Friday, 8 February 2013

Today I Learned: the Infinite Nielsen Extension

I just saw something so cool that I had to write it down.

Grab a bordered Riemann surface $$R_0$$.

Done it yet?

Goodo.

Now uniformize it with the unique (compatible) hyperbolic metric with geodesic boundaries; the universal cover of $$R_0$$ will sit as some (strict) subset of the hyperbolic plane $$\mathbb{H}$$. The fundamental group of $$R_0$$ acts discontinuously on $$\mathbb{H}$$, and the the quotient space under this action is a hyperbolic surface $$R_1\supset R_0$$. Now forget about the hyperbolic metric on $$R_1$$ and just remember its Riemann surface structure. And...

...uniformize it with a hyperbolic metric that has geodesic boundaries; ...etc.
(Just repeat what we did in the previous paragraph to get a $$R_2$$ and so forth.)

By the end of this process, we wind up with whole bunch of Riemann surfaces\begin{align}R_0\subset R_1\subset R_2\subset R_3\subset\ldots \end{align}so that when each one is uniformized to have a hyperbolic metric with geodesic boundaries, the lengths of these boundaries are strictly decreasing! In fact, take the Riemann surface obtained as the "union" of all of these and you'll have something that uniformizes to a hyperbolic surface with cusps.

And that's the infinite Nielsen extension.

It's...it's actually kinda obvious when you think about it, but I still reckon it's pretty sweet. =P

In summary:

Maan...pictures are too efficient. Gotta nerf them.