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

Abstract

We define a new notion of splitting complexity for a group $G$ along a non-trivial integral character $\varphi \in H^1(G;Z)$. If $G$ is a one-ended coherent right-angled Artin group, we show that the splitting complexity along an epimorphism $\varphi :G\to Z$ equals the $L^2$-Euler characteristic of the kernel of $\varphi$. This allows us to define a Thurston-type semi-norm ${\parallel ·\parallel }_T:H^1(G;R)\to R$ which measures the splitting complexity of integral characters. Our main tool is Friedl–Lück's $L^2$-polytope which we compute for all groups in this class. We also compute the $L^2$-Betti numbers of all kernels of characters $G\to Z$ and show that the $L^2$-homology of any finitely generated subgroup of $G$ is concentrated in the first dimension, which may be of independent interest.