The Geometry and Topology of Groups Generated by Involutions

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Under this grant, the PI will work on problems regarding the geometry and topology of groups generated by involutions. The first problem involves a new geometric criterion for when a Coxeter group is a Kleinian group. Specifically, if the defining graph of a right-angled Coxeter group can be realized as the 1-skeleton of an acute triangulation of the 2-sphere, is the Coxeter group hyperbolic? This relates the techniques of geometric group theory and classical 3-dimensional hyperbolic geometry to combinatorial problems concerning triangulations. As part of this study, the PI will investigate the moduli space of an acute triangulation of the 2-sphere. In particular, is this space connected? A second question asks when a group generated by reflections in hyperbolic 3-space contains a knot group. This is deeply related to understanding commensurability of knot complements. More broadly, the PI seeks to address the conjecture that there are at most three hyperbolic knot complements in any given commensurability class. Boileau, Boyer, Cebanu, and the PI have made significant progress on the conjecture in recent work, and this project aims to address the remaining case. In a third project, the PI seeks a CAT(0) space for the group of outer automorphisms of the free product of four copies of the integers mod 2. This group of outer automorphisms is particularly pleasing and related to the proposal as it is generated by involutions.Groups generated by involutions have long been guiding examples in furthering the understanding of the geometry of groups. They are inherently geometric and relevant to the modern theory of orbifolds. These groups have been studied in various ways by Coxeter, Thurston, Davis, and Moussong, as well as by many other celebrated mathematicians, and they continue to inform current research. Imposing some condition, such as the presence of a reflection, allows one to gain traction on a difficult problem and eventually see the whole picture. The projects proposed here have potential impact in other fields. For example, the study of triangulations, particularly acute triangulations, is important in computer-aided design and scientific computing.