For a finite <group presentation> $\langle S\mid R\rangle$ and a word $w\in F(S)$ representing the identity, the <area of a null-homotopic word> is
$$
\operatorname{Area}(w)=
\min\left\{N:w=\prod_{i=1}^N u_i r_i^{\varepsilon_i}u_i^{-1},\ r_i\in R,\ \varepsilon_i\in\{-1,1\}\right\}.
$$
The <Dehn function> of the presentation is
$$
\delta(n)=\max\{\operatorname{Area}(w):w=1\text{ in }G,\ |w|_S\leq n\}.
$$
Equivalently, 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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 133 / 1 / a / Solution

{title2=$\langle S\mid R\rangle$}
{wiki}

A group presentation $\langle S\mid R\rangle$ describes the quotient of the <free group> $F(S)$ by the normal closure of the relators $R$. It is finite when both $S$ and $R$ are finite.

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 Incoming links: Group presentation