Most of Professor Floyd's research is in geometric group theory, an area that has grown out of the interplay between combinatorial group theory and geometry. While many of the techniques come from combinatorial group theory, the prime examples and many of the important questions come from geometry. If G is a group and S is a finite generating set of G, then there is a graph, the Cayley graph, naturally associated to the pair (G,S). In geometric group theory, groups are often studied in terms of asymptotic properties of a Cayley graph of the group. A central task is to classify groups in terms of the spaces on which they can act geometrically. The central focus of his research is Cannon's Conjecture, which states that if G is a word-hyperbolic group whose Gromov boundary is homeomorphic to a 2-sphere, then G has a properly discontinuous, cocompact, isometric action on hyperbolic 3-space. Because of his approach to Cannon's Conjecture, he also works on the dynamics of Thurston maps (branched covers of the 2-sphere with only only finitely many postcritical points).
Mathematics’ William J. Floyd honored with emeritus status by Board of Visitors (Dec. 2018)