Algebraic, analytic, dynamical properties of groups actions

The project is in the domain of theoretical mathematics and seeks to enhance our understanding of the landscape of infinite groups. In mathematics, the notion of a group captures an abstraction of symmetries of a physical object or a theoretical space. An easy example of a group is the set of symmetries of the familiar three dimensional space we inhabit: these are symmetries obtained by performing compositions and inverses of translations, reflections and rotations. Here by composition we mean the successive application of these symmetries, one by one, to obtain a new symmetry. By inverse we mean a symmetry that reverses, or undoes, a given symmetry. All of this can be captured conveniently in the language of algebra: by means of a group. Group theory, broadly speaking, is the study of symmetries of objects and abstract geometric spaces. This is a central subfield of modern mathematics, and has intricate connections with other central subfields such as topology, geometry, dynamics and modern algebra. The landscape of infinite groups is vast and in a sense too large to understand completely. Much of modern group theory seeks to address this issue by systematically studying certain naturally emerging classes of groups by understanding certain properties and phenomena that they exhibit. Some of these properties are fundamental and naturally emerging, such as simplicity and finite generation. On the other hand, we also study phenomenon that are counterintuitive, such as the Banach-Tarski paradox. A major goal of this project is to study certain classes of groups that emerge as homeomorphisms of spaces such as the circle and the real line. A homeomorphism is a map from a space to itself that roughly speaking preserves the topological properties of the space. An example of a homeomorphism of the circle is simply a rotation. An example of a homeomorphism of the real line is the map f(x)=2x+1. Families of groups that emerge in this setting as compositions of such maps satisfy surprising combinations of properties. On the one hand the project aims at understanding which properties are satisfied by certain well known examples, and on the other hand it seeks to construct new examples that may exhibit surprising new features. Another goal of this project is to study a family of interconnected, well known open questions in modern group theory that relate the algebraic structure of the group with the geometry of the relevant geometric space. Many of these questions have been unanswered for several decades, and remain challenging. However, the project seeks to enhance our understanding of these questions by developing new techniques and examples.