Describing spaces of harmonic actions on the line

Abstract: Considering actions of a given group on a manifold can be seen as a nonlinear version of classical representation theory. In this context, there is a well-developed theory for actions on one-manifolds, in contrast to the situation for higher-dimensional manifolds, where the situation is still at the level of exploration. This is mainly due to the tight relation to the theory of orderable groups, which has no analogue in higher dimension.

How to describe all possible actions on the line of a given group? For finitely generated groups, one can consider the space of harmonic actions, whose existence is based on a result of Deroin-Kleptsyn-Navas-Parwani. This turns out to be a compact space endowed with a translation flow, whose space of orbits gives exactly the space of all semi-conjugacy classes of actions on the line without global fixed points.

We are able to understand the space of harmonic actions for solvable groups and many locally moving groups (including Thompson's F and generalizations): the actions of these groups which are not the obvious ones, are all obtained from actions on planar real trees fixing a point at infinity. This talk is based on a joint project with J Brum, N Matte Bon and C Rivas.

algebraic topologyfunctional analysisgroup theorygeometric topologyoperator algebras

Audience: researchers in the topic

Vienna Geometry and Analysis on Groups Seminar