The Cayley 2-complex is a simply connected space associated to a finite presentation of a group.
Filling functions capture aspects of the geometry of “filling” discs spanning loops in this space, and are important group invariants. Many natural questions arise about the interrelationships between filling functions: for example, one might ask: given knowledge about the areas of fillings, what can be said about their diameters? I will survey some work, undertaken in collaborations with Bridson and Gersten, on such questions.
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