Outer space for untwisted automorphisms of right-angled Artin groups

TL;DR

This paper constructs a contractible space called mma for the untwisted outer automorphism group of right-angled Artin groups, providing a geometric model for understanding these automorphisms and their subgroups.

Contribution

It introduces a new geometric space mma for the untwisted automorphisms of right-angled Artin groups and proves its contractibility, offering a model for these groups and their subgroups.

Findings

mma is contractible.

mma admits a proper action by the untwisted automorphism group.

A proposed geometric model for all automorphisms extends mma with more general markings.

Abstract

For a right-angled Artin group $A_\Gamma$, the untwisted outer automorphism group $U(A_\Gamma)$ is the subgroup of $Out(A_\Gamma)$ generated by all of the Laurence-Servatius generators except twists (where a {\em twist} is an automorphisms of the form $v\mapsto vw$ with $vw=wv$). We define a space $\Sigma_\Gamma$ on which $U(A_\Gamma)$ acts properly and prove that $\Sigma_\Gamma$ is contractible, providing a geometric model for $U(A_\Gamma)$ and its subgroups. We also propose a geometric model for all of $Out(A_\Gamma)$ defined by allowing more general markings and metrics on points of $\Sigma_\Gamma$.