תאריך:
ה', 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.
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.