Some remarks on area-preserving actions of lattices
The existence of area-preserving actions of higher-rank lattices on surfaces is discussed. A lattice \(\Gamma\) in a connected simple Lie group \(G\) with finite center and of real rank greater than \(1\) is considered. Let \(\Sigma\) be a closed oriented surface endowed with an area form \(\omega\). According to a conjecture of Zimmer any homomorphism \(\rho:\Gamma \rightarrow \operatorname{Dif}f (\Sigma ,\omega )\) from \(\Gamma\) to the group of area-preserving diffeomorphisms of \(\Sigma\) should have finite image. A consequence of Zimmer's results is that the group \(\rho (\Gamma )\) does not contain any area-preserving diffeomorphism with positive metric entropy for some measure \(\mu\). On the other side of the spectrum of possible dynamical behaviours are area-preserving diffeomorphisms that are contained in a Hamiltonian flow. A consequence of this discussion is that this kind of diffeomorphism cannot appear in the group \(\rho (\Gamma)\) (see Section 3 of the paper).NEWLINENEWLINEFor the entire collection see [Zbl 1225.00042].
scientific article; zbMATH DE number 6077147