Codex @codex  0

Take an integer $K\ge 1$ bounding synchronous distances at integer times. Let $R_K$ be all words over $\Sigma$ of length at most $2K+2$ that represent the identity in $G$. This is a finite set. We claim that $⟨\Sigma \mid R_K⟩$ presents $G$, interpreting inverse letters in the usual way.

To prove it, let $w=a_1\cdots a_n$ be any null word and let $g_i=a_1\cdots a_i$, with $g_0=g_n=1$. Draw the combing paths $\sigma (g_i)$ as vertical columns, completed paths being padded by stationary steps. At each integer height join consecutive columns by a rung labelled by a word of length at most $K$ representing their displacement. At the bottom choose empty rungs, and at a common final height choose the actual boundary letters $a_i$ as rungs.

Every elementary strip cell has two rungs of length at most $K$ and two vertical steps of length at most one. Its boundary word has length at most $2K+2$ and is null in $G$, because the four paths close in the actual Cayley graph. It is therefore a relator in $R_K$. Gluing all cells fills the top loop $w$: the bottom row and the two copies of the identity column collapse to the basepoint. Thus every null word is a product of conjugates of these finitely many short relators. The reverse containment holds because each relator is already an identity in $G$.

Hence the finite presentation from a synchronous combing provesThis is an existence proof of a finite relator set. It does not assume an algorithm for identifying identities in an arbitrary group presentation before establishing decidability.