Return to Revisions

The subgroup $\Gamma < \mathrm{SL}(2, \mathbb{R})$ generated by the matrices $ a = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} $ and $ b = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix} $ is free of rank two. It acts on the upper half plane model of $\mathbb{H}^2$ via Mobius transformations. The orbit of $i$ gives the vertices of the Cayley graph; translates that differ by $a$, $b$, $a^{-1}$, or $b^{-1}$ are connected by a geodesic edge. It is an exercise to show that this gives the desired embedding. [Hint: build the Voronoi domain about $i$.]