Dehn Functions:
Filling Area
in Groups
Group type:
Free group F₂ (hyperbolic) — linear
ℤ² (Euclidean) — quadratic
Heisenberg group — cubic
Baumslag-Solitar BS(1,2) — exponential
Nilpotent (degree 4) — quartic
Word length n
10
Show log scale
δ(n) = —
Dehn function δ(n):
Maximum area of a van Kampen diagram for words of length ≤n that represent the identity.
Equivalently: minimum number of relator applications needed to reduce a word of length n to the identity.
Gromov (1987):
A group is hyperbolic iff δ(n) is at most linear.
Birget-Ol'shanskii-Rips-Sapir (2002):
Dehn function characterizes complexity of the word problem.