Let $G$ be a finitely presented group with a finite presentation. Given this presentation, I want to construct via an algorithm a finite, $2$-dimensional simplicial complex $X$ with fundamental group $G$ such that the algorithm can "remember" representatives of the generators of $\pi_1 (X) \cong G$. Preferably, I would like this complex also to be embedded in some $\mathbb{R}^N$ for $N \geq 7$.
I know about presentation complexes, i.e., CW complexes that are constructed via attaching a circle to a point for every generator of $G$ and a disk for every relation (via a suitable attaching map). I also know about theorems that tell me that there is a finite simplicial complex homotopy equivalent to this CW-complex and that one can embedd these complexes in high-dimensional Euclidean space (via the Menger-Noebling-Theorem). However, it is not clear to me if/how any of this should be algorithmic, so I think one needs a different approach.
Start with the presentation complex $X$: Every 2-cell of $X$ has a natural polygonal structure; $X$ has a single vertex $v$. For each 2-cell $c$ in $X$ take a point $p=p_c$ in the interior of $c$ (the barycenter of $c$) and connect it to all the vertices of $c$ and midpoints of all the edges of $c$. This yields a simplicial set $Y$ whose vertex set consists of the vertex of $X$ and barycenters of its edges and faces. One can regard this as the 1st barycentric subdivision of $X$. But $Y$ is not yet a simplicial complex. Now, take the 2nd barycentric subdivision of $Y$. This is the required simplicial complex $Z$. The generators of $\pi_1(X,v)$ correspond to obvious loops in $Z$ (the original edges of $X$ but subdivided twice). If you want to algorithmically embed $Z$ in some $\mathbb R^N$, note that every finite simplicial complex (such as $Z$) is a subcomplex in a finite simplicial complex $W$ with the same vertex set as $Z$ and whose edge set is the complete graph on the edge set of $Z$. This $W$ is the simplicial complex underlying an $N$-dimensional simplex where $N+1$ is the number of vertices in $Z$. Now, embed this simplex in $\mathbb R^N$ anyway you like. If you want to write a formal algorithm for what I described, it is pretty straightforward and I am sure you can do this yourself.