Samuel Perales

The demo above is a sample of my work in the Texas Experimental Geometry Lab. The goal of our work in this lab is to algorithmically find 'computational proofs' to certain problems in Representation Theory. In particular, for a given group presentation and its representation in $S{L}_2\left(ℝ\right)$, we want to find a set of 'valid intervals' in $ℝP^1$ such that the Ping Pong Lemma guarantees the representation is faithful.

Unfortunately, a detailed explanation will require quite a bit of background if you're unfamiliar with representation theory. If this is you, feel free to skim through this section, but it may not make a ton of sense. If you'd like to catch up, you can try reading some of my other articles on The Ping Pong Lemma, Automatic Structures, and The Project as a Whole.

Understanding the Demo

On the left you'll find the geodesic automata which represent each of three different groups which are listed below:

  Free Product of Cyclic Groups of Orders 2 and 3: $〈a,b|a^2=b^3=1⟩$

  (3,3,4) Triangle Group $〈a,b,c|a^2=b^2=c^2=1,{\left(ab\right)}^3={\left(ac\right)}^3={\left(cb\right)}^3=1⟩$

  Surface Group: $〈a,b,c,d|ad{c}^{-1}b{a}^{-1}{d}^{-1}c{b}^{-1}=1⟩$

You can hover over nodes of this graph to see the outgoing directed edges and their labels. (For the surface group, capital labels represent the inverse of a generator).

On the right you'll find the valid intervals of $ℝP^1$ that our algorithm was sucessfully able to find. The main condition of the theorem we use here requires that when we have an edge exiting from a node, the interval associated to that node must contain the image of the node which the edge enters under the $S{L}_2\left(ℝ\right)$ which corresponds to the label of the edge. A mouthful, I know. To get a feel for what is happening, hover over the nodes on the left to see which images they need to contain.

There are two ways to view the intervals on the right; the line version, and the circle version of $ℝP^1$. They both display the same information, but when there are many intervals we need to display (like for the surface group), its useful to display each of these on separate copies of $ℝP^1$ which we cut and layout in rows.

To get a better idea of how the actions of the $S{L}_2\left(ℝ\right)$ we're using on $ℝP^1$ are working, checkout my other demo SL(2,R) Action Viewer.

Have fun exploring!