Date:

Thu, 02/12/202114:30-15:30

**Live broadcast link:**https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=2ec36750-2782-4af4-9aa7-add500c12183

**Title:**Small cancellation methods for groups and rings

**Abstract:**When we have a group $G$ given by generators and defining relations $G = \langle x_1, \ldot, x_s \mid R_1, \ldots, R_t \rangle$, in general we can not say much about a structure of $G$. However, if there are special restrictions on the relators $R_i$, it allows us to study a structure of the group. One type of such restrictions is the condition that the words $R_i$ have relatively small common subwords. Then $G$ is called a small cancellation group. This allows us to say a lot about the structure of $G$. Moreover, the idea of having small interaction between relators can be generalized. The obtained groups have a clear structure and produce examples with very interesting properties and unusual behaviour.

In our work we use small cancellation methods in two directions. First, we construct a group-like small cancellation theory for rings (https://arxiv.org/abs/1807.10070, https://arxiv.org/abs/2010.02836). These papers are the first step towards the construction of rings with unusual properties. For example, we expect that it will allow us to construct a division algebra of infinite dimension over its center with a finitely generated multiplicative group.

On the other hand, we develop a new version of an iterated small cancellation approach for groups. Let $B(m, n) = \langle x_1, \ldots, x_m \mid w^n = 1, w\in \langle x_1, \ldots, x_m\rangle \ldots \rangle$. The Burnside problem asks whether this group is finite or infinite. Using the new combinatorial ideas, we show that $B(m, n)$ is infinite for odd exponents $n \geqslant 297$.

So, one can see that the small cancellation approach is a kind of meta-framework that is applicable to problems that look totally different. In my talk I will explain a general spirit of small cancellation methods, and outline our approach for two problems noticed above. So, the aim is to show why similar ideas work both in groups and rings.