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 ).
Read more about this topic: Dehn Function
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