## sclduggery

Here’s a picture of a once-punctured genus 1 surface.

You can put it in a genus two handlebody. View the handlebody as corresponding to the free group on two generators and . Here I’m showing them as the yellow and blue cores of two handles.

Then the boundary of this once-punctured genus 1 surface may be viewed as representing an element of this free group.

Using the correspondence we can write the boundary of the surface as the product which is also denoted and known as the *commutator* of and .

Those of y’all who have learned a bit about fundamental groups know that the boundary of a (compact, orientable) once-punctured genus k surface can be expressed as the product of commutators of curves on the surface. Y’all also know that I’m being loose with basepoints, curves, and group elements.

At the Georgia Topology Fest this past May, Calegari spoke about scl, where this mix of the algebra and geometry of this can lead. He discusses it in greater detail in a recent entry of his blog. I’ll tell you a bit and then show off a fun fundamental example.

The elements of the commutator subgroup of a group can all be written as a product of commutators. That is to say if then there exists a finitely many (say, ) elements such that .

The *commutator length*, or simply *cl*, of a given is just the smallest number (the smallest ) of commutators (the ) needed to express as their product.

The *stable commutator length*, aka *scl*, of a given reflects the asymptotic behavior of *cl* as grows.

One’s first impulse is to say that *cl* is just *cl*. Take the example at the very top. Of course , but isn’t it obvious that you can’t do any better?

Here’s a few pictures even.

Go ahead. See if you can do better.

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Don’t wanna spoil the fun.

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Here’s a monkey washing a cat.

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That always gets my head a thinkin’.

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Yeah, so if by “obvious” you meant “wrong”, then you are correct.

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Thinking about Euler characteristics and irregular three-fold covers leads you to a genus 2 surface.

So *cl*. (Why is it not 1?) Hence *cl*

[...] If then the loops representing and can be thought of as the meridian and the longitude of the bounding torus. There’s some very nice pictures of this (and loads of other stuff) at the blog Sketches of Topology. [...]

The Hall-Witt identity « Geometry and the imagination said this on November 20, 2011 at 4:56 pm |