Dehn function

In the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the "area" of a "relation" in that group (that is a freely reduced word in the generators representing the identity element of the group) in terms of the length of that relation (see pp. 79–80 in ). The growth type of the Dehn function is a quasi-isometry invariant of a finitely presented group. The Dehn function of a finitely presented group is also closely connected with non-deterministic algorithmic complexity of the word problem in groups. In particular, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation of this group is recursive (see Theorem 2.1 in ). The notion of a Dehn function is motivated by isoperimetric problems in geometry, such as the classic isoperimetric inequality for the Euclidean plane and, more generally, the notion of a filling area function that estimates the area of a minimal surface in a Riemannian manifold in terms of the length of the boundary curve of that surface.
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.
Formal definition.
Let
be a finite group presentation where the "X" is a finite alphabet and where "R" ⊆ "F"("X") is a finite set of cyclically reduced words.
Area of a relation.
Let "w" ∈ "F"("X") be a "relation" in "G", that is, a freely reduced word such that "w" = 1 in "G". Note that this is equivalent to saying that "w" belongs to the normal closure of "R" in "F"("X"), that is, there exists a representation of "w" as
where "m" ≥ 0 and where "ri" ∈ "R"± 1 for "i" = 1, ..., "m".
For "w" ∈ "F"("X") satisfying "w" = 1 in "G", the "area" of "w" with respect to (∗), denoted Area("w"), is the smallest "m" ≥ 0 such that there exists a representation (♠) for "w" as the product in "F"("X") of "m" conjugates of elements of "R"± 1.
A freely reduced word "w" ∈ "F"("X") satisfies "w" = 1 in "G" if and only if the loop labeled by "w" in the presentation complex for "G" corresponding to (∗) is null-homotopic. This fact can be used to show that Area("w") is the smallest number of 2-cells in a van Kampen diagram over (∗) with boundary cycle labelled by "w".
Isoperimetric function.
An "isoperimetric function" for a finite presentation (∗) is a monotone non-decreasing function
such that whenever "w" ∈ "F"("X") is a freely reduced word satisfying "w" = 1 in "G", then
where |"w"| is the length of the word "w".
Dehn function.
Then the "Dehn function" of a finite presentation (∗) is defined as