Non-degeneracy results for (multi-)pushouts of compact groups

TL;DR

This paper investigates the structure and properties of pushouts and embeddings in the category of compact groups, providing new conditions for when these embeddings are algebraically sound and characterizing certain classes of compact groups.

Contribution

It introduces the concept of algebraically sound families of embeddings, characterizes coherently embeddable families via representation theory, and analyzes conditions for compact connected Lie groups.

Findings

Normal embeddings are algebraically sound.

Families of split embeddings are always algebraically sound.

Characterization of Lie groups with coherently embeddable normal embeddings.

Abstract

We prove that embeddings of compact groups are equalizers, and a number of results on pushouts (and more generally, amalgamated free products) in the category of compact groups. Call a family of compact-group embeddings $H\le G_i$ algebraically sound if the corresponding group-theoretic pushout embeds in its Bohr compactification. We (a) show that a family of normal embeddings is algebraically sound in the sense that $G_i$ admit embeddings $G_i\le G$ into a compact group which agree on $H$; (b) give equivalent characterizations of coherently embeddable families of normal embeddings in representation-theoretic terms, via Clifford theory; (c) characterize those compact connected Lie groups $H$ for which all finite families of normal embeddings $H\trianglelefteq G_i$ are coherently embeddable (not having central 2-tori is a sufficient, but not necessary condition), and (d) show that families of split embeddings of compact groups are always algebraically sound.