# Golden teichmüller geodesics

These computations are related to my owngoing work with Michael Boshernitzan.

A golden torus is a torus of the form $\mathbb{C} / g_t \Lambda$ where $\Lambda = \mathbb{Z} (-1,1) \oplus \mathbb{Z} (\phi,-1, \phi)$ where $\phi = \frac{\sqrt{5}+1}{2}$. 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


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