Golden teichmüller geodesics
These computations are related to my owngoing work with Michael Boshernitzan.
A golden torus is a torus of the form where where . The associated geodesics on the modular surface is known to enjoy two properties
-
it is the shortest geodesic
-
it is the geodesic that maximize the minimum of the (flat) systole
We proved together with Michael that in general, Teichmüller geodesics that enjoy the second property are exactly the ones that are obtained as ramified covers of a golden torus. During a talk in CIRM (Marseille) in July 2015, Curtis McMullen asked the following question: what do we know on the lengths of these geodesics? Indeed, they are unlikely to be shortest geodesics in their stratum component.
The number of preimages of a golden torus in a given translation surface can be effectively computed using character theory. Hence this gives the cumulated lengths of these golden geodesics. But what about the shortest? the longest?
Using the flatsurf package, I wrote up a small program and get this interesting data
Component of stratum | Nb golden sts | Nb geod. | min length | max length |
---|---|---|---|---|
H(2)^hyp | 3 | 1 | 3 | 3 |
H(1^2)^hyp | 10 | 4 | 2 | 3 |
H(4)^hyp | 18 | 2 | 3 | 15 |
H(2^2)^hyp | 57 | 7 | 3 | 18 |
H(6)^hyp | 143 | 15 | 1 | 21 |
H(3^2)^hyp | 450 | 34 | 3 | 36 |
H(8)^hyp | 1326 | 28 | 3 | 321 |
H(4^2)^hyp | 4262 | 88 | 2 | 681 |
H(8)^hyp | 1326 | 28 | 3 | 321 |
H(6)^even | 412 | 34 | 1 | 66 |
H(4, 2)^even | 2009 | 83 | 1 | 324 |
H(8)^even | 34821 | 521 | 1 | 888 |
H(2^3)^even | 1996 | 110 | 1 | 102 |
H(6, 2)^even | 151521 | 1771 | 1 | 1796 |
H(4^2)^even | 65568 | 1618 | 1 | 1390 |
H(4)^odd | 22 | 6 | 3 | 5 |
H(2^2)^odd | 69 | 11 | 1 | 9 |
H(6)^odd | 697 | 47 | 1 | 78 |
H(4, 2)^odd | 2459 | 95 | 1 | 183 |
H(2^3)^odd | 2296 | 108 | 1 | 312 |
H(6, 2)^odd | 160632 | 1900 | 1 | 1596 |
H(4^2)^odd | 69694 | 1781 | 1 | 768 |
H(3, 1)^c | 124 | 15 | 1 | 24 |
H(2, 1^2)^c | 360 | 22 | 8 | 33 |
H(5, 1)^c | 5866 | 230 | 1 | 340 |
H(3^2)^nonhyp | 2174 | 195 | 1 | 88 |
H(1^4)^c | 302 | 41 | 1 | 81 |
H(4, 1^2)^c | 16396 | 648 | 1 | 390 |
H(3, 2, 1)^c | 26416 | 540 | 1 | 762 |
H(7, 1)^c | 408956 | 4284 | 1 | 2354 |
H(5, 3)^c | 269047 | 3427 | 1 | 2332 |
H(3, 1^3)^c | 31758 | 719 | 1 | 711 |
H(2^2, 1^2)^c | 48308 | 1741 | 1 | 603 |
Here is the function that was used for these computations. It decomposes the set of golden surfaces in a given stratum into Teichmüller geodesics.
def orbits(cc):
d = cc.stratum().dimension()-1
O = cc.origamis(d)
res = []
while O:
orbit = []
o = O.pop()
orbit.append(o)
o = o.horizontal_twist().vertical_twist()
o.relabel(inplace=True)
while o in O:
O.remove(o)
orbit.append(o)
o = o.horizontal_twist().vertical_twist()
o.relabel(inplace=True)
res.append(orbit)
return res

Cet article est publié sous la licence Creative Commons Attribution-NonCommercial 4.0 International License.
This article is published under the Creative Commons Attribution-NonCommercial 4.0 International License.