Quasiconformal Maps

Ok, time to get back to Riemann surfaces! We’ve been all about hyperbolic surfaces, and so first let’s compare the two of them: every oriented hyperbolic surface is a Riemann surface, and the conformal class of a complex structure contains a unique hyperbolic metric.

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Quadratic Differentials

We’re going to get to quasi-conformal maps soon, but first, we’re going to want to build some new objects on our Riemann surfaces.  Differential forms are a fairly standard thing, and asserting that they’re holomorphic isn’t exactly a revolution.  On Riemann surfaces, of course, there’s holomorphic 1-forms only, by dimension reasons.  So now, we’ll talk about quadratic forms, and a few of the things we can do with them.

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Mapping Class Group Elements

Let’s get into the mapping class group and talk a bit about its elements and its structure.  I’m going to omit proofs, because I can’t beat Minsky’s exposition, and this is just some flavor and definitions, most of which won’t be coming up too much in the future.

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The Mapping Class Group

Last time, we touched on the mapping class group, so now, we’re going to dig in. Now, we’re not going to dig too deeply, there’s a LOT here (see the wonderful book by Farb and Margalit for a hint at what’s there) and for now, my primary reference is Minsky’s set of lectures from the PCMI summer institute in 2011 on moduli of Riemann surfaces.

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Fenchel-Nielsen Coordinates

Welcome back, and hope all you readers had a good 2014 and particularly good holidays and new year’s celebrations, if you do those things.  Today, we’re going to keep on the road to producing the moduli space of curves, by nailing down some more hyperbolic geometry.

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Hyperbolic Surfaces

Ok, with the hyperbolic plane and its metric and geodesics out of the way, we can start getting into some surface theory.

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The hyperbolic plane

So, I know I usually talk about strictly algebraic geometry stuff, but the moduli of curves lives in an interesting place.  It’s both an algebraic and an analytic object.  So we’re going to start by talking a bit about hyperbolic surfaces, as we work towards a construction of Teichmüller space, which is used to construct the moduli of curves over \mathbb{C}.

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