Between free and direct products of groups

TL;DR

This paper explores the properties of the group formed by gluing two groups at the neutral element, revealing its similarities and differences with free and direct products, and analyzing its algebraic and geometric features.

Contribution

It provides new insights into the structure and properties of the group formed by gluing two groups at the neutral element, including Property (T), CAT(0) complexes, and amenability.

Findings

The group exhibits properties akin to free and direct products.

It satisfies certain conditions for Property (T) and CAT(0) cubical complexes.

The algebraic structure and embeddability properties are characterized.

Abstract

We investigate the group $G \vee H$ obtained by gluing together two groups $G$ and $H$ at the neutral element. This construction curiously shares some properties with the free product but others with the direct product. Our results address among others Property (T), CAT(0) cubical complexes, local embeddability, amenable actions, and the algebraic structure of $G \vee H$.