Finite presentation from a synchronous combing

Attach relators for every identity word of length at most $2K+2$, where $K$ is an integer fellow-traveller constant. For any null loop, connect its consecutive combing paths at integer heights by rungs of length at most $K$. Each resulting quadrilateral has perimeter at most $2K+2$ and thus is a relator. The bottom row is collapsed at the identity and the top row is the original loop, so these finitely many relators fill every null word and present the group.

For a null loop of length $n$, a horizontal row of its combing diagram is encoded by an ordered $n$-tuple of rung words of length at most $K$. If there are $D$ such words, there are at most $D^n$ row types. Between two identical labelled rows one can delete the entire intervening annular band and glue the equal boundary paths. Repeating this leaves at most $D^n$ layers of $n$ bounded-perimeter cells. Thus a synchronous combable group has a recursive exponential isoperimetric bound. This does not require the combing language to be regular or computable.