TITLE: What is the name of this type of groups?
QUESTION [6 upvotes]: Suppose $A$ is a finite set and $\Sigma=A\cup A^{-1}$. Let $L\subseteq \Sigma^{\ast}$ be a regular language on the alphabet $\Sigma$. Is there a common name for the group $G$ presented as:
$$G=\langle A: L\rangle ?$$
Is there a reference text or article on this type of groups?

REPLY [7 votes]: By the comment of Derek Holt we can argue for the context-free case as follows:
For any $s\in \Sigma^{\ast}$, let $|s|$ denotes the length of $s$ in the monoid $\Sigma^{\ast}$. By the pumping lemma, there exists a $p\geq 1$ such that for any $s\in L$ with $|s|\geq p$ we have $s=uvxyz$,  $u, y\neq 1$ and $|vxy|\leq p$ and for all $i\geq 0$, $uv^ixy^iz\in L$. If we let $i=0$, then in $G$, we have $uxz=1$, and hence for any $i\geq 0$, 
$$ z^{-1}x^{-1}v^ixy^iz=1$$
which is equivalent to $x^{-1}vxy=1$. Hence every relation $s=1$ with $|s|\geq p$ can be replaced by a relation of the form $x^{-1}vxy=1$. Note that $|x^{-1}vxy|\leq p+1$ and hence 
$$G=\langle A: s\in L, |s|\leq p+1\rangle$$
so $G$ is f.p.