Free groups of semigroups in semi-simple Lie groups

Osvaldo Germano do Rocio and Alexandre J. Santana

Abstract: Let $G$ be a Lie group and $S \subset G$ a Lie semigroup. Neeb [{\it Glasg. Math. J.}, 34 (1992), 379--394] studied the free group on a generating Lie semigroup $(S,G)$ using the image $i_{*}({\pi}_{1}(S))$, where $i:S\to G$ is the inclusion mapping. Now, take $G$ a noncompact semi-simple Lie group, $G=KAN$ its Iwasawa decomposition and $S$ a subsemigroup which contains a large Lie semigroup. With these assumptions, San Martin--Santana [{\it Monatsh. Math.}, 136, (2002), 151--173] showed that the homotopy groups ${\pi}_{n}(S)$ and ${\pi}_{n}(K(S))$ are isomorphic, where $K(S)\subset K$ is a compact and connected subgroup. Here, using the technique developed in the above papers we extend the study of free group $G(S)$ and prove that the results of Neeb can be applied for semigroups containing a ray semigroup.

Keywords: semigroups; Lie groups; homotopy groups; free group and flag manifolds.

Classification (MSC2000): 20M20, 22E46, 57T99.

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