Final Report Summary - RAWG (Random walks and Growth of Groups)
Alternatively, under slightly stronger assumption on the lower bound for the growth function, they can be found among torsion free groups; in particular, we describe the asymptotics of growth function of any Grigorchuk torsion-free group.
Our further joint work with Bartholdi describes the groups, that lie in some sense on the borderline between exponential and intermediate growth: their growth is exponential, nevertheless any finitely supported measure on these groups has trivial Poisson-Furstenberg boundary.
In our recent work with Bartholdi we study a certain preorder, which we call “preform” on the space of isomorphism classes of finitely generated groups. We study several properties of the oriented graph, associated to this preorder. We also study the relation between algebraic and geometric properties of groups and this preorder. We show that any countable group can be imbedded as a subgroup in a group of non-uniform exponential growth.
A recent work in this domain gives a necessary and sufficient condition for a group to be imbedded as a subgroup into a group of intermediate growth. In particular, we provide first example that show that additive group of Q can be imbedded as a subgroup in such a group. In a consequent paper we study word metrics on subgroups of groups of intemediate growth and show that groups of intemediate growth can have arbitrary bad distortion for their imbeddings into Hilbert spaces.
In a joint work with Vadim Kaimanovich we study dependence of certain
characteristics of random walks on groups, considered as a function of the defining measure. We give criteria for entropy and drift to be continious as a function of the defining measure.
The work on iterated identities of groups defines and study invariants of groups, expressed in terms of iteration of given word. Such invariants provide a structure on the space of finitely generated groups, which do not necessarily satisfy group identities. We prove that iterational depth of any metabelian group, as well as any polycyclic group is finite. We formulate many open questions related to verbal dynamics and iterational identities.