Low dimensional actions of semisimple groups
This paper provides a fairly thorough understanding of the orbit structure of low dimensional actions of noncompact semisimple Lie groups. For any such group, let \(n(G)\) denote the smallest integer such that \(G\) has an almost effective action (i.e., an action with discrete kernel) on a manifold of dimension \(n(G)\). Specifically, the author shows: Theorem. Let \(G\) be a noncompact simple Lie group with finite center. Then \(n(G)\) is the minimum codimension of a maximal parabolic subgroup of \(G\). If \(G\) acts effectively on a compact manifold \(M\) of dimension \(n(G)\) then \(M\) is a finite equivariant cover of \(G/S\) for some maximal parabolic \(S \subset G\). Theorem. Let \(G\) be a connected semisimple real Lie group with finite center. Suppose \(G\) acts smoothly and almost effectively on a closed manifold \(M\) of dimension \(n(G) + 1\). Let \(K\) be the maximal compact subgroup of \(G\). Then \(M/K\) is homeomorphic to \(S^ 1\), the unit interval \(I\), or a point. In the first case the \(K\)-orbits fiber \(M\) over \(S^ 1\). Theorem. Let \(G\) be a connected semisimple noncompact Lie group with finite center acting smoothly and almost effectively on a closed manifold \(M\) of dimension \(n(G) + 1\). Let \(M/G\) be the orbit space of the action with the quotient topology. Then \(M/G\) is obtained from \(S^ 1\) or \(I\) by identifying (possibly infinitely many) open intervals to points. All of this should be considered as analogues of a result of \textit{P. S. Mostert} [Ann. Math., II. Ser. 65, 447-455 (1957; Zbl 0080.167)]: Theorem. Let \(K\) be a compact group acting on a closed manifold \(M\) with an orbit of codimension one, then the orbit space \(M/K\) is homeomorphic \(S^ 1\) or \(I\), and in the former case, the \(K\)-orbits fibre \(M\) over \(S^ 1\).
orbit strucutre of low dimensional actions of noncompact semisimple Lie groups
scientific article; zbMATH DE number 78193
scientific article