Existence of irreducible \(\mathbb R\)-regular elements in Zariski-dense subgroups
Let \(G\) be a connected semisimple algebraic group defined over \({\mathbb R}\). Recall that an element \(x\) of \(G({\mathbb R})\) is called \({\mathbb R}\)-regular if the number of eigenvalues of modulus 1 counted with multiplicity is minimal. Let \(\Gamma\) be any Zariski-dense subsemigroup of \(G({\mathbb R})\). The authors show that \(\Gamma\) contains a regular \({\mathbb R}\)-regular element \(x\) such that the subgroup generated by \(x\) is Zariski-dense in the maximal torus \(T\), where \(T\) is the one-component of \(Z_G(x)\) (Theorem 1). This in particular implies that \(Z_G(x)\) is connected, i.e., equals \(T\). To prove Theorem 1, they give some additional results on this topic, one of them a refinement of Theorem 1 of their older paper [Int. Math. Res. Not. 2001, 1229-1242 (2001; Zbl 1057.22025)]. The article is finished by some considerations about the close relation of \({\mathbb R}\)-regular and proximal elements.
scientific article; zbMATH DE number 1918959