A gallery of pullback maps

This is a hand-picked collection of pullback map drawings taken from the Hurwitz classes menu. They were chosen so that

  1. deg(f) ≤ 9,
  2. they are essentially different and
  3. the group of all symmetries of the pullback map acts on the upper half-plane as a reflection group.

Each of these files contains a guess at the form of Thurston’s pullback map σf. It contains two views of the upper half-plane. The top view corresponds to the domain of σf, and the bottom view corresponds to the range. In the top view, black hyperbolic geodesics are the reflection axes of reflections in the list of EMOD generators given in filenameMOD.output. The same is true for the bottom view with “generators” replaced by “generator images”. The top view is the upper half-plane analog of filenameEMODTree.ps. It is a fundamental domain for the action of the subgroup of liftables in the extended modular group. The bottom view shows a guess at the image of this fundamental domain under σf.

NETmap labels axes of liftable reflections, and σf respects these labels. For example, a geodesic in the domain labeled “A” maps to the geodesic in the image labeled “A”.

Brackets underneath the real axis indicate folds. For more on these, see Section 3.10 of the file NETmap.pdf. Reflection axes of extra symmetries (for which, see the end of Section 3.2 of NETmap.pdf) which are reflections are drawn in color, as are their images under σf, and σf respects colors.

Condition 3) above makes it very likely that the drawing is correct. It seems to always be possible to verify that a drawing is correct by (verifying and) using the information in the Hurwitz classes menu.