Award: DMS 1405466, Principal Investigator: Danny Calegari
Groups are the mathematical objects that formalize the idea of symmetry. These can be symmetries that mix different scales (recursive symmetries that describe the growth of trees), or that describe the aggregate behavior of vast numbers of similar objects (ideal gases, efficient markets). A complete understanding of groups is forever beyond our reach -- they exhibit phenomena of arbitrary complexity, and the simplest questions we can ask cannot be answered in the "worst case." But in the *generic* case the situation simplifies enormously. Techniques developed by the principal investigator with his collaborators allow one to construct subgroups of generic groups with a great deal of control, and to certify that these subgroups have any number of desired properties. The aim of this project is to extend these techniques to broader classes of groups, to allow more flexibility in the constructions, and to implement these theoretical tools in practical software. A key idea is to transplant and construct objects from topology and geometry -- namely *hyperbolic* geometry in dimension 2 and 3 -- into the generic combinatorial world, where a priori one would not expect to find them. The familiarity of these transplanted products provides the necessary leverage to break up the ambient group into understandable pieces.
The main aim of this project is to develop tools to construct, to certify and to organize certain classes of quasiconvex subgroups in hyperbolic groups, both in the generic case (i.e. random groups, in various models) and in general. The subgroups we are concerned with should be groups that we understand (i.e. fundamental groups of closed surfaces or compact acylindrical 3-manifolds), and their properties in the ambient group should have short certificates (which are the key to their construction in the first place). The principal investigator aims to build on techniques developed with various collaborators to construct such surface subgroups and acylindrical 3-manifold subgroups in various classes of hyperbolic groups, with the view to attacking Gromov's surface subgroup question, and various generalizations and specializations of this question. Techniques developed to construct objects in random groups can be adapted to general hyperbolic groups, where "randomness" of the ambient group is substituted for "degrees of freedom" in the method of construction. Finally, the author expects to implement these constructions on computer, proving their practical utility as a tool to understand specific classes of groups, beyond the theoretical insights they provide.
