Dehn Function - History

History

The idea of an isoperimetric function for a finitely presented group goes back to the work of Max Dehn in 1910s. Dehn proved that the word problem for the standard presentation of the fundamental group of a closed oriented surface of genus at least two is solvable by what is now called Dehn's algorithm. A direct consequence of this fact is that for this presentation the Dehn function satisfies Dehn(n) ≤ n. This result was extended in 1960s by Martin Greendlinger to finitely presented groups satisfying the C'(1/6) small cancellation condition. The formal notion of an isoperimetric function and a Dehn function as it is used today appeared in late 1980s – early 1990s together with the introduction and development of the theory of word-hyperbolic groups. In his 1987 monograph "Hyperbolic groups" Gromov proved that a finitely presented group is word-hyperbolic if and only if it satisfies a linear isoperimetric inequality, that is, if and only if the Dehn function of this group is equivalent to the function f(n) = n. Gromov's proof was in large part informed by analogy with filling area functions for compact Riemannian manifolds where the area of a minimal surface bounding a null-homotopic closed curve is bounded in terms of the length of that curve.

The study of isoperimetric and Dehn functions quickly developed into a separate major theme in geometric group theory, especially since the growth types of these functions are natural quasi-isometry invariants of finitely presented groups. One of the major results in the subject was obtained by Sapir, Birget and Rips who showed that most "reasonable" time complexity functions of Turing machines can be realized, up to natural equivalence, as Dehn functions of finitely presented groups.