Homotopy of profinite groups

TL;DR

This paper develops a homotopical framework for studying simplicial profinite groups, introducing new resolutions and filtrations, and applying spectral sequences to derive results in pro-p-group theory.

Contribution

It introduces a model category structure for simplicial pro-$rak{C}$-groups and constructs free resolutions, enabling homotopical analysis in profinite combinatorial group theory.

Findings

Established a closed simplicial model category structure.

Constructed functorial free simplicial pro-$rak{L}$-resolutions.

Applied spectral sequences to pro-p-group calculations.

Abstract

We study simplicial profinite groups with a view towards applications in profinite combinatorial group theory. This approach provides a natural framework to the concept of pro-$\mathfrak{C}$-presentation of a pro-$\mathfrak{C}$-group $G$ as a 1-truncation of its free simplicial pro-$\mathfrak{C}$-resolution. The category of simplicial pro-$\mathfrak{C}$-groups has a closed simplicial model category structure. This yields a possibility to define some old and new derived functors as left Quillen derived functors from this simplicial model category. When $\mathfrak{C}$ is L-groups, than may construct free simplical pro-L-resolution functorially. We introduce settings of $\Delta$-adic and Zassenhaus filtrations for free simplical pro-p-resolutions and derive some calculations for pro-p-groups. The usage of pro-p-Curtis-Rector spectral sequences sheds homotopical light on Golod-Shafarevitch type results.