流形上高阶格作用的不变测度与可测投射因子
Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds
摘要：
考虑流形上高秩半单李群中格的光滑作用。我们定义了两个与李群$G$的李代数的根系有关的数$r(G)$和$m(G)$。如果流形的维数小于$r(G)$，则证明了该作用保持Borel概率测度。若流形的维数至多为$m(G)$，则证明了流形上存在一个拟不变测度，使得该作用可测同构于标准边界作用上的相对保测度作用.
英文摘要：
We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers $r(G)$ and $m(G)$ associated with the roots system of the Lie algebra of a Lie group $G$. If the dimension of the manifold is smaller than $r(G)$, then we show the action preserves a Borel probability measure. If the dimension of the manifold is at most $m(G)$, we show there is a quasi-invariant measure on the manifold such that the action is measurable isomorphic to a relatively measure preserving action over a standard boundary action.
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