Final Report Summary - MODULI (Geometry of moduli spaces and mapping class groups)
The principal goal of the project was to understand geometric and dynamical properties of moduli spaces and mapping class groups.
The mapping class group of a connected sum of k copies of the product of a two-sphere and a circle essentially is the outer automorphism group of the free group. It turned out that many of its geometric properties can be understood through its isometric action on various hyperbolic graphs that can be constructed from invariants of the free group. These constructions led among others to characterizations of geometrically meaningful subgroups (so called convex cocompact subgroups) and to an understanding of asymptotic invariants of this outer automorphism group.
Geometric properties of the mapping class group of a closed surface of genus at least two are reflected in the geometry and dynamics of the moduli space of abelian and quadratic differentials of this surface. New geometric and dynamical tools were developed to obtain an understanding of the relation between geometric and algebraic properties of the group and geometric and dynamical properties of these moduli spaces.
The main results include an understanding of algebraic properties of typical stretch factors of pseudo-Anosov mapping classes, finiteness of so-called algebraically primitive Teichmüller curves in genus at least 3 and of affine invariant manifolds of rank at least two. There are also equidistribution results which were applied to obtain spectral information on typical mapping tori, which are three-dimensional manifolds which fibre over the circle.
Every closed three-manifold can be obtained by glueing two so-called handlebodies of the same genus along their boundary. The mapping class group of such a handlebody is a subgroup of the mapping class group of the boundary surface. It turned out that many of its properties are similar to the properties of the outer automorphism group of the free group. This includes its geometric properties arising from its action on various hyperbolic graphs and the fact that its so-called Dehn function is exponential.
The undertaken research led to a unified view on these different groups and spaces and contributed towards the structural understanding of the global picture.
The mapping class group of a connected sum of k copies of the product of a two-sphere and a circle essentially is the outer automorphism group of the free group. It turned out that many of its geometric properties can be understood through its isometric action on various hyperbolic graphs that can be constructed from invariants of the free group. These constructions led among others to characterizations of geometrically meaningful subgroups (so called convex cocompact subgroups) and to an understanding of asymptotic invariants of this outer automorphism group.
Geometric properties of the mapping class group of a closed surface of genus at least two are reflected in the geometry and dynamics of the moduli space of abelian and quadratic differentials of this surface. New geometric and dynamical tools were developed to obtain an understanding of the relation between geometric and algebraic properties of the group and geometric and dynamical properties of these moduli spaces.
The main results include an understanding of algebraic properties of typical stretch factors of pseudo-Anosov mapping classes, finiteness of so-called algebraically primitive Teichmüller curves in genus at least 3 and of affine invariant manifolds of rank at least two. There are also equidistribution results which were applied to obtain spectral information on typical mapping tori, which are three-dimensional manifolds which fibre over the circle.
Every closed three-manifold can be obtained by glueing two so-called handlebodies of the same genus along their boundary. The mapping class group of such a handlebody is a subgroup of the mapping class group of the boundary surface. It turned out that many of its properties are similar to the properties of the outer automorphism group of the free group. This includes its geometric properties arising from its action on various hyperbolic graphs and the fact that its so-called Dehn function is exponential.
The undertaken research led to a unified view on these different groups and spaces and contributed towards the structural understanding of the global picture.