Activities This Week

The tracial Rokhlin property for actions of finite groups is now well known, along with weakenings and versions for other classes of discrete groups. The Rokhlin property for actions of infinite infinite compact groups has also been studied. We define and investigate the tracial Rokhlin property for actions of second countable compact groups on simple unital C*-algebras. The naive generalization of the verrsion for finite groups does not appear to be good enough. We have a property which, first, allows one to prove the expected theorems, second, is “almost” implied by the version for finite groups when the group is finite, and, third, admits examples.

This is joint work with Javad Mohammadkarimi.

The dimension of the space of multilinear products of higher commutators is equal to the number of derangements, $[e^{-1}n!]$. Our search for a combinatorial explanation for this fact led us to study representations of left regular bands, whose resolution is obtained through analysis of cubical partial representations. There are applications in combinatorics, probability, and nonassociative algebra.

במרכז של תורת הקשרים עומדת השאלה: בהנתן שרוך קשור בקשר כלשהו, אשר קצוותיו תפורים זה לזה, האם ניתן “לפתוח” את הקשר בלי לחתוך את השרוך. תחום זה משתמש בטכניקות ממגוון תחומים אחרים במתמטיקה, החל מטופולוגיה ועד לתורת ההצגות.

משפט פארי מילנור הוא משפט המאפשר, בתנאים מסויימים, לתת תשובה לשאלה הנ”ל בעזרת אנליזה. בהרצאה אספר על המשפט ועל שתי הוכחות יפות שניתנו באופן בלתי תלוי על ידי פארי ומילנור.

Given a group G, an Eilenberg-MacLane space X = K(G,1) provides a topological model of both G and Aut(G). The latter is understood via Whitehead’s theorem as the group of pointed homotopy equivalences of X up to homotopy. When X has rich structure, such as the case of a closed surface group, this point of view leads to a rich understanding of Aut(G). Motivated by dynamics and mathematical physics, Biswas, Nag, and Sullivan initiated the study of the universal hyperbolic solenoid, the inverse limit of all finite covers of a closed surface of genus at least two. Following their work, Odden proved that the mapping class group of the universal hyperbolic solenoid is isomorphic to the abstract commensurator of a closed surface group. In this talk I will present a general topological analog of Odden’s theorem, realising Comm(G) as a group of homotopy equivalences of a space for any group of type F. I will then use this realisation to classify the locally finite subgroups of the abstract commensurator of a finite-rank free group. This is joint work with Daniel Studenmund.