Dehn Function - Examples

Examples

• For any finite presentation of a finite group G we have Dehn(n) ≈ n.
• For the closed oriented surface of genus 2, the standard presentation of its fundamental group

satisfies Dehn(n) ≤ n and Dehn(n) ≈ n.

• For every integer k ≥ 2 the free abelian group has Dehn(n) ≈ n2.
• The Baumslag-Solitar group

has Dehn(n) ≈ 2n (see ).

• The 3-dimensional discrete Heisenberg group

satisfies a cubic but no quadratic isoperimetric inequality.

• Higher-dimensional Heisenberg groups

,

where k ≥ 2, satisfy quadratic isoperimetric inequalities.

• If G is a "Novikov-Boone group", that is, a finitely presented group with unsolvable word problem, then the Dehn function of G growths faster than any recursive function.
• For the Thompson group F the Dehn function is quadratic, that is, equivalent to n2 (see ).
• The so-called Baumslag-Gersten group

has a Dehn function growing faster than any fixed iterated tower of exponentials. Specifically, for this group

Dehn(n) ≈ exp(exp(exp(...(exp(1))...)))

where the number of exponentials is equal to the integral part of log₂(n) (see ).