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On distortion in groups of homeomorphisms

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  • Let $X$ be a path-connected topological space admitting a universal cover. Let Homeo$(X, a)$ denote the group of homeomorphisms of $X$ preserving a degree one cohomology class $ a$.
        We investigate the distortion in Homeo$(X, a)$. Let $g\in$ Homeo$(X, a)$. We define a Nielsen-type equivalence relation on the space of $g$-invariant Borel probability measures on $X$ and prove that if a homeomorphism $g$ admits two nonequivalent invariant measures then it is undistorted. We also define a local rotation number of a homeomorphism generalizing the notion of the rotation of a homeomorphism of the circle. Then we prove that a homeomorphism is undistorted if its rotation number is nonconstant.
    Mathematics Subject Classification: 54h20; 20f, 37b, 57s.


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