James W. Cannon (born January 30, 1943) is an American mathematician working in the areas of low-dimensional topology and geometric group theory.
[3] Cannon gave an AMS Invited address at the meeting of the American Mathematical Society 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.
[1][4] Cannon was elected to the American Mathematical Society Council in 2003 with the term of service February 1, 2004, to January 31, 2007.
[6] In 1993 Cannon delivered the 30th annual Karl G. Maeser Distinguished Faculty Lecture at Brigham Young University.
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.
In general, the conjecture is false as was proved by John Bryant, Steven Ferry, Washington Mio and Shmuel Weinberger.
In particular, Cannon proved that convex-cocompact Kleinian groups admit finite presentations where the Dehn algorithm solves the word problem.
[15] Now standard proofs[17] 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.
[19][20][21] An influential paper of Cannon and William Thurston "Group invariant Peano curves",[22] that first circulated in a preprint form in the mid-1980s,[23] introduced the notion of what is now called the Cannon–Thurston map.
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,[24][25] Erica Klarreich,[26] Brian Bowditch[27] and others.
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.
[39] 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.
[39] Cannon, Floyd and Parry also applied their model to the analysis of the growth patterns of rat tissue.