Amy Herron (University at Buffalo)
Title: Triangle Presentations in ~A_2 Bruhat-Tits Buildings
Abstract: The 1-skeleton of an ~A_2 Bruhat-Tits building is isomorphic to the Cayley graph of an abstract group with relations coming from 鈥漷riangle presentations.鈥� 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 have found a different method to derive this group as well as others via perfect difference sets. This new method demonstrates a previously unknown connection between difference sets and ~A_2 buildings. Moreover, this new method makes the final computation of triangle presentations easier, which is computationally valuable for large q.
As usual, we will gather for seminar teatime in BURN 1024 at 5pm after Amy's talk.