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

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.

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.