Final Report Summary - FIDAGG (FINITE DIMENSIONAL APPROXIMATIONS OF GRAPHS, GROUPS AND ALGEBRAS)
Finite dimensional approximations of graphs, groups and algebras.
The main goal of the project was the deeper understanding of structures approximable by finite objects.The first research objective was to study the approximation of graphs and metric structures. This part is strongly related to the seminal work of Lov´asz and Szegedy on dense graph convergence. In the late nineties, Misha Gromov introduced the notion of convergence for metric measure spaces via samplings. He asked the following question: What sort of invariants of metric measure spaces can beextended to the natural limits? Is there a geometric description of such limit objects?
A related objective was the study of sparse graph sequences. The main goal was to develop the limit theory for hyperfinite graph sequences.
We identified the limit objects defined by Gromov, by quantum metric spaces, a natural extension ofthe notion of metric measure spaces. It turned out that the uniqueness theorem of Lov´asz et. al. extends to this category. Gromov defined the so-called observable invariants such as the observational diameter and the separation distance for metric measure spaces. We extended these invariants to quantum metric spaces. A particularly interesting subcase was one of the main objectives of the project: the limit of finite trees. Together with G´abor Tardos, we defined three different kind of sampling notions for finite tree (one of them is equivalent to the original notion of Gromov) and described the limit objects in each case. It turned out, that limits of trees are the well-studied real trees with some additional structures. For sparse graphs, we develop the limit theory for hyperfinite (amenable) graph sequences and proved various testability results as well as corollaries for amenable actions.
The second objective was the study of finitely approximable groups. A recent result of K. Juschenko and N. Monod states that the topological full group of a minimal Cantor-system is amenable. This group is the first example of a finitely generated simple amenable group. We proved that the example of Juschenko and Monod is finitely approximable in a very strong sense, namely, it has the LEF-property. Jointly with Nicolas Monod, we proved that the topological full group of a minimal Cantor Z2-system is not amenable. For the measurable full groups, we were able to prove that for any sofic action the associated full group is sofic itself.
The third research objective was about the finite approximations of algebras and the development of Structural L˜uck Approximation Theory. Schick and Linnell introduced the notion of regular closure for complex group algebras. They proved that if for a certain group the Strong Atiyah Conjecture then the regular closure is skewfield. We proved that for amenable groups the regular closure isa canonical object similar to the Ore-extension. However, in the case of the lamplighter group, when the group algebra is not Ore and the Atiyah Conjecture does not hold, we were able to prove that the regular closure is isomorphic to the simple ring, introduced by John von Neumann in the thirties.
The main goal of the project was the deeper understanding of structures approximable by finite objects.The first research objective was to study the approximation of graphs and metric structures. This part is strongly related to the seminal work of Lov´asz and Szegedy on dense graph convergence. In the late nineties, Misha Gromov introduced the notion of convergence for metric measure spaces via samplings. He asked the following question: What sort of invariants of metric measure spaces can beextended to the natural limits? Is there a geometric description of such limit objects?
A related objective was the study of sparse graph sequences. The main goal was to develop the limit theory for hyperfinite graph sequences.
We identified the limit objects defined by Gromov, by quantum metric spaces, a natural extension ofthe notion of metric measure spaces. It turned out that the uniqueness theorem of Lov´asz et. al. extends to this category. Gromov defined the so-called observable invariants such as the observational diameter and the separation distance for metric measure spaces. We extended these invariants to quantum metric spaces. A particularly interesting subcase was one of the main objectives of the project: the limit of finite trees. Together with G´abor Tardos, we defined three different kind of sampling notions for finite tree (one of them is equivalent to the original notion of Gromov) and described the limit objects in each case. It turned out, that limits of trees are the well-studied real trees with some additional structures. For sparse graphs, we develop the limit theory for hyperfinite (amenable) graph sequences and proved various testability results as well as corollaries for amenable actions.
The second objective was the study of finitely approximable groups. A recent result of K. Juschenko and N. Monod states that the topological full group of a minimal Cantor-system is amenable. This group is the first example of a finitely generated simple amenable group. We proved that the example of Juschenko and Monod is finitely approximable in a very strong sense, namely, it has the LEF-property. Jointly with Nicolas Monod, we proved that the topological full group of a minimal Cantor Z2-system is not amenable. For the measurable full groups, we were able to prove that for any sofic action the associated full group is sofic itself.
The third research objective was about the finite approximations of algebras and the development of Structural L˜uck Approximation Theory. Schick and Linnell introduced the notion of regular closure for complex group algebras. They proved that if for a certain group the Strong Atiyah Conjecture then the regular closure is skewfield. We proved that for amenable groups the regular closure isa canonical object similar to the Ore-extension. However, in the case of the lamplighter group, when the group algebra is not Ore and the Atiyah Conjecture does not hold, we were able to prove that the regular closure is isomorphic to the simple ring, introduced by John von Neumann in the thirties.