I’m back! Having a busy month moving and settling in @Stanford, but this hasn’t totally kept me away from making cool pictures :) Here’s a depiction of an Anosov mapping class on a torus 1/n 
Let’s unpack a bit what this means. One way to picture a homeomorphism from the torus to itself is a way of “putting clothes” on the torus: if the domain is some toroidal Christmas sweater and the codomain our torus, a homeomorphism tells us which part of the sweater goes where.
There are an infinite number of ways to put our torus in its Nordic knit - but this infinitude comes in two flavors. First - having already put the sweater on the torus, many new “clothings” can be built by bunching it up a little here, or stretching a little there.
These “different” ways of wearing a pattern don’t feel essentially distinct - as this sort of stretching / bunching / twisting happens to our clothes all day as we go about our daily lives - and we don’t feel that each time our shirt wrinkles we are wearing it “differently”
What would truly feel like a “different” way of wearing a sweater (for humans, say) would be to shove your head through an arm hole, and then figure out the rest accordingly.
For our torus, this would correspond to maybe unzipping our blue-and-black sweater into some parallelogram, and wrapping it around the torus in some new and crazy way before re-zipping it up. These truly distinct ways of getting dressed in the morning are the “mapping classes”.
Perhaps surprisingly at first - there are still an infinite number of ways for a donut to wear a sweater - even after declaring all smushing and stretching to be equivalent.
This is more believable in the universal cover, where our clothing problem lifts to wallpapering the plane. Identifying our torus with the quotient by the integer lattice, any allowable wallpaper must send this to itself. Furthermore, any wallpaper can be smushed to a linear map!
This identifies the set of ways to put clothes on our torus with the linear maps of the plane preserving the integer lattice: that is SL(2,Z)! (If we additionally allowed the torus to wear its clothes inside out, we would get twice as many)
Each of these is called a Mapping Class. This set also inherits the group structure of SL(2,Z) (maybe harder to see given the clothing analogy, but remember we are really talking about self-homeomorphisms of the torus, and those can be composed!). This is the Mapping Class Group.
Matrices in SL(2,Z) come in three types: finite order, diagonalizable, and having a generalized 1-eigenspace. Does this distinction mean anything for mapping classes? Yes! (brb - more after I teach)
Finite order are somewhat boring (like putting your shirt on backwards, if you take it off and do it again you’re back to a normal front-facing shirt) also, if you look for finite order integer matrices you’ll quickly realize there aren’t many anyway!
The main group of mapping classes for the torus correspond to the diagonalizable matrices (this is everything with trace greater than 2 in absolute value)- which are called Anosov (yay! We have finally returned to the picture!) we will talk about the remainder in another thread.
These diagonalizable mapping classes have two eigen-directions: one stretching and one compressing (because determinant is 1) so this corresponds to unzipping the sweater into a square, stretching into a parallelogram, twisting it back around the tours and zipping back up.
Here’s what that looks like for the mapping class in the original picture (corresponding to the matrix {{2,1},{1,1}}) (image from Wikipedia)
Now a little treat for those of you who have stuck with me so far. Why am I thinking about clothes on a donut? To make cool 3-manifolds to do raytracing in of course! After a little more explanation (sorry not sorry 🤓) I promise to show you some 3D virtual reality renders.
Recall to build a closed surface, we can start with a polygon and pair up the sides particular ways. We can do the same for 3 manifolds! Let’s start here with a cube
To get started, let’s pair up opposing vertical faces and identify them. This takes each horizontal plane and turns it into a torus - so the result looks kind of like a thickened torus
To get a closed 3 manifold we somehow need to still glue together the remaining surfaces - that is we need to specify a map from the inner torus to the outer! If we perturb this map a little bit it won’t affect the topology of the result, so we really need a Mapping Class.
As a crazy fact - the type of mapping class we choose determines the geometry that this 3 manifold can have! If we pick an Anosov mapping class, the resulting 3 manifold has Sol geometry.
Here’s a view from inside the resulting manifold, where we are rendering the edges of the fundamental domain by raytracing along geodesics: this is what we would actually see if we were placed inside the identified cube given the Sol metric
Some color coding may help here. Imagine standing in this manifold and looking down - we see the torus coming from the bottom of the cube (Orange). If we look up we see the torus coming from the top (Blue).
Of course, these blue and orange tori are actually just different sides of the same surface after gluing - and moving between the two by Sol isometries actually performs the Anosov gluing map! Much more on this will follow in future threads on Sol 😀
Ok - the end! The 3D pictures are from a collaboration @ICERM featuring myself @LaMiReMiMath, @Sabetta_ and @henryseg. If you would like to see some more amazing VIDEOS in Sol geometry, check out @ZenoRogue's work too!
