Amy Herron

Triangle Presentations in ~A_2 Bruhat-Tits Buildings

The 1-skeleton of an ~A_2 Bruhat-Tits building is isomorphic to the Cayley graph of an abstract group with relations coming from ”triangle presentations.” This abstract group either embeds into PGL(3, Fq((x))) or PGL(3, Qq), or else is exotic. Currently, the complete list of triangle presentations is only known for projective planes of orders q=2 or 3. However, one abstract group that embeds into PGL(3,Fq((x))) for any prime power q is known via the trace function corresponding to the finite field of order q^3. I found a new method to derive this group via perfect difference sets. This method demonstrates a previously unknown connection between difference sets and ~A_2 buildings. Moreover, this method makes the final computation of triangle presentations easier, which is computationally valuable for large q.