A higher-order generalization of group theory

TL;DR

This paper extends fundamental concepts of higher-order Fourier analysis and nilspace theory to non-commutative groups, introducing groupspaces and demonstrating their structural properties and connections to homotopy theory.

Contribution

It introduces groupspaces as a generalization of nilspaces for non-commutative groups and explores their structural properties and relations to higher category theory.

Findings

Generalized Gowers norms to non-commutative groups

Defined groupspaces and proved their structural properties

Showed abelian nature of higher structure groups

Abstract

The goal of this paper is to show that fundamental concepts in higher-order Fourier analysis can be nauturally extended to the non-commutative setting. We generalize Gowers norms to arbitrary compact non-commutative groups. On the structural side, we show that nilspace theory (the algebraic part of higher-order Fourier analysis) can be naturally extended to include all non-commutative groups. To this end, we introduce generalized nilspaces called "groupspaces" and demonstrate that they possess properties very similar to nilspaces. We study $k$-th order generalizations of groups that are special groupspaces called {\it k-step} groupspaces. One step groupspaces are groups. We show that $k$-step groupspaces admit the structure of an iterated principal bundle with structure groups $G_1,G_2,\dots,G_k$. A similar, but somewhat more technical statement holds for general groupspaces, with possibly infinitely many structure groups. Structure groups of groupspaces are in some sense analogous to higher homotopy groups. In particular we use a version of the Eckmann-Hilton argument from homotopy theory to show that $G_i$ is abelian for $i\geq 2$. Groupspaces also show some similarities with $n$-groups from higher category theory (also used in physics) but the exact relationship between these concepts is a subject of future research.