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?



What is the name of this type of groups?
Finitely presented group?


In a regular expression defining $L$, any starred term must represent the identity in $G$. For example if $uv\*w$ defines a set of relators, then $v = v^2 = u^{-1}w^{-1}$. So we can remove any starred term from the regular expression and replace it by the term itself: i.e. replace $uv\*w$ by $uw \vee v$. So we can get rid of all the stars in the regular expression and end up with an expression defining a finite set.


*Added later*: M. Shahyari has now shown that a group with a presentation $\langle X: L \rangle$ with $L$ context-free must also be finitely presentable. It might be of interest to note that the group ${\mathbb Z} \wr {\mathbb Z}$, which is a standard example of a group that is not finitely presentable, has a presentation in which $X = \{ x,y \}$ and $L = \{ y^{-n}xy^nxy^{-n}x^{-1}y^nx^{-1} : n \in {\mathbb Z}\}$, which is of course not context-free.


