I want to ask a question about universal covering of wedge space of two circles. It is known that the universal covering space is the cayley graph. I have another thing in mind which I came up with before seeing the Cayley graph as the answer to this question.
Consider the "vertical" line l (you may imagine it in $\mathbb{R}^2$ but it will be problematic). Now to each integer on l, attach a horizontal copy of $\mathbb{R}.$ Let $R_p$ be the copy of $\mathbb{R}$ attached to integer p on l. On $R_p$, to each integer k, attach another copy of $\mathbb{R}$ and call this $R_{pk}$. Note that different horizontal and vertical lines that normaly would intersect if they were in $\mathbb{R}^2$ do not intersect here. Each $\mathbb{R}$ is attached to another $\mathbb{R}$ only at one point and no other. Then the covering map is the natural covering map. That is if $a$ and $b$ are generators of loops on $S^1$ then each translate horizontally and vertically on this covering space and give the fibers as orbits of these actions.
Then this structure is also isomorphic (as a covering space) to the cayley graph that takes vertices to vertices and the segments in between to the corresponding segments right?
This also seems to generalize to arbitrary wedge product of circles but now if we have a wedge product of $n+1$ circles, you attach $n$ copies of $\mathbb{R}$ instead of $1$ copy, at each point. Right?
Thanks