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Area preserving group actions on surfaces. (English) Zbl 1036.37010
The objective is to show that there are essentially no actions of \(\text{SL} (n, \mathbb{Z})\), \(n\geq 3\), on closed oriented surfaces which are symplectic or area preserving. The key ingredient is the fact that a finite index subgroup of \(\text{SL} (n, \mathbb{Z})\) with \(n \geq 3\) always contains a subgroup isomorphic to the three-dimensional integer Heisenberg Lie group. Thus, the main result is that, if an almost simple Lie group \(G\) contains a subgroup isomorphic to the three-dimensional integer Heisenberg Lie group, then every homomorphism from \(G\) to a closed oriented surface \(S\) factors through a finite group. Some of the proofs rely on mapping class group techniques that use hyperbolic geometry. Let \(\omega\) be a smooth volume form and \(\text{Diff}_{\omega}(S)\) denote the group of diffeomorphisms preserving \(\omega\). Factorizations results for \(n \geq 4\) were proved by B. Farb and P. Shalen for \(S\neq T^2\) and by J. Rebelo for \(S=T^2\). L. Polterovich obtained a factorization result for \(n \geq 3\) and \(S\) other than \(S^2\) and \(T^2\) and there is a similar result of D. Witte.
At the end, the authors show some results for nilpotent Lie groups. They assume that \(G\) is a finitely generated nilpotent Lie subgroup of \(\text{Diff}_{\omega}(S)_0\). In this case, if \(S\neq S^2\) then \(G\) is abelian and if \(S= S^2\) then \(G\) has an abelian subgroup of index two. So, any finitely generated nilpotent subgroup \(G\) of \(\text{Diff}_{\omega}(S)\) has a finite index metabelian subgroup.

MSC:
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
57M60 Group actions on manifolds and cell complexes in low dimensions
57S25 Groups acting on specific manifolds
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
22F05 General theory of group and pseudogroup actions
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