Thurston norm for coherent right‐angled Artin groups via L2‐invariants

Abstract We define a new notion of splitting complexity for a group along a non‐trivial integral character . If is a one‐ended coherent right‐angled Artin group, we show that the splitting complexity along an epimorphism equals the ‐Euler characteristic of the kernel of . This allows us to define a Thurston‐type semi‐norm which measures the splitting complexity of integral characters. Our main tool is Friedl–Lück's ‐polytope which we compute for all groups in this class. We also compute the ‐Betti numbers of all kernels of characters and show that the ‐homology of any finitely generated subgroup of is concentrated in the first dimension, which may be of independent interest.