Final Report Summary - RAWG (Random walks and Growth of Groups)
Growth function of a finitely generated group is a function that counts the number of elements of given length in a word metric associated to some finite generating set of this group. The asymptotics of such a function does not depend on the choice of a generating set. For many classes of groups there is a following alternative: either the growth function has exponential asymptotics or it has polynomial growth. This is true for example for all solvable groups (Milnor Wolf), more generally for any elementary amenable group (Chow) and for any linear group (as a corollary of Tits alternative). First examples o groups of intermediate growth (not exponential and not polynomial growth) were discovered by Rostislav Grigorchuk in earlier eighties. Until very recently there was no example of a group of intermediate growth for which one knows the asymptotics of its growth function. In a joint work with Laurent Bartholdi we have found this asymptotics for a large class of groups of intermediate growth. We have shown in particular that any function that is in a certain sense asymptotically larger than exp(n^a) is a growth function of some group. Such groups can be found among torsion groups of intermediate growth.
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.
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.