Algebraic Geometry of Partially Commutative Groups

Solving equations and describing their solutions has always been a central part of mathematics. Here we have equations which are framed in terms of groups and for which we seek solutions in groups. We know how to describe solutions to equations over Abelian groups and to equations in one variiable in free groups. This project considers a class of groups that contains free and Abelian groups but yet is simple to describe and which is amenable to analysis. Our eventual aim is to understand solutions to equations over these groups, but to make progress we are restricting attention to equations of one variable. We allso intend to make some preliminary investigations into areas of group theory (related to equations) where there are some difficult problems that we think we may have some techniques to solve.