5 research outputs found
Invariant boundary distributions for finite graphs
Let be the fundamental group of a finite connected graph . Let be an abelian group. A {\it distribution} on the boundary
of the universal covering tree is an -valued measure defined on clopen sets. If has no
-torsion then the group of -invariant distributions
on is isomorphic to
K-theory of locally finite graph -algebras
We calculate the K-theory of the Cuntz-Krieger algebra
associated with an infinite, locally finite graph, via the Bass-Hashimoto
operator. The formulae we get express the Grothendieck group and the Whitehead
group in purely graph theoretic terms.
We consider the category of finite (black-and-white, bi-directed) subgraphs
with certain graph homomorphisms and construct a continuous functor to abelian
groups. In this category is an inductive limit of -groups of finite
graphs, which were calculated in \cite{MM}.
In the case of an infinite graph with the finite Betti number we obtain the
formula for the Grothendieck group where is the first Betti number and
is the valency number of the graph . We note, that in the
infinite case the torsion part of , which is present in the case of a
finite graph, vanishes. The Whitehead group depends only on the first Betti
number: . These allow us to provide a
counterexample to the fact, which holds for finite graphs, that is the torsion free part of .Comment: Final version, in press at the Journal of Geometry and Physics (2013