Early work Cannon's early work concerned topological aspects of embedded surfaces in
R3 and understanding the difference between "tame" and "wild" surfaces. His first famous result came in late 1970s when Cannon gave a complete solution to a long-standing "double suspension" problem posed by
John Milnor. Cannon proved that the double
suspension of a
homology sphere is a topological sphere. R. D. Edwards had previously proven this in many cases. The results of Cannon's paper an important case of the so-called
characterization conjecture for topological manifolds. The conjecture says that a
generalized n-manifold M, where n \ge 5, which satisfies the "disjoint disk property" is a topological manifold. Cannon, Bryant and Lacher established completed the proof that the characterization conjecture holds if there is even a single manifold point. In general, the conjecture is false as was proved by John Bryant, Steven Ferry,
Washington Mio and
Shmuel Weinberger.
1980s: Hyperbolic geometry, 3-manifolds and geometric group theory In 1980s the focus of Cannon's work shifted to the study of
3-manifolds,
hyperbolic geometry and
Kleinian groups and he is considered one of the key figures in the birth of
geometric group theory as a distinct subject in late 1980s and early 1990s. Cannon's 1984 paper "The combinatorial structure of cocompact discrete hyperbolic groups" was one of the forerunners in the development of the theory of
word-hyperbolic groups, a notion that was introduced and developed three years later in a seminal 1987 monograph of
Mikhail Gromov. Cannon's paper explored combinatorial and algorithmic aspects of the
Cayley graphs of Kleinian groups and related them to the geometric features of the actions of these groups on the
hyperbolic space. In particular, Cannon proved that convex-cocompact Kleinian groups admit
finite presentations where the
Dehn algorithm solves the
word problem. The latter condition later turned out to give one of equivalent characterization of being
word-hyperbolic and, moreover, Cannon's original proof essentially went through without change to show that the word problem in word-hyperbolic groups is solvable by Dehn's algorithm. Cannon's 1984 paper a notion that led to substantial further study and generalizations. An influential paper of Cannon and
William Thurston "Group invariant Peano curves", that first circulated in a preprint form in the mid-1980s, introduced the notion of what is now called the
Cannon–Thurston map. They considered the case of a closed hyperbolic 3-manifold
M that
fibers over the circle with the fiber being a closed hyperbolic surface
S. In this case the universal cover of
S, which is identified with the
hyperbolic plane, admits an embedding into the universal cover of
M, which is the
hyperbolic 3-space. Cannon and Thurston proved that this embedding extends to a continuous π1(
S)-equivariant
surjective map (now called the
Cannon–Thurston map) from the ideal boundary of the hyperbolic plane (the circle) to the ideal boundary of the hyperbolic 3-space (the
2-sphere). Although the paper of Cannon and Thurston was finally published only in 2007, in the meantime it has generated considerable further research and a number of significant generalizations (both in the contexts of Kleinian groups and of word-hyperbolic groups), including the work of
Mahan Mitra, Erica Klarreich,
Brian Bowditch and others.
1990s and 2000s: Automatic groups, discrete conformal geometry and Cannon's conjecture Cannon was one of the co-authors of the 1992 book
Word Processing in Groups which introduced, formalized and developed the theory of
automatic groups. The theory of automatic groups brought new computational ideas from
computer science to
geometric group theory and played an important role in the development of the subject in 1990s. A 1994 paper of Cannon gave a proof of the "
combinatorial Riemann mapping theorem" that was motivated by the classic
Riemann mapping theorem in
complex analysis. The goal was to understand when an
action of a group by
homeomorphisms on a
2-sphere is (up to a topological conjugation) an action on the standard
Riemann sphere by
Möbius transformations. The "combinatorial Riemann mapping theorem" of Cannon gave a set of sufficient conditions when a sequence of finer and finer combinatorial subdivisions of a topological surface determine, in the appropriate sense and after passing to the limit, an actual
conformal structure on that surface. This paper of Cannon led to an important conjecture, first explicitly formulated by Cannon and Swenson in 1998 Cannon's conjecture motivated much of subsequent work by other mathematicians and to a substantial degree informed subsequent interaction between
geometric group theory and the theory of analysis on metric spaces. Cannon's conjecture was motivated (see ) by
Thurston's Geometrization Conjecture and by trying to understand why in dimension three variable negative curvature can be promoted to constant negative curvature. Although the
Geometrization conjecture was recently settled by
Perelman, Cannon's conjecture remains wide open and is considered one of the key outstanding open problems in geometric group theory and
geometric topology.
Applications to biology The ideas of combinatorial conformal geometry that underlie Cannon's proof of the "combinatorial Riemann mapping theorem", Cannon, Floyd and Parry produced a mathematical growth model which demonstrated that some systems determined by simple
finite subdivision rules can results in objects (in their example, a tree trunk) whose large-scale form oscillates wildly over time even though the local subdivision laws remain the same. Cannon, Floyd and Parry also applied their model to the analysis of the growth patterns of rat tissue. They suggested that the "negatively curved" (or non-euclidean) nature of microscopic growth patterns of biological organisms is one of the key reasons why large-scale organisms do not look like crystals or polyhedral shapes but in fact in many cases resemble self-similar
fractals. In particular they suggested (see section 3.4 of ) that such "negatively curved" local structure is manifested in highly folded and highly connected nature of the brain and the lung tissue. ==Selected publications==