Centralizers and classifying spaces for commutativity of SL₂(ℤ[1/p])

2026/07/30 by Omar Antol\'ın-Camarena, Ramón Flores, Luis Jorge Sánchez Saldaña
Mathematics · #math.AT #math.GR #msc:55R35

paper · pdf

19 pages

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

We compute the homotopy type of the classifying space for commutativity introduced by Adem, F. Cohen and Torres-Giese for the group \SL₂(\Z[1/p]), both as a discrete topological group and with the subspace topology from \SL₂(\R). Doing so requires compiling detailed information on the maximal abelian subgroups of \SL₂(\Z[1/p]), which turn out to be precisely the centralizers of the non-central elements of the group, for which we employ methods from geometric group theory.

Citations

• The classifying space for commutativity of geometric orientable 3-manifold groups
• Multiplication is discontinuous in the Hawaiian earring group (with the quotient topology)
• Commuting elements, simplicial spaces, and filtrations of classifying spaces
• Commuting Elements and Spaces of Homomorphisms

Related

• The commuting graph and a graph associated with centralizers
• Commuting elements, simplicial spaces, and filtrations of classifying spaces
• Counting Centralizers of a Finite Group with an Application in Constructing the Commuting Conjugacy Class Graph
• Commutators, centralizers, and strong conciseness in profinite groups
• Centralizers of Commuting Elements in Compact Lie Groups