具有infinite dimensional目标的余圈超刚性和调和映射
Cocycle superrigidity and harmonic maps with infinite dimensional targets
摘要：
我们宣布推广齐默的上循环超刚性定理证明使用调和映射技术。这使得我们可以将许多关于高阶格的结果推广到具有性质$(T)$的半单群中的所有格。特别是，我们的结果适用于SP（1，n）和$F_4^{-20}$和晶格在这些群体。主要的技术步骤是证明一个非常普遍的结果，关于存在的调和映射到无限维空间，即一类单连通，齐次，非球面希尔伯特流形。这建立在Corlette-Zimmer和Korevaar-Schoen以前的工作基础上。我们的结果是新的，因为我们考虑了比以前的作者更一般的目标，并没有假设有关的行动。特别是，不假设目标空间具有非正曲率，并且在重要情况下具有显着的正曲率。我们也不做关于无穷远处不存在不动点的"还原性"假设。与以往的遍历理论证明不同，本文给出的上圈超刚性的证明是有效的。上循环的拉直显然是热流的极限。这种显式构造应该产生进一步的应用。
英文摘要：
We announce a generalization of Zimmer's cocycle superrigidity theorem proven using harmonic map techniques. This allows us to generalize many results concerning higher rank lattices to all lattices in semisimple groups with property $(T)$. In particular, our results apply to SP(1,n) and $F_4^{-20}$ and lattices in those groups. The main technical step is to prove a very general result concerning existence of harmonic maps into infinite dimensional spaces, namely a class of simply connected, homogeneous, aspherical Hilbert manifolds. This builds on previous work of Corlette-Zimmer and Korevaar-Schoen. Our result is new because we consider more general targets than previous authors and make no assumption concerning the action. In particular, the target space is not assumed to have non-positive curvature and in important cases has significant positive curvature. We also do not make a "reductivity" assumption concerning absence of fixed points at infinity. The proof of cocycle superrigidity given here, unlike previous ergodic theoretic ones, is effective. The straightening of the cocycle is explicitly a limit of a heat flow. This explicit construction should yield further applications.
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