Group presentation ($⟨S\mid R⟩$)

A group presentation $⟨S\mid R⟩$ 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.

For a word $w\in F(S)$ representing the identity in $⟨S\mid R⟩$, its area is the least $N$ for whichin $F(S)$, where $r_i\in R$ and ${\epsilon }_i\in \{-1,1\}$.

The Dehn function of a finite presentation isIt measures the worst number of relator applications needed to prove a null word of bounded length.