Pénélope Azuelos

Group structures on real trees and their products

The study of finitely generated and locally compact groups as geometric objects has been a highly productive area for some time. In this talk I will discuss what group structures (or free transitive actions) can look like on metric spaces that are typically very far from being locally finite or locally compact: real trees. These are continuous analogues of simplicial trees which play an important role in geometric group theory, and the group structures on them can be thought of as analogues of free groups. This can be made precise using work of Berestovskii—Plaut, which shows that any group with the metric structure of a length space is the (metric and group) quotient of a group with the structure of a real tree. I will describe some methods of constructing group structures on real trees and their products, and discuss some of the surprising behaviours that show up.