Spectral Rigidity and the Culler-Vogtmann Outer Space
Professor Kapovich works in the area of geometric group theory. A group is a central notion in mathematics, capturing the properties of the set of symmetries of a geometric shape or of a physical system. Although the concept of a group was born in the late nineteenth century in a geometric context, for the next 100 years groups were studied mainly as abstract algebraic objects. Geometric group theory, which arose as a distinct field in early 1990s, recaptures some of the classic ideas and seeks to exploit the relationship between algebraic properties of groups on one side and geometric properties of spaces on which these groups act by symmetries on another side.
During his Center appointment Professor Kapovich will study the property of spectral rigidity for subsets of free groups, exploring a new and deeper layer of the phenomenon known as marked length spectrum rigidity. This phenomenon allows, in many interesting cases, to recover the action of an abstract group by symmetries on some metric space from the marked length spectrum of the action, which assigns to every element of the group the number known as the translation length of that element. In such situations one can then look at smaller subsets of the group such that knowing the translation lengths of the elements from that subset is sufficient for decoding the entire marked length spectrum and thus recovering the entire group action. Such subsets of the group are said to be spectrally rigid.
Much of Professor Kapovich’s recent research has concerned the study of Out(Fn), the outer automorphism group of a free group Fnof finite rank n ≥ 2. In a joint paper with colleagues he proved that the set Pn of all primitive elements (where an element of Fn is primitive if it belongs to some free basis of Fn) is spectrally rigid in Fn. This result and others have raised a number of questions, including two he will study in his Center project:
Problem 1. Is the set Pn of all primitive elements in Fn strongly spectrally rigid?
Problem 2. What subsets of Pn are spectrally rigid?
He also plans to explore a conjecture involving Schottki-type free subgroups.