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Pascal's Triangles in Abelian and Hyperbolic Groups

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Type
Preprint
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Submission Date
Identifiers
arXiv ID: math/9611206
Source
arXiv
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Abstract

Pascal's triangle will give the number of geodesics from the identity to each point of ${\bf Z}^2$ if you write it in each of the quadrants. Given a group $G$ and generating set $\cal G$ we take the {\it Pascal's function} $p_{\cal G}: G \to {\bf Z}_{\ge 0}$ to be the function which assigns to each $g\in G$ the number of geodesics from $1$ to $g$. We give a general method for calculating this in hyperbolic groups and discuss the generic case in abelian groups.

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