Towards a group-like small cancellation theory for rings
Let a group G be given by generators and defining relations. It is known that we cannot, in general, extract specific information about the structure of G using the defining relations. However, if the defining relations satisfy small cancellation conditions, then we possess a great deal of knowledge about G. In particular, such groups are hyperbolic, that is, we can express the multiplication in the group by means of thin triangles. It seems of interest to develop a similar theory for rings.
Let kF be the group algebra of the free group F over some field k. Let F have a fixed system of generators. Then its elements are reduced words in these generators that we call monomials. Let I be an ideal of kF generated by a set of polynomials, and let kF / I be the corresponding quotient algebra. In the present work we state conditions on these polynomials that will enable a combinatorial description of the quotient algebra similar to small cancellation quotients of the free group. In particular, we construct a linear basis of kF / I and describe a special system of linear generators of kF / I for which the multiplication table amounts to a linear combination of thin triangles.
Constructions of groups with exotic properties make extensive use of small cancellation theory and its generalizations. In a similar way, generalizations of our approach allow one to construct various examples of algebras with exotic properties.
This is a joint work with A. Kanel-Belov, E. Plotkin and E. Rips.