Finiteness Conditions for the $n$-fold Tensor Product of Groups

TL;DR

The paper investigates when the $n$-fold non-abelian tensor product $G^{\otimes n}$ preserves finiteness and finite presentability, relating it to the lower central series term $\gamma_n(G)$ for finitely generated or finitely presented groups. It develops the theory of $G^{\otimes n}$ via the canonical map $\lambda^G_n: G^{\otimes n} \to \gamma_n(G)$ and studies the central extension $1 \to \ker(\lambda^G_n) \to G^{\otimes n} \to \gamma_n(G) \to 1$, drawing on homological tools such as $H_2(G)$ and the nilpotent multiplier $M_n(G)$. The main contributions are: (i) for finitely generated $G$, $G^{\otimes n}$ is finite iff $G$ is finite, and $G^{\otimes n}$ is polycyclic iff $G$ is polycyclic; (ii) for finitely presented $G$, $G^{\otimes n}$ is finitely presented iff $\gamma_n(G)$ is finitely presented; (iii) if the derivative $[G,H]$ is finitely presented, then $G \otimes H$ is finitely presented; and (iv) a Schur-Baer type framework for finitely presented groups yielding closure properties for classes under non-abelian tensor products. Together, these results connect tensor constructions with fundamental finiteness properties and central-extension phenomena in group theory, with implications for higher nilpotent invariants and extension theory.

Abstract

Let $G$ be a finitely generated group. We prove that the $n$-fold tensor product $G^{\otimes n}$ is finite (resp. polycyclic) if and only $G$ is finite (resp. polycyclic). Further, assuming that $G$ is finitely presented, we show that $G^{\otimes n}$ is finitely presented if and only if $γ_n(G)$ is finitely presented. We also examine some finiteness conditions for the non-abelian tensor product of groups.

Theorems & Definitions (11)

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• proof : Proof of Theorem \ref{['thm:fp']}
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• proof : Proof of Theorem \ref{['thm.finite.polycyclic']}
• proof : Proof of Theorem \ref{['thm:fpresult']}
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