Some new results on Golod-Shafarevich algebras

One of the most beautiful and useful parts of noncommutative ring 
theory is the Golod-Shafarevich theorem (1964), which shows that free  algebras defined by homogeneous relations are infinite dimensional,  provided that the number of defining relations of each degree is not  too large. Golod-Shafarevich algebras were introduced by Golod and  Shafarevich in 1964, and were later used to solve several interesting  open problems in several different areas of mathematics, namely the  Burnside problem in group theory, the Kurosh Problem in noncommutative  algebra and the Class Field Tower in number theory.

The results concerning Golod-Shafarevich algebras are related to 
Golod-Shafarevich groups, and the results often mirror one another 
even though the proofs are often different. For example, in 2006 the 
author solved  a question of Zelmanov by showing that there are 
Golod-Shafarevich algebras that have no infinite-dimensional 
homomorphic images of polynomial growth. It was shown by Ershov that  there exist Golod-Shafarevich groups without infinite images of 
polynomial growth. In 2000, Zelmanov showed that Golod-Shafarevich 
groups contain non-abelian free pro-groups and later related results 
on free subgroups were obtained by Kassabov. Recently, it was shown by  the author that finitely presented Golod-Shafarevich algebras contain  free noncommutative subalgebras  under mild assumptions on the number  of generating relations of each degree, and that such 
Golod-Shafarevich algebras can be mapped onto algebras with linear 
growth.

There are many inspiring open questions in this area; for example, it 
is an open question whether the converse of the Golod-Shafarevich 
theorem is true (Anick's question). Very interesting results related 
to this question were obtained by Wisliceny and recently by Iyudu and 
Shakarin.

In this talk we mention recent results on Golod-Shafarevich algebras 
and some open questions in this area.