Hyperbolic groups admit finite Dehn presentations

Choose a finite generating set $S$ for $G$ and an integer $L$ as in [A1]. Let $R$ be the finite set of freely reduced words over $S^{\pm 1}$ of length at most $2L$ that represent the identity. Finiteness follows from [A2], and $⟨S\mid R⟩$ presents $G$ once the Dehn property below is proved.

Suppose a nonempty freely reduced trivial word $w$ contains no subword longer than half of a cyclic conjugate of a member of $R^{\pm 1}$. If a subword of $w$ of length at most $L$ were nongeodesic, choose one of minimal length and call it $u$, and choose a shorter geodesic word $v$ with the same endpoints. Minimality of $u$ implies that $u$ and $v$ share neither an initial nor a terminal edge: deleting such a common edge would give a shorter nongeodesic subword. Hence the loop word $u{v}^{-1}$ is freely and cyclically reduced. It belongs to $R$, has length $\mathrm{\mid }u\mathrm{\mid }+\mathrm{\mid }v\mathrm{\mid }<2\mathrm{\mid }u\mathrm{\mid }\le 2L$, and contains $u$ as more than half of a cyclic conjugate, a contradiction. Thus every length-at-most-$L$ subword of $w$ is geodesic, so the closed path labelled by $w$ is an $L$-local geodesic.

Fact [A1] forbids a nonempty closed $L$-local geodesic, contradicting step 2.1. Hence every nonempty freely reduced trivial word has the required long relator subword. This is the Dehn property, so the finite set $R$ presents $G$ and gives a finite Dehn presentation.