Between free and direct products of groups

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 $G \vee H$.

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