Main progress in understanding behaviour of the growth is a joint work of the P.I. with Tianyi Zheng, that shows sharp lower estimates for Grigorchuk groups, including the most studied first Grigorchuk group. The above mentioned work evaluates the limit log log v(n)/log n.
A figure below shows the Schreier graph of the Grigorchuks group, a labelled graph with an interesting self-similar behaviour for its labelling.
A model example for Poisson boundary behaviour is a well-known class of random walks on free groups. I this case the Cayley graph of the group is a tree, and the trajectory of the random walks converge almost surely to its geometric boundary (which can be described as end points of the tree). Main progress in describing the Poisson boundary of random walks, obtained during the project, is a result of the PI and Vadim Kaimanovich that describes and arboreal struncture (a forest) definted by any group of superpolynomial growth.
An asymptotic property of the groups that have strong algebraic manifestation in case when the group is amenable (the property meaning that the isoperimetric profile is not bounded) is so called Shalom's property. Results achieved during the project include cautiousness of random walks criterion for Shalom's property and description of thin subgroups for random walks on groups (joint work of the PI and Narutaka Ozawa); new isoperimetric inequality and new properties of groups with Shalom's property (joint work of the P.I. and Tianyi Zheng);
New invariants of metric spaces related to Travelling Salesmen Property, Ordering Ratio function for hyperbolic spaces and Gap theorem for orders on groups of polynomial Growth is a subject of a joint with Ivan Mitrofanov. The Gap theorem for orders provides a new information about the metric (in terms of its invariants in terms of orders) even in such classical and basic examples as Abelian groups.
Isoperimetry was also studied in the work of Bogdan Stankov, his results include exact evaluation of the Foelner function and seems to be the first group of superpolynomial growth where such evaluation is done. His other work studies Poisson boundary for groups of piecewise projective homeomorphisms over the integers. The Poisson boundary was also studied by Johannes Cuno. His joint work with Ecaterina Sava-Huss describes the boundary for Baumslag-Solitar groups. This is a well-known class of groups, introduced by Baumslag and Solitar in the sixties, who enjoy unusual algebraic properties. While some algebraic properties of these groups were well-known, it was a challenge to gain a precise understanding of their asymptotic geometry and behaviour of random walks on these groups.
The work of Arman Darinyan focus on algorithmic and algebraic properties of groups. A group is said to be simple if it does not admit any quotient groups. It is a challenge to understand the asymptotic behaviour of simple groups.
The joint work of Arman Darbinyan with Markus Steenbock relates the situation of ordered group to that of simple ordered groups. They
shows that every countable left-ordered group embeds into a finitely generated left-ordered simple group.
For many notions studied in the project there two different types of phenomena that can be observed, depending on the fact whether the group is "small" (being Abelian, of subexponential growth, amenable) or large (being free group or in some class generalising the notion of freeness including hyperbolic groups, relatively hyperbolic groups and their further generalisations). The work of Nima Hoda deals with hyperbolic like groups and the notions inspired by hyperbolicity.
Rachel's Skipper research concerns groups of self-similar nature, including group of automorphisms and almost automorphisms of trees.
