice in which is not cocompact. In fact, the compact subsets of are described by the so-called Mahler’s compactness criterion asserting that is relatively compact if and only if .

Example 3 (Siegel)Let be a non-degenerate quadratic form in , , with for all . In this setting, is a lattice in .

Remark 2It is possible to prove that is cocompact if and only if doesn’t represent zero (i.e., ).Nevertheless, this information is not very useful to produce cocompact lattices because it is possible (to use Hasse’s principle) to show that if and is not definite, then represents zero.

- is a lattice in .
- can be viewed as a lattice in via the map where is Galois conjugation.

Historically, the basic idea of the previous example was adapted to produce the first examples of discrete *cocompact* subgroups of :

Example 5Let . Then, can be viewed as a lattice in via the map .Note that is definite and, hence, it doesn’t represent zero. Therefore, is a discretecocompactsubgroup of (thanks to Mahler’s compactness criterion).

In the sequel, we will put all examples above in a single framework.

**2. Arithmetic groups**

Let be an algebraic subgroup (i.e., a subgroup described by the zeros of polynomial functions of the matrix entries of elements of ).

Definition 3

- is
simpleif its Lie algebra is simple in the sense that its ideals are trivial.- is
semi-simpleif where are simple ideals.

If is semi-simple, the adjoint map has finite kernel and finite index image.

In other terms, if is semi-simple, then equals to the group of matrices up to finite index. In particular, the adjoint map of a semi-simple group allows to replace the “extrinsic” algebraic structure by the “intrinsic” algebraic structure modulo finite index.

Definition 4A -form of is the choice of -vector subspace of such that

- is a Lie subalgebra;
- .

In other words, a -form of is a choice of basis where the Lie bracket is described by a matrix with rational coefficients.

Definition 5Anarithmetic subgroupof is for some choice of .

It is possible to check that Examples 2, 3, 4, 5 above describe arithmetic subgroups of semisimple Lie groups . In particular, the fact that these examples provide lattices in can be viewed as concrete applications of Borel–Harish-Chandra theorem:

Theorem 6 (Borel–Harish-Chandra)An arithmetic subgroup of an algebraic semisimple group is a lattice.

Remark 3One can prove that is cocompact if and only if doesn’t contain nilpotent elements, i.e., there is no such that the matrix is nilpotent.

Remark 4This theorem naturally leads to the question of the existence of non-arithmetic lattices. As it turns out, this question is answered by the so-called Margulis arithmeticity theorem.

Let us now pursue the discussion of the statement of Theorem 1 by introducing the notion of irreducible subgroups.

Definition 7Let be an algebraic semisimple group with Lie algebra , where are simple ideals. A discrete subgroup isirreducibleif

is finite for all .

Remark 5Any is irreducible when is simple.

Finally, let us close this section by noticing that Theorem 1 is a sort of “converse” to the so-called Borel density theorem:

Theorem 8 (Borel)Let be a connected semisimple algebraic group with Lie algebra , where are simple ideals. Assume that none of the factors of (associated to ‘s) is compact. Then, any lattice is Zariski dense (i.e., is not included in a proper algebraic subgroup of ).

**3. Horospherical group**

The last bit of information needed to understand the statement of Theorem 1 is the concept of horospherical group. In the literature, this notion is usually phrased in terms of unipotent radicals of parabolic subgroups. For the sake of exposition, we will give an alternative elementary definition of this notion.

Definition 9Let be a semisimple group with neutral element .

- an element is
unipotentwhenever such that ;- a non-trivial subgroup is
horosphericalwhen such that

Example 6Let , , and where and is the identity matrix.Note that if , then . Hence,

is a horospherical subgroup.

As it was shown by Kazhdan and Margulis, the cocompactness of lattices is detected by horospherical groups:

Theorem 10 (Kazhdan–Margulis)Let be a lattice in a semisimple Lie group . Then, is not cocompact contains unipotent elements horospherical such that is cocompact in .

This completes our discussion of the statement of Theorem 1. Next time, we will start the proof of Theorem 1 in the particular case of Example 6, i.e., , , and .

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