Tim Riley (Yale University)

Title: Filling functions and their applications

Abstract: The Word Problem transcends its origins in group theory, decidability and complexity theory, and forms a bridge to the geometric world of soap films and isoperimetry. To a finite presentation of a group we can associate a topological space, the cayley 2-complex, in which recognising a word to be trivial amounts to spanning a loop with a disc (a van Kampen diagram). Filling functions capture geometrical features of this disc and serve both as measures of the algorithmic complexity of the Word Problem ans as windows onto the large scale geometry of the group –they are a source of quasi-isometry invariants. The best know filling function is the Dehn function, which concerns area, but others such as the filling length, gallery length, and diameter functions also play important roles. I will describe how these filling functions interact and give applications (e.g. to nilpotent groups and to asymptotic cones), examples, and open problems.