An example of group which is regularly presented, but not finitely presented

Question

Suppose $F_n$ is a free group with $n$ generators.

Suppose $G$ is a finitely generated group. We call $G$ finitely presented iff $\exists n \in \mathbb{N}$ and finite $A \subset F_n$ such that $G \cong \frac{F_n}{\langle \langle A \rangle \rangle}$. We call $G$ regularly presented iff $\exists n \in \mathbb{N}$ and $A \subset F_n$, which is regular as a formal language over the alphabet of generators and their inverses, such that $G \cong \frac{F_n}{\langle \langle A \rangle \rangle}$.

Does there exist a group, which is regularly presented, but not finitely presented?

If there is, I would like to see an example.

The things I managed to find:

-The cardinalities of classes of finitely presented groups and regularly presented groups are the same (they are countable)

-Any regularly presented group is recursively presented and thus isomorphic to a subgroup of a finitely presented group by Higman embedding theorem.

Answer 1

There is the following theorem of C. Frougny, J. Sakarovitch and P.E. Schupp (in: Finiteness conditions on subgroups and formal language theory, Proc. Lond. Math. Soc. 58 (1989), 74–88; here cited from this nice survey on groups, languages etc., Theorem 3.20):

Let $G$ be a finitely generated group and let $N⊂G$ be a normal subgroup of $G$. Then $N$ is finitely generated as a normal subgroup (that is, $N$ equals the normal closure of a finite set of elements of $G$) if and only if $N$ has a context-free enumeration (that is, it can be understood as the image of a context-free language in the alphabet of generators of $G$).

Now, the normal closure of every regular language $A\subset F_n$ has a context-free enumeration, therefore by the result above a regularly presented group is finitely presented.