Pascal’s Triangles in Abelian and Hyperbolic Groups
نویسنده
چکیده
We are used to imagining Pascal’s triangle as extending forever downwards from a vertex located at the top. But it is interesting to see it as occupying the first quadrant of the plane with it’s vertex at (0, 0). Imagine further that the plane is made of graph paper — that is, that we have embedded into it the Cayley graph of Z × Z with respect to the standard generating set. If we place the entries of Pascal’s triangle at the vertices of this Cayley graph, they now measure something about this graph. The entry at each point gives the number of geodesics from (0, 0) to that point. This leads us to the following definition.
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