The sphere complex of a locally finite graph

Abstract

For a locally finite graph $\Gamma$, we consider its mapping class group $\text{Map}(\Gamma)$ as defined by Algom-Kfir-Bestvina. For these groups, we prove a generalization of the results of Laudenbach and Brendle-Broaddus-Putman, producing a $3$-manifold $M_{\Gamma}$ whose mapping class group surjects onto $\text{Map}(\Gamma)$ with kernel a compact abelian group of sphere twists so that the corresponding short exact sequence splits. Along the way we obtain an induced faithful action of $\text{Map}(\Gamma)$ on the sphere complex $\mathcal{S}(M_{\Gamma})$ of $M_{\Gamma}$, which is the simplicial complex whose simplices are isotopy classes of finite collections of spheres in $M_{\Gamma}$ which are pairwise disjoint. When $\Gamma$ has finite rank, we further show that the action of $\text{Map}(\Gamma)$ on a certain natural subcomplex has elements with positive translation length, and also consider a candidate for an Outer space of such a graph. As another application, we prove that for many $\Gamma$, $\text{Map}(\Gamma)$ is quasi-isometric to a particular subgraph of $\mathcal{S}(M_{\Gamma})$, following Schaffer-Cohen. We also deduce analogs of the results of Domat-Hoganson-Kwak.