Tag - Small cancellation theory

Agatha Atkarskaya: Introduction to group-like small cancellation theory for rings

18th November, 2022

The structure of small cancellation groups is well known. Тhey are widely used in construction of groups with unusual properties (for example Burnside groups and Tarskii monster). We were interested in developing a similar theory for rings. However, such theory meets significant difficulties because, unlike groups, rings have two operations: addition and multiplication. I will speak about small cancellation conditions for rings that we introduced. These conditions provide the desired properties. I will discuss our way towards these conditions, examples and possible applications of small cancellation rings.

Olga Kharlampovich: Universal theory of random groups

22nd April, 2021

We will use Gromov's density model of randomness. A random group at density d satisfies some property (of groups) P if the probability of occurrence of P tends to 1 as the length of relations goes to infinity. Julia Knight conjectured that the limit of the theories of random groups should converge to the theory of a free group. We will show that this is true for the universal theory of a random group at density d<1/16. Namely, every universal and every existential axiom of the free group is also true in a random group. Notice that a random group at density d<1/16 satisfies a small cancellation condition C'(1/8). We will also show that a random group at density d<1/2 is not a limit group (for a few relations model this was proved by Ho when the number of generators is less than the number of relations). These are joint results with R. Sklinos.

Agatha Atkarskaya: Small cancellation rings

4th December, 2020

The theory of small cancellation groups is well known. In this paper we introduce the notion of Group-like Small Cancellation Ring. This is the main result of the paper. We define this ring axiomatically, by generators and defining relations. The relations must satisfy three types of axioms. The major one among them is called the Small Cancellation Axiom. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. It turns out that the defined ring possesses a kind of Gröbner basis and a greedy algorithm. Finally, this ring can be used as a first step towards the iterated small cancellation theory which hopefully plays a similar role in constructing examples of rings with exotic properties as small cancellation groups do in group theory. Joint results with A. Kanel-Belov, E. Plotkin, E. Rips.

Agatha Atkarskaya: Group-like small cancellation theory for rings

25th June, 2020

(joint work with Prof. A.Kanel-Belov, Prof. E.Plotkin, Prof. E.Rips) It is well known that small cancellation groups and especially their generalizations provide a very powerful technique for constructing groups with unusual, and even exotic, properties, like for example, infinite Burnside groups and Tarski monster groups. In the present work we develop a small cancellation theory for associative algebras with a basis of invertible elements. We introduce a list of small cancellation conditions for a presentation of such associative algebras. We construct an explicit linear basis for these algebras. In parallel we show that our algebras possesses algorithmic properties similar to ones for groups with small cancellation groups. Namely, the equality problem in these algebras is solvable with the use of a certain analogue of Dehn's algorithm. We do hope that being of interest as rings of new type by itself, these algebras will inherit useful practical properties known for small cancellation groups and, thus, they can be used for obtaining complicated algebras with the very specific properties.