The Artin-Out-ME-OA project is at the interface of group theory, in its geometric and ergodic aspects, and operator algebras, more specifically von Neumann algebras.
On the one side, group theory provides the mathematical framework for the concept of symmetry. The symmetries of a geometric object forms what mathematicians call a group: this means that symmetries can be composed, and every symmetry has an inverse (one can always apply the symmetry in reverse). Conversely, any group G (defined formally as an algebraic structure with a composition and inverse operations) can be realized as a set of symmetries of a geometric object. Geometric group theory, which was developed throughout the 20th century in the works of Dehn, Gromov, and many others, studies groups through the lens of geometry.
On the other side, von Neumann algebras, defined as operator algebras on Hilbert spaces, emerged in the 1930s as a mathematical framework for quantum mechanics. An important observation by Murray and von Neumann, which connects two seemingly unrelated theories, is that one can associate a von Neumann algebra L(G) to every group G.
A central question in the Artin-Out-ME-OA project is to understand whether the symmetries of the algebra L(G) contain enough information to recover the group G. This question, known as the rigidity problem for L(G), is now approachable thanks to recent breakthroughs in operator algebras, such as Popa's deformation/rigidity theory. Our goal is to investigate this question for several important families of groups of geometric or combinatorial origin: Artin groups, groups of diffeotopies of surfaces, automorphism groups of free groups.
A third theory interacts with geometric group theory and operator algebras in the project, namely probability theory. The idea here is that a group G can also be studied through the lens of its actions on probability spaces X. Following the works of Dye, Ornstein-Weiss, Zimmer, and many others, we explore the following question: to what extent does the orbit partition of the action of G on X remembers the group and the action? This is known as the orbit equivalence (or measure equivalence) rigidity problem for the group G and its actions. A central objective of the project is to tackle this problem for all the aforementioned groups. And since every group action on a probability space also gives rise to a von Neumann algebra, this question tightly relates to the previous one.
