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.
