Filling radii of finitely presented groups

The filling radius function R of Gromov measures the minimal radii of van Kampen diagrams filling edge-circuits omega in the Cayley 2-complex of a finite presentation P. It is known that the Dehn function can be bounded above by a double exponential in R and the length of the loop, and it is an open question whether a single exponential bound suffices. We define the upper filling radius (R) over bar(omega) of omega to be the maximal radius of minimal area fillings of omega and let (R) over bar be the corresponding filling function, so (R) over bar (n) is the maximum of (R) over bar(omega) over all edge-circuits omega of length at most n. We show that the Dehn function is bounded above by a single exponential in (R) over bar and the length of the loop. We give an example of a finite presentation P where R is linearly bounded but (R) over bar grows exponentially.