Pascal's Triangles in Abelian and Hyperbolic Groups

Abstract

Pascal's triangle will give the number of geodesics from the identity to each point of ${\mathbf{Z}}^2$ if you write it in each of the quadrants. Given a group $G$ and generating set $\mathcal{G}$ we take the {\it Pascal's function} $p_{\mathcal{G}}:G\to {\mathbf{Z}}_{\ge 0}$ 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.

DVI file only, 7 pages

References in corpus (1)

• Automatic structures, rational growth and geometrically finite hyperbolic groups