Thurston norm for coherent right-angled Artin groups via $L^2$-invariants

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 $\phi \text{\hspace{0.17em}⁣}:G\to \mathbb{Z}$ equals the $L^2$-Euler characteristic of the kernel of $\phi$. This allows us to define a Thurston-type semi-norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_T\text{\hspace{0.17em}⁣}:H^1(G;\mathbb{R})\to \mathbb{R}$ that measures the splitting complexity of integral characters. Our main tool is Friedl–Lück’s $L^2$-polytope.