# Growth, Isoperimetry and Random walks on Groups

## Periodic Reporting for period 3 - GroIsRan (Growth, Isoperimetry and Random walks on Groups)

Growth of groups is an important invariant of groups, which provides insight for asymptotic geomtry of group and geometry.
Poisson boundary is a fundamental tool to study random walks. The study of Poisson boundary for random walks on groups is on a crossroads of group theory and probability theory.

In a joint work of Anna Erschler with Tianyi Zheng one give an aswer to a long standing question about growth of Grigorchuk groups.
Groups are algebraic objects, an example of group operations are additions for numbers (real numbers, integer numbers), multiplication of numbers. These are
commutative groups, since the operation satisfies a*b = b*a. Example of non-commutative groups are provided for example by multiplication of mathrices.
An area in in group theory, in active develpment in recent decades, allows to use geometric methods to study these algebraic objects. A basic geometric
invariant of a group is its growth: the growth function measure the size of balls of a given radius in an associated geometric space.
The growth can be polynomial (this is for example the case for any finitely generated commutative group), or it can be exponential (this is the case for many
groups of mathrices). It is harder to construct the examples of intermediate (not polynomial and not exponential growht). A rich class of such groups is
constructed in the early eighties by Grigorchuk. Still, a well-known question which remained open since their construction was to evaluate
the growth of these groups. In particular, for the most well-known construction of the first Grigorchuk group known lower bound was
quite far from the upper bounds for the growth funciton. In a work with Tianyi Zheng we find a nearly matchin lower bound for this function.

A joint work of Anna Erschler and Ivan Mitrofanov studies invariants of metric spaces related to Travelling Salesman Problem.
Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city and returns to the origin city? This is a well-known problem in mathematics and computer science. Universal Travelling Salesman problem asks for a universal strategy, which can be applied for an arbitrary subset of cities among a given list of cities. In the early eighties an algorithm based on self-filling curves is introduced for cities on the plane. It has been studied later both for cities on the plane, and for sequences of finite graphs, not necesarily planar. In our work we study new relations between invariants expressed in terms of Universal Travelling Salesman problem, geomertic invariants of metric spaces and algebraic and geometric invariants of groups.

In a joint work of Anna Erschler with Vadim Kaimanovich we study arboreal structures in groups.
For any countable group with infinite conjugacy classes one has constructed a family of forests on the group. For each of them there is a random walk on the group with the property that its sample paths almost surely converge to the geometric boundary of the forest in a way that resembles the simple random walks on trees. It allows us to identify the Poisson boundary of the random walk with the boundary of the forest and to show that the group action on the Poisson boundary is free (which, in particular, implies non-triviality of the Poisson boundary). As a consequence one obtains that any countable group carries a random walk such that the stabilizer of almost every point of the Poisson boundary coincides with the hyper-FC-centre of the group, and, more generally, we characterize all normal subgroups which can serve as the pointwise stabilizer of the Poisson boundary of a random walk on a given countable group. The work is a development of a recent result of Frisch - Hartman - Tamuz - Vahidi Ferdowsi who proved that any group which is not hyper-FC-central admits a measure with a non-trivial Poisson boundary.
Main results achived so far are

-cautiosness of random walks, criterion for Shalom's property H_FD, thin subgroups for random walks on groups (Anna Erschler, Narutaka Ozawa)

-New isoperimetric inequality and new properties of groups with Shalom's property H_FD (Anna Erschler, Tianyi Zheng)

-New lower bounds for Grigorchuk groups, the answer of a long standing question about growth of the first Grigorchuk goup (Anna Erschler, Tianyi Zheng)

-Essential freeness of action of groups for some meaures on groups, for their action on Poisson boundary. Complete description of the Poisson
boundary for these measures (Anna Erschler, Vadim Kaimanovich).

-New invariants of metric spaces related to Travelling Salemen Property. Ordering Ratio function for hyperbolic spaces.
Gap theorem for orders on groups of polynomial Growth (Anna Erschler, Ivan Mitrofanov, in preparation).
We plan to work on new criteria for Poisson boundary triviality on groups, applications to Stability problem, limit theorem for random walks on groups and Travelling Salesman problem in groups.