Automorphisms of the sphere complex of an infinite graph

TL;DR

This paper establishes an isomorphism between the automorphism group of the sphere complex of a doubled handlebody and the proper homotopy equivalences of an associated infinite graph, revealing deep geometric group structure.

Contribution

It proves the automorphism group of the sphere complex corresponds to the proper homotopy equivalences of the graph, extending known results to infinite graphs and constructing rigid set exhaustions.

Findings

Automorphism group of the sphere complex is isomorphic to the graph's proper homotopy group.

Constructs an exhaustion of the sphere complex by finite rigid sets.

Extends results to infinite graphs with finite rank and rays.

Abstract

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 $\operatorname{Aut}(S(M_\Gamma)) \cong \operatorname{Map}(\Gamma)$, where $\mathcal{S}(M_\Gamma)$ is the sphere complex of the doubled handlebody $M_\Gamma$ associated to $\Gamma$. We also construct an exhaustion of $S(M_\Gamma)$ by finite strongly rigid sets when $\Gamma$ has finite rank and finitely many rays, and an appropriate generalization otherwise.