For a locally finite, connected graph $\Gamma$, let $\operatorname{Map}(\Gamma)$ denote the group of proper homotopy equivalences of $\Gamma$ up to proper homotopy.
Excluding sporadic cases, we show $\operatorname{Aut}(\mathcal{S}(M_\Gamma)) \cong \operatorname{Map}(\Gamma)$, 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. This is joint work with Michael Kopreski, Rebecca Rechkin, George Shaji, and Brian Udall.

preprint: https://arxiv.org/abs/2410.06531