In the previous post of this series, we gave the statements of some of the results of Breuillard and Sert on the definition and basic properties of the joint spectrum, and we promised to discuss the proofs in subsequent posts.

Today, after a long hiatus, I’ll try to accomplish part of this promise. More precise, I’ll transcript below my notes for the two talks (by Rodolfo Gutiérrez-Romo and myself) aiming to explain to the participants of our *groupe de travail* the proof of the first portion of Theorem 5 in the previous post, i.e., the convergence of (cf. Theorem 5 below), the convergence of (cf. Theorem 7 below), and the equality of the limits (cf. Theorem 9 below).

Evidently, all mistakes in what follows are my sole responsibility.

**1. Spectral radius formula revisited**

Let be a reductive linear algebraic group. Recall that the Cartan projection and Jordan projection were defined in the previous post via the Cartan decomposition and the Jordan–Chevalley decomposition with elliptic, unipotent, and “hyperbolic” conjugated to .

The *semisimple rank* of is where is a maximal torus of and is the center of . We denote by , a system of roots such that is a base of simple roots.

Each induces a weight satisfying and for all , where is the Lie subalgebra of and is a fixed extension of the Killing form on the Lie subalgebra of to the Lie algebra of such that becomes an orthogonal decomposition.

The weights , , are the highest weights of distinguished representations , of . One has

where is a choice of -invariant norm with diagonalisable in an orthonormal basis and stable under the adjoint operation, and denotes the top eigenvalue of a matrix . In particular, the Cartan projection is represented by a vector of logarithms of norms of matrices, the Jordan projection is represented by a vector of the logarithms of the moduli of top eigenvalues of matrices, and, *a fortiori*, the usual formula for the spectral radius implies that:

for every .

**2. Proximal elements and Cartan projections**

As we indicated in § 2.1 of the previous post of this series, the convergence of relies on the notion of *proximal matrices*.

Definition 2Let be the Fubini-Study metric on the projective space of a finite-dimensional real vector space equipped with an Euclidean norm .Given , we say that is a -proximal matrix whenever:

- has an unique eigenvalue of maximal modulus with eigendirection and -invariant supplementary hyperplane ;
- ;
- for all with and .

In general, we say that an element of a reductive linear algebraic group with distinguished representations , , is –*proximal* when the matrices are -proximal for all .

A basic feature of proximal elements is the fact their Cartan and Jordan projections are comparable (cf. Lemmas 2.15 and 2.16 of Breuillard–Sert paper extracted from Benoist’s paper).

Lemma 3There is a constant such that

and

for all .For each , there is a constant such that the Cartan and Jordan projections of any -proximal element satisfy

Another crucial feature of proximal elements (discovered by Abels, Margulis and Soifer, see Theorem 4.1 of their paper) is their ubiquity in Zariski dense monoids:

Theorem 4 (Abel–Margulis–Soifer)Let be a connected, reductive, real Lie group. Suppose that is a Zariski dense monoid. Then, there exists such that, for all , there exists a finite subset with the property: given , there exists so that is -proximal.

At this point, we are ready to prove the convergence of Cartan projections:

Theorem 5Let be a connected reductive real Lie group and suppose that is a compact subset generating a Zariski dense subgroup. Then,

converges in the Hausdorff topology as .

*Proof:* By Lemma 2 of the previous post, our task is reduced to show that , , stays in a compact region of , and for each , there exists such that for all and .

By Lemma 3, there exists a constant such that

for all . It follows that for all , that is, is confined in a compact region of .

Let us now estimate for , say with . By Abels–Margulis–Soifer theorem 4, we can select a *finite* subset of the monoid generated by such that for each , there exists so that is -proximal. In particular, we can take such that is -proximal for all . By Lemma 3, we have

and

Since , it follows from the triangular inequality that

Therefore, if we fix and we write , , we can use the Euclidean division , to obtain an element of via the formula

It follows from the definitions and Lemma 3 that

Since

by taking (or equivalently ) we derive that

Hence, given , there exists such that

for all , . This completes the proof.

**3. Twisting and Jordan projections**

A Zariski dense monoid of matrices is *twisting* in the sense that it always contains an element putting a finite configuration of lines and hyperplanes in general positions:

Lemma 6Let be a connected, reductive Lie group and suppose that is a Zariski dense monoid. Given a finite collection , , of irreducible representations of and finite configurations and , , of points and hyperplanes in , there is an element such that

for all and .

*Proof:* Since are irreducible, the sets are non-empty and Zariski open in . Thus,

is Zariski open in and non-empty (because is connected). Since is Zariski dense,

This completes the argument.

Remark 1The conclusion of this lemma can be reinforced as follows (cf. Remark 2.22 of Breuillard–Sert paper): it is possible to select from a finite subset of depending only on and (but not on and ).

At this stage, we can start the discussion of the convergence of Jordan projections:

Theorem 7Let be a connected reductive real Lie group and suppose that is a compact subset generating a Zariski dense subgroup. Then,

converges in the Hausdorff topology as .

*Proof:* Similarly to the previous section (on convergence of Cartan projections), our task consists into showing that for each , there exists such that

for all , . In this direction, let us fix and let us take , . By the formula for the spectral radius (cf. Lemma 1), we can fix with

By Abels–Margulis–Soifer theorem 4, we can fix a finite subset of the monoid generated by such that for some , we have that is -proximal.

By Lemma 3,

where , and

for all .

Consider the distinguished representations , , of . Note that the dominant eigendirection and the dominated hyperplane for the actions of the proximal matrices on are the same for all .

We fix . By the twisting property in Lemma 6, there exists , say , such that

for all and .

The dynamics of projective actions of the iterates of a proximal matrix is easy to describe: any direction transverse to is attracted towards . By rendering this argument slightly more quantitative (with the aid of the so-called *Tits proximality criterion*), Breuillard and Sert proved in Lemma 3.6 of their paper that

Lemma 8If is -proximal and is a finite subset such that

for all and , then there exists such that for all and , one has that is -proximal for all .

By applying this lemma with , we can select such that is -proximal for all , and .

Once again, it follows from Lemma 3 that

and

for all and .

By Euclidean division, we can write with and define

From our discussion above, we derive that

Since and

by letting (or equivalently, ) we conclude that

for all . This completes the proof.

**4. Coincidence of the limits**

Let be a connected, reductive real Lie group and let be a compact subset generating a Zariski dense monoid. By Theorems 5 and 7, we have that

as .

*Proof:* By the formula for the spectral radius (cf. Lemma 1), for all , one has as . In particular, .

In order to derive the other inclusion, we recall that the proof of Theorem 5 about the convergence of Cartan projections revealed that there exists and a constant such that for all and , there exists with and

Therefore,

Since , we have that is bounded and, *a fortiori*,

as . This shows that , as desired.

**5. Realization of the joint spectrum by sequences**

Closing this post, let us further discuss Theorem 5 from the previous post by showing that with is realized by a single sequence in the sense that .

For this sake, we use Abels–Margulis–Soifer theorem 4 and the strong version in Remark 1 of the twisting property in Lemma 6 to select a finite subset of and some constants such that for each there are with the property that is a *Schottky family* in the sense that is -proximal, and and for all . (This nomenclature comes from the fact that the projective actions of the elements in a Schottky family resemble the classical Schottky groups.) Note that where .

Let us now choose a rapidly increasing sequence so that

for all , and we define by

By definition, any finite word has the form where and is a prefix of . Observe that

By Lemma 3, . Moreover, Lemma 2.17 in Breuillard–Sert paper ensures that the Schottky property for the family makes that is a -proximal element with

Therefore, it follows from Lemma 3 that

Since converges to , we conclude that converges to as .

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