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 $C{V}_n$ 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 $\rho :X\to {\mathbb{S}}_{\Gamma }$ 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 $\text{Out}(A_{\Gamma })$.

13 pages, no figures. Comments welcome