Emerging Topics on Coherence and Quasi-Convex Subgroups

Support for this workshop has been generously provided by Friends Founders’ Circle members Cynthia and Robert Hillas.

Organizers:  Daniel Groves, University of Illinois at Chicago; Alan Reid, Rice University, and Genevieve Walsh, Tufts University

Participants:  Kasia Jankiewicz, Rob Kropholler, Lars Louder, Jason Manning, Kim Ruane, Emily Stark, Matthew Stover, Henry Wilton

Summary:  Hyperbolic groups are one of the most influential classes of groups in geo-metric group theory. The geometry of subgroups of a hyperbolic group is rich, interesting, and extremely important to understanding the total group. Their best-behaved subgroups are quasi-convex, that is, their geometry is undistorted in the whole group. These subgroups are themselves hyperbolic, and hence finitely presented. When a hyperbolic group is also the fundamental group of a 3-manifold with boundary, then every finitely generated subgroup is quasi-convex. In contrast, the fundamental group of a closed hyperbolic 3-manifold contains geometrically infinite surface groups, and these are exactly the virtual fibers. Quasi-convex subgroups have recently seen dramatic use in Wise’s theory of hyperbolic groups with a quasi-convex hierarchy, which led to Agol’s solution of the Virtual Haken Conjecture (among many other important results). There are a number of new tools and approaches which could shed light on some important problems about hyperbolic groups and their subgroups and the purpose of this workshop is to bring together diverse experts to tackle some of these problems. There are a number of reasons why the time is ripe to work intensely on these problems:

  • Understanding hyperbolic groups with Sierpinski carpet boundary is currently a stumbling block towards understanding hyperbolic groups with 2-sphere boundary and hyperbolic groups with planar boundary, which are conjectured to be virtually Kleinian.
  • Recent advances towards showing that one-relator groups are coherent points to the prospect of using combinatorial tools to possibly understand coherence in more complicated groups.
  • Techniques from complex hyperbolic manifolds and higher-rank lattices may shed light on other types of questions, such as proving the incoherence of groups which act geometrically on a product of trees.

Working Group Report