Groups, growth, and rationality
Time and place
1 PM on Tuesday, February 17th, 2015; NAC 6/327
Moon Duchin (Tufts University)
Abstract
Given a group and a finite generating set, a very natural question is to study the "volume growth," or the number of words that can be spelled in at most n letters, which is the size of the ball of radius n in the word metric. Growth of groups has been a much-studied topic since the 1950s. In the 1980s it was shown that two big classes of groups, hyperbolic groups and virtually abelian groups, have rational growth in all generators (i.e., the growth series describes a rational function). Rationality is equivalent to the existence of recursions describing the growth, and it inspired Thurston's study of finite state automata for decision problems in groups. In recent work with Mike Shapiro, we show that the Heisenberg group shares this strong property, settling a long-standing open question.
