Isometry groups of skewed $\Gamma$-complexes

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 $\text{Out}(A_\Gamma)$.