Combing of a group

Choose a finite symmetric generating alphabet $\Sigma$ and one word $\sigma (g)$ representing each group element, with $\sigma (1)$ the empty word. The associated paths in the Cayley graph start at the identity and traverse one edge per unit time, remaining at their endpoint after completion. A synchronous combing has a constant $K$ such that paths to adjacent endpoints stay within $K$ at equal times. An asynchronous version permits increasing reparametrizations. The formal language need not be a regular language; regular synchronous representative languages define automatic groups.

A finitely generated group is synchronously combable if it admits a combing of a group with the synchronous fellow-traveller property. Without a regular-language assumption, this is more general than an automatic group. A bounded-rung strip argument gives a finite group presentation, and finite-state row shortening gives a recursive exponential filling bound and a soluble word problem for a group.

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.

Two paths satisfy the synchronous fellow-traveller property with constant $K$ if $d(p(t),q(t))\le K$ for all equal times, with completed paths held at their endpoints. Checking integer times suffices up to increasing $K$ by at most two for unit-speed edge paths. A synchronous combing of a group requires a uniform constant for all adjacent endpoints.