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The question entails that I should choose two finite groups, then construct a 'biregular' tree, and show that the action of the free product of the two finite groups on the biregular tree will have a fundamental domain that consists of a single edge and two vertices. What I have so far is the two finite groups. The first group is $A = C_2$, and the second group is $B = D_4$. I know the group presentations of these groups. I understand that the free product of A and B is a group of symmetries of the biregular tree. I am lost on how to construct the biregular tree. If anyone can offer some suggestions or help it would be greatly appreciated. Thanks in advance.

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You graph should consist of (infinitely many) vertices with $4$ incident edges and (infinitely many) vertices with $2$ incident edges, joined in such a way that to get from any vertex with valency $4$ to any other with valency $4$ you have to travel through a vertex with valency $2$. That is, your graph consists of vertices with valency $4$ connected only to vertices with valency $2$, and vertices with valency $2$ connected only to vertices with valency $4$. $C_2$ acts on the vertices with valency $2$, while $D_4$ acts on the vertices with valency $4$. –  user1729 Jan 15 '12 at 20:17
    
Thank you very much! –  Aggie Jan 16 '12 at 0:44
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