AlgTop - Relationship between boundary-word group, free group, and fundamental group? Consider a regular polygon with an even number of sides. If we label pairs of edges ($a,b,c,...$) and weight them all individually with orientations $(\rightarrow\text{ or } \leftarrow)$, we can create compact, orientable or non-orientable surfaces by means of a total quotients (identifications).
Starting at any given vertex, one may read CW or CCW around the perimeter to attain the boundary word. If we fix an orientation, the edge $a_i$ with opposite orientation is to be labeled formally $a_i^{-1}$. Some examples of boundary words are $aa$, $aa^{-1}$, $aba^{-1}b^{-1}$, $abcdeff^{-1}e^{-1}d^{-1}c^{-1}a^{-1}b^{-1}$, etc. We may create a group under string concatentation, $*$, of boundary words, that corresponds to connected sum in the geometry. There are some technicalities with reduced words etc. that I forget at the moment (I came across this in Algebraic Topology back in 2018).
My question: We may also create a free group on the letters that make up the boundary word and use the boundary word as the relation. Ex: $\langle a,b\text{ }|\text{ }aba^{-1}b^{-1} = e\rangle$. Are these two groups related? And by a long shot, is the fundamental group of a particular surface related to the free group I just mentioned?
Since the first one corresponds to connected sum, I don't expect relation to $\pi_1(X)$.
Also taking the example of the sphere says $\langle a\text{ }|\text{ }aa^{-1} = e \rangle \cong \mathbb{(Z,+,0)}$. But $\pi_1(S^2)$ is trivial, so they don't seem to be related?

 A: What is missing in your discussion is that fact that the set of equivalence classes of vertices of your fundamental polygon $P$ might consist of more than one element. The standard case you remember reading in your AT textbook 4 years ago was when all vertices are equivalent to each other. Then, indeed, $\pi_1(X)\cong G$, where the group $G$ is defined by the presentation given by the "boundary word" as in your question. What can you do if there is more than one equivalence class? You form the quotient space $X'$ of $X$ identifying all the equivalence classes of vertices of $P$ in one point. The fundamental groups of $X$ and $X'$ are related by
$$
\pi_1(X')\cong \pi_1(X) * F_n,
$$
where $n+1$ is the number of equivalence classes of vertices of $P$. There is also a natural isomorphism $\pi_1(X')\to G$, where $G$ is the 1-relator group as above. (This is an application of the Seifert- van Kampen theorem.)
In the special case, when the boundary word is $a a^{-1}$, you have two equivalence classes, $n=1$, hence $\pi_1(X')\cong F_1={\mathbb Z}\cong G$.
