Automorphisms of the sphere complex of an infinite graph

2024/10/09 by Hill, Thomas, Kopreski, Michael C., Rechkin, Rebecca +2 · 2 citations
#57M07 #57S05 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2410.06531

Abstract

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

Cited by

• Dense Conjugacy Classes and Stability of Locally Finite Graphs
• On a lemma of Milnor and Schwarz, après Rosendal

Related

• The sphere complex of a locally finite graph
• Automorphisms of the 3-sphere that preserve spatial graphs and handlebody-knots
• Embeddings of mapping tori for end-periodic graph maps
• Surfaces proper homotopy equivalent to graphs and their Dehn-Nielsen-Baer maps
• On the homotopy type of complexes of graphs with bounded domination number