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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 $$\mu$$.

##### MSC:
 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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