Limit Sets of Punctured-Surface Groups.

The choice of generators is based on the theory described in

I. Kra, Horocyclic coordinates for Riemann surfaces and moduli spaces. I: Teichmüller and Riemann spaces of Kleinian group, J.A.M.S., v. 3, 1990, 499-578.

Free on two parabolic generators

a = [ 1 , 2 ; 0 , 1]
b = [ 1 , 0 ; mu, 1]

This means:

Nearly Fuchsian: mu = 1.91 +.05 i

The blue arrows label certain limit points with their associated infinite word. Keep reading to learn what the notation means. The red curves are flowlines associated to certain elements of the group, flowing from the repelling fixed point to the attractive fixed point.

Notation for labelling limit points by infinite words:

where each letter is a generator or an inverse and the word in between brackets is repeated without end. Thus, if z is the attractive fixed point of y1 .... ym then the associated limit point is

A capital letter always represents the inverse of the generator represented by the corresponding small case letter.

Near other main cusp of Riley slice: mu = 0.05 + 0.93 i

Maskit-Kra's T(1,2) (twice-punctured tori)

a = [ 1 + mu1*mu2 , 2*mu1 ; mu2/2 , 1]
b = [ 1 , 2 ; 0 , 1]
c = [ 1 , 0 ; 2 , 1]

Side-pairing

mu1 = mu2 = -.08 + 1.95 i:

Same Picture: Focus on Bottom Half

Same Picture: Invariant domain filled

Relevant artwork

It should be emphasized that this algorithm draws these sets as continuous curves, not as unions of circles. To see the curve-drawing process clearly, one must view the original PostScript file via a previewer such as ghostview. Click (or shift-click in Netscape) on the image below to load the corresponding PostScript file. It is a small piece of the above limit set, namely, all limit points corresponding to infinite words beginning with aa. Note: this is a 0.6 Mbyte compressed PostScript file (.eps.gz).

A piece of the previous limit set

Embedded Riley Slice Group

Riley Slice group inside T_1,2 group

mu1 = mu2 = 2 i:

mu1 = mu2 = 2.1 i:

Maskit-Kra's T(1,3) (thrice-punctured tori)

• mu1=mu2=mu3=2.02 i
• mu1=mu1=mu3= 0.08 + 1.93 i, mu2= -0.08 + 1.93 i
• mu1=0.08 + 1.96 i, mu2 = 2 i, mu3 = -0.08 + 2.02 i
• mu1=mu2=mu3= 2 i

Maskit-Kra's T(0,5) (five-punctured spheres)

a = [ 1 + mu1 , -mu1^2/2 ; 2 , 1 - mu1 ]
b = [ 1 , 2 ; 0 , 1]
c = [ 1 , 0 ; 2 , 1]
d = [ 1 + 2*mu2 , 2 ; -2*mu2^2 , 1 - 2*mu2 ]

mu1 = mu2 = 2.02 i:

mu1 = mu2 = .08 + 1.95 i:

Maskit-Kra's T(0,6) (six-punctured spheres)

a = [ 1 -2*mu1 -2*mu1^2*mu2 , 2 + 4*mu1*mu2 + 2*mu1^2*mu2^2 ;
-2*mu1^2, 1+2*mu1+2*mu1^2*mu2 ]
b = [ 1 + 2*mu2 , -2*mu2^2 ; 2 , 1 - 2*mu2 ]
c = [ 1 , 2 ; 0 , 1]
d = [ 1 , 0 ; 2 , 1]
e = [ 1 + 2*mu3 , 2 ; -2*mu3^2 , 1 - 2*mu3 ]

mu1 = mu3 = 2.02 i, mu2= 1.25 i:

Cusp: mu1 = mu3 = 2 i, mu2= 1.25 i:

Embedded five-times punctured sphere

Click here to see this picture

Maskit-Kra's T(2,1) (once-punctured genus two)

a = i* [ mu1*mu2*mu3 + mu1 + mu2 , mu1*mu3 + 1 ; mu2*mu3 + 1 , mu3 ]
b = [ 1 + 2*mu1 , -2*mu1^2 ; 2 , 1 - 2*mu1]
c = [ -mu1*mu4 - mu4 - 1 , mu1*mu4 + mu4 - mu1 ; -mu4 , mu4 - 1]
d = [ 1 , 2 ; 0 , 1]

mu1 = mu2 = mu3 = mu4 = 2.02 i:

mu1 = mu2 = mu3 = mu4 = .08 + 1.95 i:

Pinching to a Cusp: mu1 = mu2 = mu3 = mu4 = 2 i:

At this cusp, the surface has had two curves pinched to points, resulting in a five-times punctured sphere. We see a set of equivalent components on which the group represents this 5-punctured sphere. The stabilizer of any of these components is a quasifuchsian group. We have highlighted in color the limit set of one such subgroup below.

Click to enlarge

The next picture has the corresponding domain completely colored in.

Click to enlarge

Based on the above illustrations, you may want to try to figure out generators for this subgroup and a side-pairing relation.

Click here for answer.

It is also instructive to identify fundamental regions on the above illustrations. We have identified limit points which serve as vertices for fundamental regions. You simply have to connect the dots in the right way.