Homotopy patterns in group theory

TL;DR

This survey explores the homotopical and homological aspects underlying classical problems in group theory, Lie rings, and group rings, revealing deep connections with homotopy theory and derived functors.

Contribution

It highlights the role of homotopy theory in understanding the complexity of dimension subgroups and related structures in group theory.

Findings

Homotopy theory explains the complexity of dimension subgroups.

Derived functors and homotopy groups naturally appear in group-theoretical problems.

The survey illustrates the rich variety of structures through diverse examples.

Abstract

This is a survey. The main subject of this survey is the homotopical or homological nature of certain structures which appear in classical problems about groups, Lie rings and group rings. It is well known that the (generalized) dimension subgroups have complicated combinatorial theories. In this paper we show that, in certain cases, the complexity of these theories is based on homotopy theory. The derived functors of non-additive functors, homotopy groups of spheres, group homology etc appear naturally in problems formulated in purely group-theoretical terms. The variety of structures appearing in the considered context is very rich. In order to illustrate it, we present this survey as a trip passing through examples having a similar nature.