Isometry groups of skewed $\Gamma$-complexes

TL;DR

This paper investigates the isometry groups of skewed $ ext{Gamma}$-complexes, showing that isometries homotopic to the identity are in the identity component and that the isometry group has finitely many components related to $ ext{Out}(A_ extGamma)$.

Contribution

It establishes the structure of the isometry group of skewed $ extGamma$-complexes, linking it to the outer automorphism group of the associated right-angled Artin group.

Findings

Any isometry homotopic to the identity is in the identity component.

The group of path components of the isometry group is finite.

The path components inject into $ ext{Out}(A_ extGamma)$.

Abstract

Let $A_\Gamma$ be a right-angled Artin group. Charney, Vogtmann and the author constructed an outer space for $\text{Out}(A_\Gamma)$ generalizing both $CV_n$ for $\text{Out}(F_n)$ and the symmetric space $\text{SL}_n(\mathbb{R})/\text{SO}_n(\mathbb{R})$ for $\text{GL}_n(\mathbb{Z})$. Points in this space are equivalence classes of pairs $(X,\rho)$ where $\rho\colon X\rightarrow \mathbb{S}_\Gamma$ is a homotopy equivalence from $X$ to the Salvetti complex $\mathbb{S}_\Gamma$ and $X$ is a locally CAT(0) space called a skewed $\Gamma$-complex. In this note we show that any isometry of a skewed $\Gamma$-complex which is homotopic to the identity lies in the identity component of $\text{Isom}(X)$. As a corollary, we prove that the group of path components of $\text{Isom}(X)$ is finite and injects into $\text{Out}(A_\Gamma)$.