Actions of Relatively Hyperbolic Groups on Cube Complexes

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In mathematics, a "group" is an algebraic object which encodes the symmetries of an object. As such, group theory provides an algebraic language and a set of techniques for studying problems throughout mathematics. In recent years, some of the most important problems in 3-dimensional geometry (such as the Virtual Haken Conjecture) have been solved by Agol, using results from geometric group theory, and particularly the work of Wise, as well as that of Kahn, Markovic, Haglund, Hsu, Bergeron, Manning and the principal investigator. A key organizational principle is to understand spaces by cutting them along subspaces of one lower dimension. For example, one understands 3-dimensional spaces by cutting along surfaces (two dimensional spaces). This is the context of "Haken" 3-manifolds and of the Virtual Haken Conjecture. It turns out that the way to arrange the cutting subspaces of one lower dimension is via spaces made out of cubes. The major focus of this project is to understand the symmetry groups of these cube complexes, to build new tools to study them and to apply these new tools to many problems in geometry and group theory. The award provides support of training graduate students through research.In the first two parts of the project, we study relatively hyperbolic groups acting cocompactly on CAT(0) cube complexes. In the case the action is proper, Manning and the principal investigator intend to prove a version of Agol's Theorem, that all such groups are virtually special. Along with Einstein, the principal investigator will develop the theory of "relatively proper" actions of relatively hyperbolic groups on CAT(0) cube complexes, a context much more broadly applicable than the setting of proper and cocompact actions. We intend to prove versions of Agol's Theorem, and of Wise's Quasi-convex Hierarchy Theorem in this setting, and also to show that the proper and cocompact theory is a subset of the relatively proper theory. In the final part of this work, we will pursue applications of this theory to questions about hyperbolic and relatively hyperbolic groups with planar boundary.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.