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18 Translation Surfaces and the Veech Group18.1 Affine Automorphisms . . . . . . . . . . . .18.2 The Diffential Representation . . . . . . . .18.3 Hyperbolic Group Actions . . . . . . . . . .18.4 Proof of Theorem 18.1 . . . . . . . . . . . .18.5 Triangle Groups . . . . . . . . . . . . . . . .18.6 Linear and Hyperbolic Reflections . . . . . .18.7 Behold, The Double Octagon! . . . . . . . .201. 201. 202. 204. 205. 207. 208. 21019 Continued Fractions19.1 The Gauss Map . . . . . . . . . . . . . . .19.2 Continued Fractions . . . . . . . . . . . .19.3 The Farey Graph . . . . . . . . . . . . . .19.4 Structure of the Modular Group . . . . . .19.5 Continued Fractions and the Farey Graph19.6 The Irrational Case . . . . . . . . . . . . .225. 225. 226. 228. 229. 231. 232.234. 234. 235. 238. 239.242. 242. 243. 244. 245. 246. 24820 Teichmüller Space and Moduli Space20.1 Parallelograms . . . . . . . . . . . . .20.2 Flat Tori . . . . . . . . . . . . . . . .20.3 The Modular Group Again . . . . . .20.4 Moduli Space . . . . . . . . . . . . .20.5 Teichmüller Space . . . . . . . . . . .20.6 The Mapping Class Group . . . . . .21 Topology of Teichmüller Space21.1 Pairs of Pants . . . . . . . . .21.2 Pants Decompositions . . . .21.3 Special Maps and Triples . . .21.4 The End of the Proof . . . . .22 The22.122.222.322.422.522.6.Banach–Tarski TheoremThe Result . . . . . . . . . . . . . .The Schroeder–Bernstein TheoremThe Doubling Theorem . . . . . . .Depleted Balls . . . . . . . . . . . .The Depleted Ball Theorem . . . .The Injective Homomorphism . . .8.215215216218219220223

23 Dehn’s Dissection Theorem23.1 The Result . . . . . . . . .23.2 Dihedral Angles . . . . . .23.3 Irrationality Proof . . . .23.4 Rational Vector Spaces . .23.5 Dehn’s Invariant . . . . .23.6 Clean Dissections . . . . .23.7 The Proof . . . . . . . . .24 The24.124.224.324.424.524.624.711.1.252. 252. 253. 254. 255. 256. 257. 258Cauchy Rigidity TheoremThe Main Result . . . . . . . . . .The Dual Graph . . . . . . . . . .Outline of the Proof . . . . . . . .Proof of Lemma 24.3 . . . . . . . .Proof of Lemma 24.2 . . . . . . . .Euclidean Intuition Does Not WorkProof of Cauchy’s Arm Lemma . .260260261261263265267268Book OverviewBehold, the Torus!The Euclidean plane, denoted R2 , is probably the simplest of all surfaces.R2 consists of all points X (x1 , x2 ) where x1 and x2 are real numbers.One may similarly define Euclidean 3-space R3 . Even though the Euclideanplane is very simple, it has the complicating feature that you cannot reallysee it all at once: it is unbounded.Perhaps the next simplest surface is the unit sphere. Anyone who hasplayed ball or blown a bubble knows what a sphere is. One way to definethe sphere mathematically is to say that it is the solution set, in R3 , to theequationx21 x22 x23 1.The sphere is bounded and one can, so to speak, comprehend it all at once.However, one complicating feature of the sphere is that it is fundamentallycurved. Also, its most basic definition involves a higher-dimensional space,namely R3 .9

The square torus is a kind of compromise between the plane and thesphere. It is a surface that is bounded like the sphere yet flat like the plane.The square torus is obtained by gluing together the opposite sides of a square,in the manner shown in Figure 1.1.Figure 1.1. The square torusWe will not yet say exactly what we mean by gluing, but we say intuitivelythat a 2-dimensional being–call it a bug–that wanders off the top of the squarewould reappear magically on the bottom, in the same horizontal position.Likewise, a bug that wanders off the right side of the square would magicallyreappear on the left side at the same vertical position. We have drawn acontinuous curve on the flat torus to indicate what we are talking about. In§3.1 we give a formal treatment of the gluing construction.At first it appears that the square torus has an edge to it, but this is anillusion. Certainly, points in the middle of the square look just look like theEuclidean plane. A myopic bug sitting near the center of the square wouldnot be able to tell he was living in the torus.Consider what the bug sees if he sits on one of the horizontal edges. Firstof all, the bug actually sits simultaneously on both horizontal edges, becausethese edges are glued together. Looking “downward”, the bug sees a littlehalf-disk. Looking “upward”, the bug sees another little half-disk. These 2half-disks are glued together and make one full Euclidean disk. So, the bugwould again think that he was sitting in the middle of the Euclidean plane.The same argument goes for any point on any of the edges.The only tricky points are the corners. What if the bug sits at oneof the corners of the squares? Note first of all that the bug actually sitssimultaneously at all 4 corners, because these corners are all glued together.10

As the bug looks in various directions, he sees 4 little quarter-disks that gluetogether to form a single disk. Even at the corner(s), the bug thinks that heis living in the Euclidean plane.Modulo a ton of details, we have shown that the square torus has noedges at all. At every point it “looks locally” like the Euclidean plane. Inparticular, it is perfectly flat at every point. At the same time, the squaretorus is bounded, like the sphere.The torus is such a great example that it demands a careful and rigoroustreatment. The first question that comes to mind is What do we mean bya surface? We will explain this in §2. Roughly speaking, a surface a spacethat “looks like” the Euclidean plane in the vicinity of each point. We do notwant to make the definition of “looks like” too restrictive. For instance, alittle patch on the sphere does not look exactly like the Euclidean plane, butwe still want the sphere to count as a surface. We will make the definitionof “looks like” flexible enough so that the sphere and lots of other examplesall count.1.2Gluing PolygonsIn §3 we give many examples of surfaces and their higher-dimensional analogues, manifolds. One of the main tools we use is the gluing construction.The square torus construction above is the starting point for a whole zoo ofrelated constructions.321132Figure 1.2. Another torusImagine, for example, that we take the hexagon shown in Figure 1.2 andglue the sides in the pattern shown. What we mean is that the 2 edges11

labelled 1 are glued together, according to the direction given by the arrows,and likewise for the edges labelled 2 and 3. We can think of Figure 1.2 as adistorted version of Figure 1.1. The hexagon has a left side, a right side, atop, and a bottom. The top is made from 2 sides and the bottom is madefrom 2 sides. The left and right sides are glued together and the top is gluedto the bottom. The resulting surface retains some of the features of the flattorus: a bug walking around on it would not detect an edge. On the otherhand, consider what happens when the bug sits at the point of the surfacecorresponding to the white dots. Spinning around, the bug would noticethat he turns less than 360 degrees before returning to his original position.What is going on is that the sum of the interior angles at the white dots isless than 360 degrees. Similarly, the bug would have to spin around by morethan 360 degrees before returning to his original position were he to sit atthe point of the surface corresponding to the black points. So, in general,the bug would not really feel like he was living in the Euclidean plane. Ourgeneral definition of surfaces and gluing will be such that the example wegave still counts as a surface.Figure 1.3 shows an example based on the regular octagon, in which theopposite sides of the octagon are glued together.12433421Figure 1.3. Gluing an octagon togetherThis example is similar to the square torus, except that this time 8 corners, rather than 4, are glued together. A myopic bug sitting anywhere onthe surface except at the point corresponding the 8 corners might think thathe was sitting in the Euclidean plane. However, at the special point, the bugwould have to turn around 720 degrees (or 6π radians) before returning to12

his original position. We will analyze this surface in great detail. One canview it as the next one in the sequence that starts out sphere, torus, . . . .At least for this introductory section, we will call it the octagon surface. (Itis commonly called the genus 2 torus.) We can construct similar examplesbased on regular 2n-gons, for each n 5, 6, 7 . . .1.3Drawing on a SurfaceOnce we have defined surfaces and given some examples, we want to workwith them to discover their properties. One natural thing we can do is dividea surface up into smaller pieces and then count them. Figure 1.4 shows 2different subdivisions of the square torus into polygons. We have left off thearrows in the diagram, but we mean for the left/right and top/bottom sidesto be glued together.Figure 1.4. Dividing the torus into facesIn the first subdivision, there are 4 faces, 8 edges, and 4 vertices. It firstappears that there are more edges, but the edges around the boundary areglued together in pairs. So each edge on the boundary only counts for halfan edge. A similar thing happens with the vertices. We make the countfaces edges vertices 4 8 4 0.In the second example, we get the countfaces edges vertices 8 14 6 0.The same result holds for practically any subdivision of the square torusinto polygons. This result is known as the Euler formula for the torus. Wediscuss this formula in more detail in §3.4.13

You can probably imagine that you would get the same result for a torusbased on a rectangle rather than a square. Likewise, we get the same resultfor the surface based on the hexagon gluing in Figure 1.2. All these surfaceshave an Euler characteristic of 0.Things turn out differently for the sphere. For instance, thinking of thesphere as a puffed-out cube, we get the countfaces edges vertices 6 12 8 2.Thinking of the sphere as a puffed-out tetrahedron, we get the countfaces edges vertices 4 6 4 2.Thinking of the sphere as a puffed-out icosahedron, we get the countfaces edges vertices 20 30 12 2.The Euler formula for the sphere says that the result of this count is always2, under very mild restrictions. You can probably see that we would get thesame result for any of the “sphere-like” surfaces mentioned above.Were we to make the count for any reasonable subdivision of the octagonsurface, we would get an Euler characteristic of 2. Can you guess the Eulercharacteristic, as a function of n, for the surface obtained by gluing togetherthe opposite sides of a regular 2n-gon?Another thing we can do on a surface is draw loops—meaning closedcurves—and study how they move around. The left side of Figure 1.5 shows3 different loops on the square torus.Figure 1.5. Loops on the torus14

One of the loops, the one represented by the thick vertical line, is differentfrom the others. Imagine that these loops are made from rubber bands,and are allowed to compress in a continuous way. The first 2 loops canshrink continuously to points, whereas the third loop is “stuck”. It can’tmake itself any shorter no matter how it moves. Such a loop is commonlycalled essential . There are many essential loops on the torus. The right sideof Figure 1.5 shows another essential loop. In contrast, the sphere has noessential loops at all.We will see in §4 that there is an algebraic object we can associate toa surface (and many other kinds of spaces) called the fundamental group.The fundamental group organizes all the different ways of drawing loops onthe surface into one basic structure. The nice thing about the fundamental group is that it links the theory of surfaces to algebra, especially grouptheory. Beautifully, it turns out that 2 (compact) surfaces have the sameEuler characteristic if and only if they have the same fundamental group.The Euler characteristic and the fundamental group are 2 entry points intothe vast subject of algebraic topology.For the most part, studying algebraic topology is beyond the scope of thisbook, but we will study the fundamental group and related constructions, ingreat detail. After defining the fundamental group in §4, we will compute anumber of examples in §5.1.4Covering SpacesThere is a nice way to unwrap the essential loops on a torus. The idea is thatwe remember that the square torus is made from a square, which we think ofas the unit square with vertices (0, 0), (0, 1), (1, 0) and (1, 1). We draw a linesegment in the plane that starts out at the same point as the loop and hasthe same length. We think of this path starting at the point (0, 0). Figure1.6 shows an example. In this example, the unwrapped path joins (0, 0) to(3, 2).The process can be reversed. Starting with a line segment that joins (0, 0)to (m, n), a point with integer coordinates, we can wrap the segment aroundthe torus so that it makes an essential loop. In fact, the essential loops thatstart at (0, 0) are, in the appropriate sense, in one-to-one correspondence withthe points of Z 2 , the integer grid in the plane. The basic result is that any 2essential loops L1 and L2 , corresponding to points (m1 , n1 ) and (m2 , n2 ), canbe continuously moved, one into the other, if and only if (m1 , n1 ) (m2 , n2 ).15

(3,2)(0,0)Figure 1.6: Unwrapping a loop on the torusAs we will explain in §6 and §7, this unwrapping construction can bedone for any surface. In the case of the torus, we see that the (equivalenceclasses of) essential simple loops are in exact correspondence with the pointsof the integer grid in the plane. One might wonder if a similarly nice pictureexists in general. The answer is “yes”, and in fact the picture becomesmore interesting when we consider surfaces, such as the octagon surface.However, in order to “see” the picture in these cases, you have to drawit in the possibly unfamiliar world of hyperbolic geometry. The idea is thathyperbolic geometry does for the octagon surface (and most other surfaces aswell) what Euclidean geometry does for the square torus and what sphericalgeometry does for the sphere.We will discuss Euclidean, spherical, and hyperbolic geometry in §8, §9,and §10 respectively. Our main goal is to understand how these geometriesinteract with surfaces, but we will also take time out to prove some classicalgeometric theorems, such as Pick’s Theorem (a relative of the Euler formula)and the angle-sum formula for hyperbolic and spherical triangles.The Euclidean, spherical, and hyperbolic geometries are the 3 most symmetrical examples of 2-dimensional Riemannian geometries. To put the 3special geometries into a general context, we will discuss Riemannian geometry in §11.1.5Hyperbolic Geometry and the OctagonNow let us return to the question of unwrapping essential loops on the octagon surface. The octagon surface looks a bit less natural than the square16

torus, thanks to the special point. However, it turns out that the octagonsurface “wears” hyperbolic geometry very much in the same way that thesquare torus “wears” Euclidean geometry.We already mentioned that we will study hyperbolic geometry in detailin §10. Here we just give the barest of sketches, in order to give you a tasteof the beauty that lies in this direction. One of the many models for thehyperbolic plane is the open unit disk. There is a way to measure distancesin the open unit disk so that the shortest paths between points are circulararcs that meet the boundary at right angles. These shortest paths are knownas geodesics. The left-hand side of Figure 1.7 shows some of the geodesicsin the hyperbolic plane. The boundary of the unit disk is not part of thehyperbolic plane and the lengths of these geodesics are all infinite. A bugliving in the hyperbolic plane would see it as unbounded in all directions.Figure 1.7. Gluing the octagon togetherThe hyperbolic plane shares many features with the Euclidean plane.There is a unique geodesic joining any 2 distinct points, and any 2 distinctgeode

MostlySurfaces Richard Evan Schwartz August 21, 2011 Abstract hich isVolume60intheA.M.S.StudentLibraryseries .