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

TL;DR

This paper introduces a Thurston-type semi-norm for coherent right-angled Artin groups, linking splitting complexity along characters to $L^2$-invariants, and utilizes the $L^2$-polytope as a key tool.

Contribution

It defines a new splitting complexity measure for these groups and connects it to $L^2$-Euler characteristics, extending Thurston norm concepts to this algebraic setting.

Findings

Splitting complexity equals the $L^2$-Euler characteristic of the kernel.

A Thurston-type semi-norm is defined on $H^1(G; \\mathbb{R})$.

The $L^2$-polytope is used as a main analytical tool.

Abstract

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.