Geometric Group Theory via Geometric Combinatorics

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

The goal of this project is to bring various classes of groups commonly studied by geometric group theorists (such as Artin groups, one-relator groups, small cancellation groups, word-hyperbolic groups, fundmental groups of ample twisted face pairing 3-manifolds, etc.) within range of one of the standard general theories (such as piecewise Euclidean spaces of nonpositive curvature, conformal nonpositive curvature, or Garside structures) by constructing appropriate complexes on which they act. For example, there are emerging strategies for constructing high-dimensional nonpositively curved cube complexes for metric small cancellation groups as well as for fundamental groups of ample twisted face pairing 3-manifolds, for constructing 2-dimensional conformally CAT(0) complexes for one-relator groups, and for constructing Garside-like structures for arbitrary Artin groups. Geometric and enumerative combinatorics play a prominent role in the constructions themselves as well as in the establishment of their major properties. In each case, the complexes constructed and the approaches themselves are new and innovative and their key properties are in the process of being established. Early indications are that these complexes carry geometric and combinatorial structures which would resolve such longstanding conjectures as the coherence of one-relator groups, and the solution of the word problem for Artin groups. This project lies at the interface between geometric/combinatorial group theory and geometric/enumerative combinatorics. The former studies algebraic structures associated with geometric objects (such as their group of symmetries) while the latter can be roughly defined as the study of things which can be described using only a finite amount of data. This is precisely the type of mathematics that computers can do. Computational mathematics might seem far removed from geometric considerations, but there is a growing collection of combinatorial phenomena which can best be viewed as finite analogues of facts about the curvature of smooth spaces. The primary goal of this project is to use these computationally discovered combinatorial phenomena to construct complexes which carry a geometric/topological structure which then explain the observed algebraic behavior of the original groups.