Double Cusp GroupDavid J. Wright

1332 NOTICES OF THE AMS VOLUME 51, NUMBER 11

People have long been fascinated with repeatedpatterns that display a rich collection of symmetries.The discovery of hyperbolic geometries in the nine-teenth century revealed a far greater wealth of pat-terns, some popularized by Dutch artist M. C.Escher in his Circle Limit series of works. The coverillustration on this issue of the Notices portrays apattern which is symmetric under a group gener-ated by two Möbius transformations a(z) and b(z)of the form αz+β

γz+δ where α, β , γ, and δ and z are allcomplex numbers. These are not distance-preserving, but they do preserve angles betweencurves and they map circles to circles.

This black lace-work (Figure 1) isthe set of ‘limitpoints’ of thegroup, meaningany point that maybe approached byan infinite se-quence of distincttransformations ap-plied to some otherpoint. Euclideanwallpaper patternshave only one limitpoint at infinity. Es-cher’s Circle Limitworks have a circleof limit points. Thelimit set here iscomposed of infi-nitely many circles

and the accumulation points of these circles.Figure 2 is a finite web of tangent disks from

the cover picture that defines how the transfor-mations a and b operate. The violet disks aremoved from one to the next by the transformationa, indicated by red arrows. The two light blue disksare invariant by the transformation b, shown by theblue arrows. Any Möbius transformation that leavesinvariant two tangent disks is called ‘parabolic’,and has a unique fixed point at the point of

tangency. The trace of the matrix corresponding tobmust be ±2, giving a polynomial condition on thegenerators a and b. On the other hand, the a trans-formation moves points in infinite double spiralsfrom its ‘repelling’ fixed point to its ‘attractive’fixed point.

The complement of the circle web consists offour white regions or ‘blobs’ labelled a, A , b andB. We stipulate that the a transformation map the

David J. Wright is professor of mathematics at Oklahoma State University. His email address iswrightd@math.okstate.edu.

Figure 1.

Figure 2.

Figure 3.

DECEMBER 2004 NOTICES OF THE AMS 1333

ring of circles around the A blob into the ring ofcircles around the a blob in reverse order, andsimilarly for the b transformation and blobs. Also,disks must be mapped to disks of the same color.

Figure 3 shows some closed loops of arrowsbetween the disks. Following the arrows corre-sponding to the composition a ◦ b−1 ◦ a−1 ◦ b ,namely, a forward blue arrow, then a backwardred arrow, then a backward blue arrow, then a for-ward red arrow, we see that this transformationfixes the large outer ‘disk’ containing ∞ (whichwe’ll denote D1) as well as the largest purple diskat the left (which we’ll denote D2). That implies thata ◦ b−1 ◦ a−1 ◦ b is also parabolic, another poly-nomial condition on a and b.

Furthermore, if we start at D2 and proceed byexactly 15 applications of a (red arrows), we arriveat the symmetrical large purple disk at the right,and then we can jump back to D2 by b−1 (back-wards blue arrow). This same composition alsofixes the disk a−1(D2) just one red arrow prior toD2, implying that b−1 ◦ a15 is parabolic. These par-abolic conditions uniquely determine the pattern(up to conjugacy).

The coloring of disks works as follows. Apply-ing b doesn’t change the color; applying a changesthe color of the disk to the next one in a cycle offifteen colors. That this is consistent depends onsome properties of this group.

Figure 4 simply shows the two equivalenceclasses of disks under the action of the group. Itextends the coloring of the disks in the original cir-cle web. Each disk is fixed by a sizeable subgroupof transformations. For example, we have seen theouter disk D1 containing ∞ is invariant under b anda ◦ b−1 ◦ a−1 ◦ b; these two ‘words’ generate thestabilizer subgroup of this disk. All the words inthis subgroup have the sum of the exponents of the

a terms equal to 0. Thus, the color of the image ofthis disk under any word in a and b is determinedjust by the sum of the exponents of the a terms.

Similarly, the disk D2 has stabilizer generatedby a ◦ b−1 ◦ a−1 ◦ b and b−1 ◦ a15. Any word inthis subgroup has the exponents of the a’s sum-ming to a multiple of 15. Hence, our coloring willbe consistent if we use 15 colors.

Choosing the same color for the D1 and D2, wesee the following rule throughout the pattern: twotangent disks belong to different classes if andonly if they have the same color.

Figure 5 is a zoom into the center of the pic-ture, where the fixed points of the a transfor-mation appear like hypnotic eyes. The title “dou-ble cusp group” refers to this group’s origin as anextreme ‘deformation’ of two-generator ‘quasi-fuchsian’ groups. Some discussion of this may befound in Chapter 9 of [2].

About similar kinds of groups and their limitsets, Klein wrote in 1894 [1]:

“The question is, what will be the configurationformed by the totality of all the circles, and in par-ticular what will be the position of the limitingpoints. There is no difficulty in answering thesequestions by purely logical reasoning; but the imag-ination seems to fail utterly when we try to forma mental image of the result.”

All the pictures were rendered using a program“kleinian” written by the author in collaborationwith David Mumford. This note was prepared whilethe author was on sabbatical at the University ofOklahoma in Norman, and the author wishes tothank the department there for indulging his ac-tivities.

References[1] FELIX KLEIN, Lectures on Mathematics, American Math.

Soc., 2000.[2] DAVID MUMFORD, CAROLINE SERIES, and DAVID WRIGHT,

Indra’s Pearls: The Vision of Felix Klein, Cambridge Uni-versity Press, 2002.

Figure 4.

Figure 5.