James W. Cannon (born January 30, 1943) is an American mathematician working in the areas of low-dimensional topology and geometric group theory. He was an Orson Pratt Professor of Mathematics at Brigham Young University.
Biography James W. Cannon was born on January 30, 1943, in Bellefonte, Pennsylvania. Cannon received a Ph.D. in mathematics from the University of Utah in 1969, under the direction of C. Edmund Burgess. He was a professor at the University of Wisconsin, Madison from 1977 to 1985. In 1986 Cannon was appointed an Orson Pratt Professor of Mathematics at Brigham Young University. He held this position until his retirement in September 2012. Cannon gave an American Mathematical Society (AMS) Invited address at the meeting of the AMS in Seattle in August 1977, an invited address at the International Congress of Mathematicians in Helsinki 1978, and delivered the 1982 Mathematical Association of America Hedrick Lectures in Toronto, Canada. Cannon was elected to the American Mathematical Society Council in 2003 with the term of service February 1, 2004, to January 31, 2007. In 2012 he became a fellow of the American Mathematical Society. In 1993 Cannon delivered the 30th annual Karl G. Maeser Distinguished Faculty Lecture at Brigham Young University. Cannon is a member of the Church of Jesus Christ of Latter-day Saints.
Mathematical contributions
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 were used by Cannon, Bryant and Lacher to prove (1979) an important case of the so-called characterization conjecture for topological manifolds. The conjecture says that a generalized n-manifold M {\displaystyle M} , where n ≥ 5 {\displaystyle n\geq 5} , which satisfies the "disjoint disk property" is a topological manifold. Cannon, Bryant and Lacher established that the conjecture holds under the assumption that M {\displaystyle M} be a manifold except possibly at a set of dimension ( n − 2 ) / 2 {\displaystyle (n-2)/2} . Later Frank Quinn 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 also introduced an important notion a cone type of an element of a finitely generated group (roughly, the set of all geodesic extensions of an element). Cannon proved that a convex-cocompact Kleinian group has only finitely many cone types (with respect to a fixed finite generating set of that group) and showed how to use this fact to conclude that the growth series of the group is a rational function. These arguments also turned out to generalize to the word-hyperbolic group context. Now standard proofs of the fact that the set of geodesic words in a word-hyperbolic group is a regular language also use finiteness of the number of cone types. Cannon's work also introduced an important notion of almost convexity for Cayley graphs of finitely generated groups, 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.
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