We define a new notion of splitting complexity for a group $G$ along a non-trivial integral character $\phi \in H^1(G; \mathbb{ Z})$. If $G$ is a one-ended coherent right-angled Artin group, we show that the splitting complexity along an epimorphism $\phi \colon G \to \mathbb{ Z}$ equals the $L^2$-Euler characteristic of the kernel of $\phi$. This allows us to define a Thurston-type semi-norm $\| \cdot \|_T \colon H^1(G ; \mathbb{ R}) \to \mathbb{ R}$ that measures the splitting complexity of integral characters. Our main tool is Friedl--L\"{ u}ck's $L^2$-polytope.
Categories: math.GR math.GT