Ping pong and beyond

Spring 2021
Undergraduate researchers: Jordan Grant, Abhay Katyal, Samuel Perales
Mentors: Max Riestenberg, Teddy Weisman
Faculty advisor: Jeff Danciger

 

Groups are abstract mathematical objects that aide in the construction of many foundational subfields of physics, chemistry, and other sciences; perhaps most importantly, they are a central object of interest throughout numerous fields of mathematics, so aquiring knowledge about the structure and behavior of different groups can be very meaningful. In this project, we study the actions of the matrix group \(SL(2, \mathbb{R})\) on \(\mathbb{R} \mathbb{P} ^{1}\). Given a finitely presented group in \(SL(2, \mathbb{R})\), we want to algorithmically determine whether it has a faithful action on \(\mathbb{R} \mathbb{P} ^{1}\). It is known that free groups have faithful actions, so if the group \(G\) of \(n\)-generators is isomorphic to a free group of rank \(n\), then \(G\) has a faithful group action. Given \(n\) generators in \(SL(2, \mathbb{R})\), we search for subsets of the unit circle that satisfy the conditions of the Ping Pong Lemma, which allows us to determine whether our group is free, and hence has a faithful group action.