Dehn Function - Basic Properties

Basic Properties

• If G and H are quasi-isometric finitely presented groups and some finite presentation of G has an isoperimetric function f(n) then for any finite presentation of H there is an isoperimentric function equivalent to f(n). In particular, this fact holds for G = H, where the same group is given by two different finite presentations.
• Consequently, for a finitely presented group the growth type of its Dehn function, in the sense of the above definition, does not depend on the choice of a finite presentation for that group. More generally, if two finitely presented groups are quasi-isometric then their Dehn functions are equivalent.
• For a finitely presented group G given by a finite presentation (∗) the following conditions are equivalent:
  • G has a recursive Dehn function with respect to (∗)
  • There exists a recursive isoperimetric function f(n) for (∗).
  • The group G has solvable word problem.

In particular, this implies that solvability of the word problem is a quasi-isometry invariant for finitely presented groups.

• Knowing the area Area(w) of a relation w allows to bound, in terms of |w|, not only the number of conjugates of the defining relations in (♠) but the lengths of the conjugating elements u_i as well. As a consequence, it is known that if a finitely presented group G given by a finite presentation (∗) has computable Dehn function Dehn(n), then the word problem for G is solvable with non-deterministic time complexity Dehn(n) and deterministic time complexity Exp(Dehn(n)). However, in general there is no reasonable bound on the Dehn function of a finitely presented group in terms of the deterministic time complexity of the word problem and the gap between the two functions can be quite large.