Filling Radii of Finitely Presented Groups

The filling radius function R of Gromov measures the minimal radii of van Kampen diagrams filling edge‐circuits w 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 R̄ (w) of w to be the maximal radius of minimal area fillings of w and let R̄ be the corresponding filling function, so R̄ (n) is the maximum of R̄ (w) over all edge‐circuits w of length at most n. We show that the Dehn function is bounded above by a single exponential in R̄ and the length of the loop. We give an example of a finite presentation P where R is linearly bounded but R̄ grows exponentially.