Integral foliated simplicial volume and S 1-actions

`1-Homology and Simplicial Volume

Introduction A pervasive theme of contemporary mathematics is to explore rigidity phenomena caused by the symbiosis of algebraic topology and Riemannian geometry on manifolds. In this context, the term " rigidity " refers to the astounding fact that certain topological invariants provide obstructions for geometric structures. Consequently , topological invariants of this type serve as interface...

Simplicial Volume and Fillings of Hyperbolic Manifolds

Let M be a hyperbolic n–manifold whose cusps have torus crosssections. In [FM10], the authors constructed a variety of nonpositively and negatively curved spaces as “2π–fillings” of M by replacing the cusps of M with compact “partial cones” of their boundaries. These 2π–fillings are closed pseudomanifolds, and so have a fundamental class. We show that the simplicial volume of any such 2π–fillin...

Integral Homology of Random Simplicial Complexes

Characterizing rigid simplicial actions on trees

We extend Forester’s rigidity theorem so as to give a complete characterization of rigid group actions on trees (an action is rigid if it is the only reduced action in its deformation space, in particular it is invariant under automorphisms preserving the set of elliptic subgroups). Let T be a simplicial tree with a cocompact action of a group G (i.e. the BassSerre tree associated to a decompos...

Stable complexity and simplicial volume of manifolds

Let the Δ-complexity σ(M) of a closed manifold M be the minimal number of simplices in a triangulation of M . Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the infimum on all finite coverings of M normalized by the covering degree, we can promote σ to a multiplicative invariant, a characteristic number already considered by Milnor and Thurston, whi...