Group Actions on 1-Manifolds: A List of Very Concrete Open Questions

Abstract

Over the last four decades, group actions on manifolds have deserved much attention by people coming from different fields, as for instance group theory, low-dimensional topology, foliation theory, functional analysis, and dynamical systems. This text focuses on actions on 1-manifolds. We present a (non exhaustive) list of very concrete open questions in the field, each of which is discussed in some detail and complemented with a large list of references, so that a clear panorama on the subject arises from the lecture.

Finitely generated infinite simple groups of homeomorphisms of the real line

“…Finally, we remark that the groups we construct are also interesting test cases for the following well known open question (see Question 7.5 in [1] and Question 3 in [17]). Question 1.6.…”

The Structure of Homeomorphism and Diffeomorphism Groups

“…Navas posed this question explicitly for onedimensional manifolds in a 2017 problems list [Nav18]. As he notes, questions of distinguishing regularity have a long history among dynamicists that can be traced back to work of Denjoy on the relationship between regularity and topological dynamics.…”

Singular Stationary Measures for Random Piecewise Affine Interval Homeomorphisms

“…Determining which of the two possibilities occur is a well-known problem, especially in the context of groups of smooth diffeomorphisms acting on the circle (see e.g. [Nav17,Question 18]). Up to now, an answer has been given only in some particular cases.…”

“…The fact that the maps are piecewise affine is especially interesting, since such systems are studied intensively and often serve as models for smooth systems (see e.g. [Nav17,Questions 12 and 16]). In a forthcoming paper [BS19] we prove that the stationary µ for an AM-system is singular and has Hausdorff dimension smaller than 1 for an open set of parameters, including also non-resonant cases.…”