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Schur's theorem and its relation to the closure properties of the non-abelian tensor product

Guram Donadze, Manuel Ladra, Pilar Páez-Guillán

TL;DR

The paper demonstrates that the Schur multiplier of a Noetherian group need not be finitely generated and establishes closure properties for the non-abelian tensor product: if a polycyclic (resp. polycyclic-by-finite) group $G$ and a Noetherian group $H$ act compatibly, then $G\otimes H$ remains polycyclic (resp. polycyclic-by-finite). It further develops generalized Schur-type theorems via central extensions, showing that $H\otimes H$ controls the finiteness of $[G,G]$ in many classes, and connects these results to the (non-)finiteness of Schur multipliers, including Ol'shanskiı's example. Overall, the work links closure properties of the non-abelian tensor product to fundamental finiteness phenomena in group theory and clarifies when Noetherianity is preserved under tensor products.

Abstract

We show that the Schur multiplier of a Noetherian group need not be finitely generated. We prove that the non-abelian tensor product of a polycyclic (resp. polycyclic-by-finite) group and a Noetherian group, is a polycyclic (resp. polycyclic-by-finite) group. We also prove new versions of Schur's theorem.

Schur's theorem and its relation to the closure properties of the non-abelian tensor product

TL;DR

The paper demonstrates that the Schur multiplier of a Noetherian group need not be finitely generated and establishes closure properties for the non-abelian tensor product: if a polycyclic (resp. polycyclic-by-finite) group and a Noetherian group act compatibly, then remains polycyclic (resp. polycyclic-by-finite). It further develops generalized Schur-type theorems via central extensions, showing that controls the finiteness of in many classes, and connects these results to the (non-)finiteness of Schur multipliers, including Ol'shanskiı's example. Overall, the work links closure properties of the non-abelian tensor product to fundamental finiteness phenomena in group theory and clarifies when Noetherianity is preserved under tensor products.

Abstract

We show that the Schur multiplier of a Noetherian group need not be finitely generated. We prove that the non-abelian tensor product of a polycyclic (resp. polycyclic-by-finite) group and a Noetherian group, is a polycyclic (resp. polycyclic-by-finite) group. We also prove new versions of Schur's theorem.
Paper Structure (3 sections, 22 theorems, 22 equations)

This paper contains 3 sections, 22 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Non-abelian tensor product of Noetherian groups
  3. Connection with Schur's theorem

Key Result

Proposition 2.3

Let $G$ and $H$ be groups acting on each other compatibly. If $G$ and $H$ are finitely generated, then $G\otimes H$ is also finitely generated if and only if so are $D_H(G)$ and $D_G(H)$.

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 30 more