# Colloquium: Agatha Atkarskaya (HUJI)

Date:
Thu, 02/12/202114:30-15:30
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.
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 $n \geqslant 297$.