What is the universal cover of a torus minus one point?
Same as this question but I did not find the answer to be understandable.
I know a universal covering must be simply connected, i.e. have trivial $\pi_1$.
I also know that the fundamental group of the torus minus a point is $\mathbb{Z} \ast \mathbb{Z} \cong F_2$.
The wedge $S^1 \vee S^1$ has the same $\pi_1$, and universal cover this thing
A comment on the post linked above mentioned thinking of a "fattened up" version of the above Cayley graph, but I'm not sure what this refers to.
When I try to imaging the graph thickened, it seems like it would become a plane, or some sort of snowflake looking thing?
The answer given in the same post was not understandable to me since I'm not familiar with holomorphisms or Riemann uniformization.
Is there an explanation for how to compute this universal cover that relies primarily upon basic Algebraic Topology (e.g. Hatcher)?