Periodic Reporting for period 3 - LIMITS (Limits of Structures in Algebra and Combinatorics)
Berichtszeitraum: 2022-04-01 bis 2023-09-30
We are focused on several problems spanning the boundary between algebra and combinatorics. Let me briefly list some of the themes. 1) We are investigating rank functions and approximations of infinite groups with respect to rank functions. This is an exciting area where many new developments have been happening over the last few years. We particularly would like to prove the so-called stability of approximations with respect to the rank metric for new classes of groups. 2) We are investigating graphings and their generalisations (discrete measured groupoids), with particular focus on expansion/Property (T) and the Aldous-Lyons Conjecture. 3) We are investigating equidecompositions of sets in Euclidean spaces, with a particular focus on the interplay between equidecompositions and the expanding properties of sets.
In the case of rank approximations for new classes of groups, we are currently developing the techniques which allow to pass from commutative groups to noncommutative groups. In particular we now have finished proving an effective variant of "Nullstellensatz" for the group ring of the Heisenberg group, and we are in the process of developing the analogues of the other commutative algebra techniques which were used successfully in the case of group rings of commutative groups.
In graphings, we have finished generalising the theorem of Hutchcroft and Pete from the context of group actions to the more general context of graphings and equivalence relations with property (T). We have also found interesting new examples of graphings with property (T). This has required very considerable development of general theory of groupoids with property (T) (for example, we generalised the Connes-Weiss theorem characterising property (T) groups, and the Kazhdan theorem on lattices in groups with property (T) to the context of random countable subsets in groups with property (T)). In a separate development, we have suggested "directed analogues" of expanders and hyperfinite graphs sequences, which are the most important notions in the theory of graph limits. We hope that these notions will be investigated further and that we will find interesting applications.
In equidecompositions, we have developed a model of random sets, modeled after the Sierpinski gasket, to find examples of sets which are not equidecomposable, but whose "obvious" equidecomposability invariants are the same.
In graphings, we have finished generalising the theorem of Hutchcroft and Pete from the context of group actions to the more general context of graphings and equivalence relations with property (T). We have also found interesting new examples of graphings with property (T). This has required very considerable development of general theory of groupoids with property (T) (for example, we generalised the Connes-Weiss theorem characterising property (T) groups, and the Kazhdan theorem on lattices in groups with property (T) to the context of random countable subsets in groups with property (T)). In a separate development, we have suggested "directed analogues" of expanders and hyperfinite graphs sequences, which are the most important notions in the theory of graph limits. We hope that these notions will be investigated further and that we will find interesting applications.
In equidecompositions, we have developed a model of random sets, modeled after the Sierpinski gasket, to find examples of sets which are not equidecomposable, but whose "obvious" equidecomposability invariants are the same.
The progress so far has been summarised in the previous paragraph. In terms of further work, we will focus further on the following topics.
1) Graphings and groupoids, in particular in the context of measured manifolds. We would like to show that the Aldous-Lyons conjecture holds for graphings which are "2-dimensional", extending some of the recent results of Conley, Gaboriau, Marks and Tucker-Drob. We would like to also find further examples of graphings with property (T) in the context of measured manifolds, paritcularly such graphings which do not arise from group actions.
2) Rank approximations, in particular we hope to complete the proof of stability of rank approximations in the case of the Heisenberg group.
3) Borel combinatorics, in particular we hope for some further results on Borel and measurable versions of the Lovasz Local Lemma.
1) Graphings and groupoids, in particular in the context of measured manifolds. We would like to show that the Aldous-Lyons conjecture holds for graphings which are "2-dimensional", extending some of the recent results of Conley, Gaboriau, Marks and Tucker-Drob. We would like to also find further examples of graphings with property (T) in the context of measured manifolds, paritcularly such graphings which do not arise from group actions.
2) Rank approximations, in particular we hope to complete the proof of stability of rank approximations in the case of the Heisenberg group.
3) Borel combinatorics, in particular we hope for some further results on Borel and measurable versions of the Lovasz Local Lemma.