Can every finitely generated group be realized as the fundamental group of some finite topological space?
The motivation for my question is that the topological space on the finite point set $X=\{v_1,v_2,e_1,e_2\}$, with the topology generated by $\{e_1,e_2,\{v_1,e_1,e_2\},\{v_2,e_1,e_2\}\}$, has a fundamental group of $\mathbb Z$. The idea is that $X$ resembles a circle in a loose sense; $v_1$ and $v_2$ are like vertices, and $e_1$ and $e_2$ are disjoint paths connecting $v_1$ to $v_2$. (I think this is called the digital circle? Its universal covering space is the digital line).
Using this same idea, you can make a finite topological space which resembles any finite graph $G=(V,E)$. The points are $V\cup E$, and the topology is the finest topology for which every $e\in E$ is open and every $v\in V$ is closed. This is equivalent to saying that sets of the form $\{e\}$ for $e\in E$ and $v\cup E_v$ for $v\in V$ are open, where $E_v$ are the edges which have $v$ as an endpoint. This means every finitely generated free group is the fundamental group of a finite topological space.
In the preceding paragraph, we made a finite topological space corresponding to any $1$ dimensional CW complex. I know that there is a $2$ dimensional cell complex whose fundamental group is any desired group (there is one vertex, the $1$-cells are generators, the $2$-cells are relations). It therefore seems like the goal is to extend this construction to make arbitrary finite $2$-complexes. I could not figure out how to do this.
Okay, I figured it out. For every finite simplicial complex, there is a finite topological space with the same fundamental group. A simplicial complex defines a partial order on the simplices based on inclusion; the points of the topological space are the simplices, and the open sets are the upper sets of this partial order (so for each simplex $s$ there is an open set consisting of all simplices containing $s$ as a sub-simplex). Then you can mimic the CW complex construction using a simplicial complex to construct a simplicial complex with any given fundamental group.