Automorphisms of the sphere complex of an infinite graph

Abstract

For a locally finite, connected graph $\Gamma$, let $\mathrm{Map}(\Gamma )$ denote the group of proper homotopy equivalences of $\Gamma$ up to proper homotopy. Excluding sporadic cases, we show $\mathrm{Aut}(\mathcal{S}(M_{\Gamma }))\cong \mathrm{Map}(\Gamma )$, where $\mathcal{S}(M_{\Gamma })$ is the sphere complex of the doubled handlebody $M_{\Gamma }$ associated to $\Gamma$. We also construct an exhaustion of $\mathcal{S}(M_{\Gamma })$ by finite strongly rigid sets when $\Gamma$ has finite rank and finitely many rays, and an appropriate generalization otherwise.