Manchester Geometry Seminar 2003/2004

Let X be a finite connected graph. The fundamental group Γ of X is a free group and acts on the universal covering tree Δ and on its boundary ∂ Δ, endowed with a natural topology. The action of Γ on ∂ Δ is ``bad'', in the sense that the quotient space Γ\ ∂ Δ is not Hausdorff. However, this action may be studied by means of the crossed product C^*-algebra C(∂ \Delta) צ Γ. Similar algebras may be constructed for boundary actions on affine buildings of dimension ≥ 2. The K-theory of these algebras can be computed explicitly in some cases. Moreover, the class [1] of the identity element in K₀ always has torsion. This talk will outline some of the geometry and algebra involved.