Preface xiii

Part 1: Surfaces and Topology. In this part, we define such

concepts as surface, Euler characteristic, fundamental group, deck

group, and covering space. We prove that the deck group of a surface

and its fundamental group are isomorphic. We also prove, under some

conditions, that a space has a universal cover.

Part 2: Surfaces and Geometry. The first 3 chapters in this

part introduce Euclidean, spherical, and hyperbolic geometry, respec-

tively. (In the Euclidean case, which is so well known, we concentrate

on nontrivial theorems.) Following this, we discuss the notion of a

Riemannian metric on a surface. In the final chapter, we discuss

hyperbolic surfaces, as special examples of Riemannian manifolds.

Part 3: Surfaces and Complex Analysis. In this part, we give a

rapid primer on the main points taught in the first semester of com-

plex analysis. Following this, we introduce the concept of a Riemann

surface and prove some results about complex analytic maps between

Riemann surfaces.

Part 4: Flat Cone Surfaces. In this part, we define what is meant

by a flat cone surface. As a special case, we consider the notion of

a translation surface. We show how the “aﬃne symmetry group” of

a translation surface, known as the Veech group, leads right back to

complex analysis and hyperbolic geometry. We end this part with an

application to polygonal billiards.

Part 5: The Totality of Surfaces. In this part, we discuss the

basic objects one considers when studying the totality of all flat or

hyperbolic surfaces, namely moduli space, Teichm¨ uller space, and the

mapping class group. As a warmup for the flat-surface case, we discuss

continued fractions and the modular group in detail.

Part 6: Dessert. In this part, we prove 3 classic results in geome-

try. The Banach–Tarski Theorem says that—assuming the Axiom of

Choice—you can cut up a ball of radius 1 into finitely many pieces

and rearrange those pieces into a (solid) ball of radius 2. Dehn’s

Theorem says that you cannot cut up a cube with planar cuts and re-

arrange it into a regular tetrahedron. The Cauchy Rigidity Theorem

says roughly that you cannot flex a convex polyhedron.