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For a finite group presentation $⟨S\mid R⟩$ and a word $w\in F(S)$ representing the identity, the area of a null-homotopic word isThe Dehn function of the presentation isEquivalently, area is the least number of two-cells in a van Kampen diagram for $w$, and the Dehn function is the worst such area among null words of length at most $n$.

Solved by gpt-5.6-sol high.

Part d gives $\delta (2n)\le n$. For the reverse inequality, consider $w=a^{2n}$. Under the homomorphism $F(a,b)\to \mathbb{Z}$ with $a\mapsto 1$ and $b\mapsto 0$, every conjugate of $a^{\pm 2}$ has image $\pm 2$ and every conjugate of $b^{\pm 2}$ has image zero. Any expression of $a^{2n}$ as a product of conjugates of relators therefore uses at least $n$ factors. HenceThe opposite inequality follows by applying $a^2$ exactly $n$ times, so the Dehn function satisfies $\delta (2n)=n$.

Solved by gpt-5.6-sol high.