Periodic Reporting for period 1 - Artin-Out-ME-OA (Artin groups, mapping class groups and Out(Fn): from geometry to operator algebras via measure equivalence)
Période du rapport: 2022-09-01 au 2025-02-28
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.
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.
We tackled the measure equivalence rigidity problem for several important families of groups of geometric and combinatorial origin: right-angled Artin groups, and more generally, graph products.
A right-angled Artin group has a simple description: it can be thought of as the set of all words that can be written on an alphabet with finitely many letters a,b,... (and their inverses), with the extra rule that some of the letters commute (e.g. writing ab may be considered the same as writing ba). Right-angled Artin groups are of particular importance in geometric group theory because their geometry is in close relation to cube complexes. This geometry gave them a prominent role in low-dimensional topology in the works of Haglund-Wise, Agol.
Graph products are a generalization of right-angled Artin groups. Starting from a finite graph X, with a group Gv attached to every vertex v of X, the graph product is the group obtained from the groups Gv by only imposing that Gv and Gw commute whenever v and w are adjacent.
With Jingyin Huang, we solved the measure equivalence rigidity problem for many right-angled Artin groups, upon imposing a (necessary) integrability condition on a cocycle naturally associated to the problem.
With Amandine Escalier, we proved that the measure equivalence classification problem for graph products reduces in many situations to the measure equivalence classification problem within the vertex groups Gv. As a consequence of our work, we obtained the first example of a group G which is rigid in measure equivalence, but not rigid from the viewpoint of its asymptotic geometry (in quasi-isometry). This means that there are many groups H that have the same geometry as G on a large scale, but none of these groups can act on a probability space with the same orbits as G.
Another achievement of the project is the computation, with Damien Gaboriau and Yassine Guerch, of some of the l^2-Betti numbers of Out(Wn), the outer automorphism group of a free Coxeter group. These are important measure equivalence invariants.
A right-angled Artin group has a simple description: it can be thought of as the set of all words that can be written on an alphabet with finitely many letters a,b,... (and their inverses), with the extra rule that some of the letters commute (e.g. writing ab may be considered the same as writing ba). Right-angled Artin groups are of particular importance in geometric group theory because their geometry is in close relation to cube complexes. This geometry gave them a prominent role in low-dimensional topology in the works of Haglund-Wise, Agol.
Graph products are a generalization of right-angled Artin groups. Starting from a finite graph X, with a group Gv attached to every vertex v of X, the graph product is the group obtained from the groups Gv by only imposing that Gv and Gw commute whenever v and w are adjacent.
With Jingyin Huang, we solved the measure equivalence rigidity problem for many right-angled Artin groups, upon imposing a (necessary) integrability condition on a cocycle naturally associated to the problem.
With Amandine Escalier, we proved that the measure equivalence classification problem for graph products reduces in many situations to the measure equivalence classification problem within the vertex groups Gv. As a consequence of our work, we obtained the first example of a group G which is rigid in measure equivalence, but not rigid from the viewpoint of its asymptotic geometry (in quasi-isometry). This means that there are many groups H that have the same geometry as G on a large scale, but none of these groups can act on a probability space with the same orbits as G.
Another achievement of the project is the computation, with Damien Gaboriau and Yassine Guerch, of some of the l^2-Betti numbers of Out(Wn), the outer automorphism group of a free Coxeter group. These are important measure equivalence invariants.
The solution (with Jingyin Huang) of the measure equivalence rigidity problem for many right-angled Artin groups establishes new connections, but also highlights several key differences, with the case of lattices in rank 1 Lie groups that had been studied by Bader, Furman and Sauer (2013). It has important geometric consequences, e.g. to the classification of locally compact groups in which right-angled Artin groups can embed as lattices.
The solution (with Amandine Escalier) of the measure equivalence classification problem for many graph products suggests a general framework in which one can hope to establish further rigidity and classification results. It also paves the way towards tackling the analogous problem at the von Neumann algebra level.
The solution (with Amandine Escalier) of the measure equivalence classification problem for many graph products suggests a general framework in which one can hope to establish further rigidity and classification results. It also paves the way towards tackling the analogous problem at the von Neumann algebra level.