The Trace $2\sqrt{2}$ slice

This is a study instigated by Epstein, Marden and Markovic into a slice related to bending-and-earthquake coordinates. The groups are defined by the matrices

$$a= \begin{pmatrix} \sqrt{2} \, e^{z/2} & (1+\sqrt{2})\, e^{z... ... b=\begin{pmatrix}\sqrt2-1 & 0 \\ 0 & \sqrt 2 + 1 \end{pmatrix}$$

These correspond to traces

$$\mathop{\text{Tr}}\nolimits a = 2\sqrt2 \cosh(z/2) \qquad \ma... ...mits b = 2\sqrt 2 \qquad \mathop{\text{Tr}}\nolimits abAB = -2$$

(using the upper case convention for inverses).

We use a boundary tracing method for exploring the values of $\mathop{\text{Tr}}\nolimits a$for which such a group is discrete. The p/q word in a and b is defined by

$$w_{0/1} = a \qquad w_{1/0} = B \qquad w_{(p+r)/(q+s)} = w_{p/q} w_{r/s} \text{for Farey neighbor fractions}$$

We solve in order of p/q for values of $\mathop{\text{Tr}}\nolimits a$ where w_p/q has trace 2.

The resulting plot of $i\, \mathop{\text{Tr}}\nolimits a$ is below, for cusps ranging from -3/1 just off to the left to 3/1 just off to the right.

The PostScript for this is at

epstein-slice-clr.epsi

Then we apply (in Maple) the map $z= 2 \,\text{arccosh}( \mathop{\text{Tr}}\nolimits a / (2\sqrt2) )$ to the boundary values calculated above to get the slice in earthquake coordinates. This is below:

The PostScript for this is at

epstein-maple.ps

We then computed the limit set for a point at the apparent ``maximum'' of the upper curve. This appears to be at the limit of the Fibonacci cusps F_n/F_n+1 as $n\to \infty$. The limit set is

The PostScript for this is at

epstein-lset.epsi