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Distortion elements in group actions on surfaces. (English) Zbl 1088.37009
Here, \(S\) is a closed orientable surface, \(\mu\) is a Borel probability measure on \(S\) and \(\text{Diff}(S)_0\) is the group if \(C^1\)-diffeomorphisms of \(S\) that are isotopic to the identity. Let \(G\) be a finitely generated group. An element \(f\in G\) is said to be a distorsion element of \(G\) if \(f\) has infinite order and if we have \(\lim\inf_{n\to\infty} | f^n| / n=0\) where \(| f^n| \) denotes the length of the element \(f^n\) of \(G\) with respect to the word metric associated to some finite system of generators of \(G\). (The definition obviously does not depend on the chosen system of generators). The group \(G\) is said to satisfy property \(T\) if the identity element of \(G\) is isolated in the unitary dual \(\widehat G\). The group \(G\) is said to be almost simple if every normal subgroup of \(G\) is finite or has finite index. The main results of this paper are the following theorems:
Theorem 1: If \(f\) is a distorsion element of \(\text{Diff}(S)_0\) and if \(\mu\) is \(f\)-invariant, then the support of \(\mu\) is contained in the fixed-point set of \(f\), provided the genus of \(S\) is \(\geq 2\).
The authors also give related results in the case where \(S\) is a torus or a 2-sphere.
Theorem 2: If \(S\) has positive genus, if \(G\) is almost simple and possesses a distorsion element \(u\), and if either the Borel measure \(\mu\) has infinite support or \(G\) satisfies property \(T\), then any homomorphism from \(G\) into the group of \(C^1\)-diffeomorphisms of \(S\) that preserve \(\mu\) has finite image.
Again, the authors give a related result in the case where \(S\) is the 2-sphere.
Theorem 3: If \(G\) is almost simple and has a subgroup isomorphic to the three-dimensional Heisenberg group, then any homomorphism from \(G\) into the group of \(C^1\)-diffeomorphisms of \(S\) that preserve \(\mu\) has finite image.
Several corollaries are obtained, in particular concerning the images of homomorphisms from subgroups of higher-rank lattices in the group of \(C^1\)-diffeomorphisms of \(S\) that preserve \(\mu\).

37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
57M60 Group actions on manifolds and cell complexes in low dimensions
22F10 Measurable group actions
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