Fine asymptotics of the distribution of periodic
orbits for the Teichmüller flow on strata of quadratic
or abelian differentials can be related to dynamical zeta functions.
A Borel conjugacy of the Teichmüller flow on the moduli space of quadratic differentials into the Weil-Petersson flow will be used to analyze dynamical properties of the Weil-Petersson flow.
The handlebody group is a finitely presented
subgroup of the mapping class group which however
is not quasi-isometrically embedded.
A new geometric model for the group will be used towards obtaining a comprehensive understanding of the geometry of this group, in particular with respect to calculating the Dehn function and quasi-isometric rigidity.
A similar geometric model for the outer automorphism group of a free group may yield hyperbolicity of the electrified sphere graph on which this group acts by simplicial automorphisms.
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