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I'm trying to find the Bass-Serre tree corresponding to the Baumslag-Solitar group $BS(1,2)$. I could find only one reference from the Internet, which I don't understand, and although I have the corresponding graph of groups, I cannot reconstruct this tree.

Edit: from the comments below, the corresponding tree would be an infinite binary tree but I'm missing the action of the group on that tree.

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    Well, what I am certain about is that the tree *looks* like what I described (so, yes, an infinite binary tree for your case). The action...I am less sure about. Perhaps the thing to do is to work out what the action is so that the fundamental domain is what we want it to be (a single vertex and a loop edge).2012-09-11

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I believe it works like this:

Your tree $T$ is an infinite binary tree, where each vertex has one edge going in, and two edges going out.

Let your group be given by $G=\langle a,b\mid bab^{-1}=a^2\rangle$, and let it act on the right for this tree $T$, and pick a "starting vertex".

Find a copy of the real line in this tree, with the "starting vertex" at coordinate $0$. So at each vertex, $b$ sends some vertex along one of its children, left or right. $a$ acts by swapping these choices.

For example, if $b$ pushes the starting vertex along its left child, then $ab$ pushes it along its right child. $b^{-1}ab$ will send a vertex up to its parent, and down the other branch.

A nice set of pictures for this is in Meier's Groups, Graphs, and Trees. He doesn't discuss this action directly, but he shows how the Cayley graph of $G$ projects to the trivalent tree. It is pages 117--118 in my copy.

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    Yes, I think taking "left all the time" works fine. I think that the action I described is correct though, because $a^2$ should act trivially.2012-09-17