Let $M$ be a non-compact connected surface. One can assume that $\partial M = \varnothing$.

It follows from the results of the paper

• Epstein, D. B. A. Curves on 2-manifolds and isotopies. Acta Math. 115 1966 83–107.

that

1. $\pi_1 M$ is locally free, i.e. every finitely generated subgroup $G$ of $\pi_1 M$ is free;

2. moreover, $\pi_1 M$ is a union of an increasing countable sequence of finitely generated free subgroups.

However, I do not know how to deduce that $\pi_1 M$ is free.

That paper contains elementary proofs of basic results on curves on surfaces and homemorphisms surfaces.

Recall that a connected subsurface $N \subset M$ is incompressible if the homomorphism $\pi_1 N \to \pi_1 M$ is injective.

Lemma 2.2 of Epstein's paper. Let $X \subset M$ be a compact subset and $G$ be a finitely generated subgroup of $\pi_1 M$. Then there is a compact incompressible subsurface $N \subset M$ such that

• $X \subset int(N)$

• $G \subset \pi_1 N \subset \pi_1 M$.

Corollary 1. *$\pi_1 M$ is locally free.

Proof. Indeed, since $N$ is compact and has non-empty boundary, $N$ can be deformed onto a finite graph, and therefore $\pi_1 N$ is free and contains $G$. Hence, by Nielsen–Schreier theorem, $G$ is free as well.

Corollary 2. $\pi_1 M$ is a union of an increasing countable sequence of finitely generated free subgroups.

Proof. Represent $M$ as a countable union of compact subsets $X_1 \subset X_2 \subset \cdots $ such that $M = \cup_i X_i$. Let also $G_0 = 1$ be the unit subgroup of $\pi_1 M$, and $N_0 \subset M$ be an incompressible subsurface such that

• $X_0 \subset int(N_0)$ and $G_0 \subset \pi_1 N_0 \subset \pi_1 M$.

Denote $G_1 = \pi_1 N_0$, and let $N_1 \subset M$ be an incompressible subsurface such that

• $X_1 \subset int(N_1)$ and $G_1 \subset \pi_1 N_1 \subset \pi_1 M$,

repeating this process we will obtain that an increasing sequence of incompressible compact subsurfaces $N_0 \subset N_1 \cdots $ such that $M = \cup_i N_i$.

Since every loop in $M$ is contained in some compact subset and therefore in some $N_i$, it follows that $\pi_1 M $ is a union of its finitely generated free subgroups $\pi_1 N_0 \subset \pi_1 N_1 \subset \cdots $

Let $M$ be a non-compact connected surface. One can assume that $\partial M = \varnothing$. It follows from Lemma 2.2 this paper

• Epstein, D. B. A. Curves on 2-manifolds and isotopies. Acta Math. 115 1966 83–107.

that

1. $\pi_1 M$ is locally free, i.e. every finitely generated subgroup $G$ of $\pi_1 M$ is free;

2. $\pi_1 M$ is a union of an increasing countable sequence of finitely generated free subgroups.

However there are locally free groups that are not free.

Recall that a connected subsurface $N \subset M$ is incompressible if the homomorphism $\pi_1 N \to \pi_1 M$ is injective, so one can regard $\pi_1 N$ as a subgroup of $\pi_1 M$.

Lemma 2.2 of Epstein's paper. Let $X \subset M$ be a compact subset and $G$ be a finitely generated subgroup of $\pi_1 M$. Then there is a compact incompressible subsurface $N \subset M$ such that

• $X \subset int(N)$

• $G \subset \pi_1 N \subset \pi_1 M$.

The proof is elementary and is based on Jordan curve theorem and properties of covering spaces.

Proof that $\pi_1 M$ is locally free. Indeed, since $N$ is compact and has non-empty boundary, $N$ can be deformed onto a finite graph, and therefore $\pi_1 N$ is free and contains $G$. Hence, by Nielsen–Schreier theorem, $G$ is free as well.

Proof that $\pi_1 M$ is a union of an increasing countable sequence of finitely generated free subgroups. Represent $M$ as a countable union of compact subsets $X_1 \subset X_2 \subset \cdots $ such that $M = \cup_i X_i$. Let also $G_0 = 1$ be the unit subgroup of $\pi_1 M$, and $N_0 \subset M$ be an incompressible subsurface such that

• $X_0 \subset int(N_0)$ and $G_0 \subset \pi_1 N_0 \subset \pi_1 M$.

Denote $G_1 = \pi_1 N_0$, and let $N_1 \subset M$ be an incompressible subsurface such that

• $X_1 \subset int(N_1)$ and $G_1 \subset \pi_1 N_1 \subset \pi_1 M$,

repeating this process we will obtain that an increasing sequence of incompressible compact subsurfaces $N_0 \subset N_1 \cdots $ such that $M = \cup_i N_i$.

Since every loop in $M$ is contained in some compact subset and therefore in some $N_i$, it follows that $\pi_1 M $ is a union of its finitely generated free subgroups $\pi_1 N_0 \subset \pi_1 N_1 \subset \cdots $