Centralizers and classifying spaces for commutativity of $SL_2(\mathbb{Z}[1/p])$

Omar Antolín-Camarena; Ramón Flores; Luis Jorge Sánchez Saldaña

Centralizers and classifying spaces for commutativity of $SL_2(\mathbb{Z}[1/p])$

Algebraic Topology 2026-07-30 v1 Group Theory

Authors: Omar Antolín-Camarena , Ramón Flores , Luis Jorge Sánchez Saldaña

Abstract

We compute the homotopy type of the classifying space for commutativity introduced by Adem, F. Cohen and Torres-Giese for the group $\SL_2(\Z[1/p])$ , both as a discrete topological group and with the subspace topology from $\SL_2(\R)$ . Doing so requires compiling detailed information on the maximal abelian subgroups of $\SL_2(\Z[1/p])$ , 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.

Keywords

classifying space for commutativity homotopy type maximal abelian subgroup centralizer geometric group theory

Cite

@article{arxiv.2607.28001,
  title  = {Centralizers and classifying spaces for commutativity of $SL_2(\mathbb{Z}[1/p])$},
  author = {Omar Antolín-Camarena and Ramón Flores and Luis Jorge Sánchez Saldaña},
  journal= {arXiv preprint arXiv:2607.28001},
  year   = {2026}
}

Comments

19 pages