The canonical trace on the reduced C*-algebra of a discrete group gives rise to a homomorphism from theK-theory of this C*-algebra to the real numbers. Thi s paper studies the range of this homomorphism. For torsion-free groups, the Baum-Connes conjecture, together with Atiyah's L2-index theorem, implies that the range consists of the integers.
We give a direct and elementary proof that if G acts on a tree and admits a homomorphism α to another group H whose restriction α|Gv to every stabilizer group of a vertex is injective, then
This follows from a general relative Fredholm module technique.
Examples are in particular HNN-extensions of H where the stable letter acts by conjugation with an element of H, or amalgamated-free products G=H*UH of two copies of the same groups along a subgroup U.
Thomas Schick. "The trace on the K-theory of group C*-algebras." Duke Math. J. 107 (1) 1 - 14, 1 March 2001. https://doi.org/10.1215/S0012-7094-01-10711-4