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$$\mathbb{R}$$-regular elements in Zariski-dense subgroups. (English) Zbl 0828.22010
Let $$G$$ be a connected semi-simple algebraic group defined over $$\mathbb{R}$$ and $$G(\mathbb{R})$$ the group of real points of $$G$$. Then $$G(\mathbb{R})$$ is a real Lie group. Let $$\text{Ad}$$ denote the adjoint representation of $$G(\mathbb{R})$$ and suppose that $$G(\mathbb{R})$$ is non-compact. An element $$g \in G(\mathbb{R})$$ is called $$\mathbb{R}$$-regular if the number of eigenvalues (counted with multiplicities) of modulus 1 of $$\text{Ad} (g)$$ is minimum possible. The author shows that any subsemigroup of $$G(\mathbb{R})$$ which is Zariski dense in $$G$$ contains an $$\mathbb{R}$$-regular element. The proof depends on a characterization of $$\mathbb{R}$$-regular elements in terms of exterior powers of the adjoint representation proved by the author and M. S. Raghunathan [Ann. Math., II. Ser. 96, 296-317 (1972; Zbl 0245.22013)].

##### MSC:
 2.2e+16 General properties and structure of real Lie groups
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