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

Abstract

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

30 pages, final version, to appear in Journal of Topology