The computational aspects of property (T)

Abstract: Kazhdan's Property (T) is a well known concept in the theory of group actions. Its numerous applications include finite generation of lattices, Fixed-point properties of isometric actions, constructions of expanding graphs and product replacement algorithm. However a complicated notion requires a serious fire-power to be established. Indeed to prove that a group has property (T) requires a non-trivial effort even in the case of most classical examples, such as SL(3,Z).

We hope to ease the effort by drawing from the field of semi-definite programming and cone-optimisation. Using the Positivestellensatz and following the work of Ozawa and Netzer&Thom we will show how to translate property (T) into a semi-definite optimisation problem. Given an explicit generating set S of a finitely presented group G this will (possibly) allow us to produce a "witness" for the property (T) and simultaneously estimate the Kazhdan's constant for (G,S).