The sphere complex of a locally finite graph

2024/07/10 by Udall, Brian
#FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2407.07976

Abstract

For a locally finite graph Γ, we consider its mapping class group Map(Γ) 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_Γ whose mapping class group surjects onto Map(Γ) 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 Map(Γ) on the sphere complex S(M_Γ) of M_Γ, which is the simplicial complex whose simplices are isotopy classes of finite collections of spheres in M_Γ which are pairwise disjoint. When Γ has finite rank, we further show that the action of Map(Γ) 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 Γ, Map(Γ) is quasi-isometric to a particular subgraph of S(M_Γ), following Schaffer-Cohen. We also deduce analogs of the results of Domat-Hoganson-Kwak.

Related

• Automorphisms of the sphere complex of an infinite graph
• Automorphisms of the 3-sphere that preserve spatial graphs and handlebody-knots
• On large groups of symmetries of finite graphs embedded in spheres
• Topological symmetry groups and mapping class groups for spatial graphs
• Sphere geometry and invariants