Tag - Mapping class groups

Harry Hyungryul Baik: Normal generators for mapping class groups are abundant in the fibered cone

17th July, 2020

We show that for almost all primitive integral cohomology classes in the fibred cone of a closed fibred hyperbolic 3-manifold, the monodromy normally generates the mapping class group of the fibre. The key idea of the proof is to use Fried’s theory of suspension flow and dynamic blow-up of Mosher. If the time permits, we also discuss the non-existence of the analogue of Fried’s continuous extension of the normalized entropy over the fibered face in the case of asymptotic translation lengths on the curve complex.

Mark Hagen: Hierarchical hyperbolicity from actions on simplicial complexes

5th June, 2020

The notion of a "hierarchically hyperbolic space/group" grows out of geometric similarities between CAT(0) cubical groups and mapping class groups. Hierarchical hyperbolicity is a "coarse nonpositive curvature" property that is more restrictive than acylindrical hyperbolicity but general enough to include many of the usual suspects in geometric group theory. The class of hierarchically hyperbolic groups is also closed under various procedures for constructing new groups from old, and the theory can be used, for example, to bound the asymptotic dimension and to study quasi-isometric rigidity for various groups. One disadvantage of the theory is that the definition - which is coarse-geometric and just an abstraction of properties of mapping class groups and cube complexes - is complicated. We therefore present a comparatively simple sufficient condition for a group to be hierarchically hyperbolic, in terms of an action on a hyperbolic simplicial complex. I will discuss some applications of this criterion to mapping class groups and (non-right-angled) Artin groups.

Federico Berlai: From hyperbolicity to hierarchical hyperbolicity

5th June, 2020

Hierarchically hyperbolic groups (HHGs) and spaces are recently introduced generalizations of (Gromov-) hyperbolic groups and spaces. Other examples of HHGs include mapping class groups, right-angled Artin/Coxeter groups, and many groups acting properly and cocompactly on CAT(0) cube complexes. After a substantial introduction and motivation, I will present a combination theorem for hierarchically hyperbolic groups. As a corollary, any graph product of finitely many HHGs is itself a HHG.