We calculate the Novikov homology of right-angled Artin groups and certain HNN–extensions of these groups. This is used to obtain information on the homological Sigma invariants of Bieri–Neumann–Strebel–Renz for these groups. These invariants are subsets of all homomorphisms from a group to the reals containing information on the finiteness properties of kernels of such homomorphisms. We also derive information on the homotopical Sigma invariants and show that one cannot expect any symmetry relations between a homomorphism and its negative regarding these invariants. While it was previously known that these invariants are not symmetric in general, we give the first examples of homomorphisms which are symmetric with respect to the homological invariant, but not with respect to the homotopical invariant.
"Novikov homology of HNN–extensions and right-angled Artin groups." Algebr. Geom. Topol. 9 (2) 773 - 809, 2009. https://doi.org/10.2140/agt.2009.9.773