Periodic Reporting for period 4 - GroIsRan (Growth, Isoperimetry and Random walks on Groups)
Reporting period: 2022-03-01 to 2023-08-31
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 e.g. by multiplication of matrices.
An area in group theory, in active development in recent decades, allows to use geometric methods to study these algebraic objects.
The growth can be polynomial (this is for example the case for any commutative group), or it can be exponential (this is the case for many
groups of matrices). It is harder to construct the examples of intermediate (not polynomial and not exponential growth). 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 function. In a work with Tianyi Zheng we find a nearly matching lower bound for this function (to a long known upper bound due to Bartholdi).
Growth of groups which measures the volume of the balls is an important invariant of groups, which provides insight for asymptotic geometry of group and geometry.
Poisson boundary is a fundamental tool to study random walks. Isoperimetric profile measure the size of the boundary of a finite set.
In a joint work of Anna Erschler with Tianyi Zheng one give an answer to a long standing question about growth of Grigorchuk groups.
A joint work of the P.I. 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. We studied its relation to geometry.
A joint work of Anna Erschler with Vadim Kaimanovich studies 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 boundary of the random walk with the boundary of the forest. 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. The work is a development of a recent breakthrough 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.
In a joint work with Josh Frisch introduced a criterion for Liouville property (triviality of Poisson boundary) for group extensions. As an application, we reduced the problem whether random walks on a linear group is Liouville to the question about 2 times 2 metabelian groups. We have then obtained characterisation of Liouville property for linear groups over the field of positive characteristics, providing a positive answer to the Stability problem. In a work with Josh Frisch and Mark Rychnovsky, we study linear groups over a field of characteristics 0.
An area in group theory, in active development in recent decades, allows to use geometric methods to study these algebraic objects.
The growth can be polynomial (this is for example the case for any commutative group), or it can be exponential (this is the case for many
groups of matrices). It is harder to construct the examples of intermediate (not polynomial and not exponential growth). 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 function. In a work with Tianyi Zheng we find a nearly matching lower bound for this function (to a long known upper bound due to Bartholdi).
Growth of groups which measures the volume of the balls is an important invariant of groups, which provides insight for asymptotic geometry of group and geometry.
Poisson boundary is a fundamental tool to study random walks. Isoperimetric profile measure the size of the boundary of a finite set.
In a joint work of Anna Erschler with Tianyi Zheng one give an answer to a long standing question about growth of Grigorchuk groups.
A joint work of the P.I. 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. We studied its relation to geometry.
A joint work of Anna Erschler with Vadim Kaimanovich studies 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 boundary of the random walk with the boundary of the forest. 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. The work is a development of a recent breakthrough 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.
In a joint work with Josh Frisch introduced a criterion for Liouville property (triviality of Poisson boundary) for group extensions. As an application, we reduced the problem whether random walks on a linear group is Liouville to the question about 2 times 2 metabelian groups. We have then obtained characterisation of Liouville property for linear groups over the field of positive characteristics, providing a positive answer to the Stability problem. In a work with Josh Frisch and Mark Rychnovsky, we study linear groups over a field of characteristics 0.
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.
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.
The ERC project GroIsRan is finished now, and the progress is described in the previous section.