Existence of irreducible ${\mathbb R}$-regular elements in Zariski-dense subgroups

Abstract

Let G be a connected semisimple algebraic group defined over the field R of real numbers. An element x of G(R) is called R-regular if the number of eigenvalues, counted with multiplicity, of modulus 1 of Ad x is minimum possible. (If G is R-anisotropic, i.e., the group G(R) is compact, every element of G(R) is Rregular.) The existence of R-regular elements in an arbitrary subsemigroup Γ of G(R) which is Zariski-dense in G was proved by Y. Benoist and F. Labourie [3] using Oseledet's multiplicative ergodic theorem, and then reproved by the firstnamed author [15] by a direct argument. Recently G.A. Margulis and G.A. Soifer asked us a question, which arose in their joint work with H. Abels on the Auslander problem, about the existence of R-regular elements with some special properties. The purpose of this note is to answer their question in the affirmative. Before formulating the result, we recall (cf. [16], Remark 1.6(1)) that an R-regular element x is necessarily semisimple, so if in addition it is regular, then T := Z G (x) • is a maximal torus; moreover, x belongs to T (see [4], Corollary 11.12). Theorem 1. Let G be a connected semisimple real algebraic group. Then any Zariski-dense subsemigroup Γ of G(R) contains a regular R-regular element x such that the cyclic subgroup generated by it is a Zariski-dense subgroup of the maximal torus T := Z G (x) •. Remark 1. Let Γ, x and T = Z G (x) • be as in Theorem 1. Let T s (resp., T a) be the maximal R-split (resp., R-anisotropic) subtorus of T. Then T = T s • T a (an almost direct product), T s is a maximal R-split torus of G since x is R-regular (see [16], Lemma 1.5), and T a (R) (R/Z) r , where r = dim T a. There is a positive integer d such that x d = y • z with y ∈ T s (R) and z ∈ T a (R). Then the cyclic group C generated by z is dense in T a in the Zariski-topology and since T a (R) is a compact Lie group, C is actually dense in T a (R) in the usual compact Hausdorff topology on the latter. Thus, in particular, if G(R) is compact, then any dense subsemigroup contains a Kronecker element, i. e. an element such that the closure of the subsemigroup generated by it is a maximal torus. Also, since the cyclic subgroup generated by x is dense in T in the Zariskitopology, Z G (x) = Z G (T) = T. Thus the centralizer of x is connected.

Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups

“…We remark here that whenever Γ is a co-compact and torsion free lattice of a real algebraic group G without compact factors, there exists a free abelian subgroup of Γ that is singular in Γ. In fact the authors in [RS10] proved that under these hypothesis, the Cartan subgroup constructed in [PR03] H ⊂ G is such that Λ 0 := Γ ∩ H is isomorphic to Z rk R (G) and Λ 0 is malnormal in Γ. So we can use this Cartan subgroup in the proof of the above proposition to get that Λ 0 has finite index in a singular subgroup of Γ and since Λ 0 is malnormal in Γ, then we must have that Λ 0 is singular in Γ.…”

Stable laws and spectral gap properties for affine random walks

“…We say that Γ satisfies condition i-p if Γ is strongly irreducible and contains a proximal element γ. It is proved in [39] that condition i-p for Γ and its Zariski closure Zc(Γ) are equivalent. Since Zc(Γ) is a closed Lie subgroup of G with a finite number of connected components, condition i-p can be checked in examples (see Section 5 for some examples).…”

Number-theoretic techniques in the theory of Lie groups and differential geometry

“…These results fit into a broader project of constructing elements with special properties in a given Zariski-dense subgroup dealt with in our papers [17], [19]- [20]. The starting point of this project was the following question asked independently by G.A.…”

“…[15]). It should be noted that even the existence of an R-regular element without any additional requirement in an arbitrary Zariski-dense subgroup Γ is a nontrivial matter: this was established by Benoist and Labourie [4] using the multiplicative ergodic theorem, and then by Prasad [14] by a direct argument; we will not, however, discuss this aspect here. The real problem is that the above argument for the existence of a regular semisimle element in Γ which generates a Zariski-dense subgroup of its centralizer does not extend to the case where Γ is not arithmetic.…”

“…Proposition ( [19]). Let K be a finitely generated field of characteristic zero, R ⊂ K be a finitely generated ring.…”