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Flexibility of surface groups in classical simple Lie groups

Inkang Kim, Pierre Pansu (2015)

Journal of the European Mathematical Society

We show that a surface group of high genus contained in a classical simple Lie group can be deformed to become Zariski dense, unless the Lie group is S U ( p , q ) (resp. S O * ( 2 n ) , n odd) and the surface group is maximal in some S ( U ( p , p ) × U ( q - p ) ) S U ( p , q ) (resp. S O * ( 2 n - 2 ) × S O ( 2 ) S O * ( 2 n ) ). This is a converse, for classical groups, to a rigidity result of S. Bradlow, O. García-Prada and P. Gothen.

Foliations by planes and Lie group actions

J. A. Álvarez López, J. L. Arraut, C. Biasi (2003)

Annales Polonici Mathematici

Let N be a closed orientable n-manifold, n ≥ 3, and K a compact non-empty subset. We prove that the existence of a transversally orientable codimension one foliation on N∖K with leaves homeomorphic to n - 1 , in the relative topology, implies that K must be connected. If in addition one imposes some restrictions on the homology of K, then N must be a homotopy sphere. Next we consider C² actions of a Lie group diffeomorphic to n - 1 on N and obtain our main result: if K, the set of singular points of the...

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