Linear isoperimetric characterisation of hyperbolic groups

If $G$ is hyperbolic, [F1] gives a finite presentation with the Dehn reduction property. For every nonempty null word, one reduction replaces a subword longer than half a defining relator by the complementary shorter subword, using one conjugate of that relator; the resulting freely reduced word is strictly shorter. Iterate. There are at most $\mathrm{\mid }w\mathrm{\mid }$ reductions before the empty word, so reversing them gives an expression of $w$ as at most $\mathrm{\mid }w\mathrm{\mid }$ conjugated relators. By [F3] it has a van Kampen diagram with at most $\mathrm{\mid }w\mathrm{\mid }$ cells. The empty word has an empty diagram. Thus the displayed linear inequality holds, with $C=1$ (or any larger positive constant).

Conversely suppose a finite presentation has diagrams with at most $C\mathrm{\mid }w\mathrm{\mid }$ cells for every null word. The boundary word of any finite disc diagram is a product of conjugates of its face relators: choose a spanning tree of its edges, cut along that tree, and peel cells from the exterior; each peel contributes one conjugated relator and the cut-tree traversals cancel freely. Repeated vertices and edges are treated by their separate directed occurrences. Thus the algebraic relator area of $w$ is at most $C\mathrm{\mid }w\mathrm{\mid }$. If the given bound is real, take $K=\max \{0,C\}$; the area is integral, so the same inequality holds. Since the relator set is finite, its lengths have a finite bound $L$.

By [F2] and [A1], the bound in step 1.2 makes the labelled geometric Cayley graph uniformly slim. Hence $G$ is hyperbolic. Together with step 1.1 this proves the equivalence.