Geometry of Subgroups

Tufts University, Medford MA

Investigators

Abstract

A group is an algebraic object that is the collection symmetries of a geometrical object. For instance, for a square the set of successive rotations by 90 degrees in the plane is a collection of symmetries and yields a group of size 4. One useful way of describing a group is via what is known as a presentation, with generators and relations between the generators completely describing the group. For example, the collection of all integers is an infinite group that is generated by a single element, namely the integer 1. If a group has a finite presentation (with a finite number of generators and a finite number of relations) then it is easier to understand, and easier to solve problems about that group. In addition, a group is called coherent if it enjoys the stronger property that every finitely generated subgroup is finitely presented. Many of the best understood groups are coherent: free groups, surface groups, and 3-manifold groups are all coherent. This project aims to understand when groups are coherent and to find geometric indicators of subgroups that are witnesses to incoherence. This project advances the field by investigating geometrically wild subgroups of seemingly well-behaved groups, which can be difficult to find. The project includes work in education, and the PI is helping to organize conferences and schools that enhance the training of the next generation. The PI plans to investigate the coherence of several interesting classes of groups, including groups that are expected to be coherent and groups which are expected to be incoherent. This project addresses fundamental and difficult problems about incoherence and the geometry of subgroups. One of the projects is a key step to a well-known conjecture that hyperbolic groups with Sierpinski carpet boundary are virtually Kleinian. Another is a newer conjecture, that groups which act geometrically on the product of two trees are incoherent. Solving these conjectures will shed substantial light on understanding when a group is the fundamental group of a 3-manifold, and also on the intricate subgroup structure of important classes of groups. 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.