Some considerations on the nonabelian tensor square of crystallographic groups

TL;DR

This paper explores the structure and growth of the nonabelian tensor square of polycyclic and pro--p groups, focusing on Hirsch length behavior and restrictions on the Schur multiplier.

Contribution

It investigates the Hirsch length growth of the nonabelian tensor square for polycyclic and pro--p groups, extending understanding of their algebraic properties.

Findings

Hirsch length of G⊗G relates to that of G in polycyclic groups.

Analysis of nonabelian tensor products of pro--p groups of finite coclass.

Restrictions on the Schur multiplier for these groups.

Abstract

The nonabelian tensor square $G\otimes G$ of a polycyclic group $G$ is a polycyclic group and its structure arouses interest in many contexts. The same assertion is still true for wider classes of solvable groups. This motivated us to work on two levels in the present paper: on a hand, we investigate the growth of the Hirsch length of $G\otimes G$ by looking at that of $G$, on another hand, we study the nonabelian tensor product of pro--$p$--groups of finite coclass, which are a remarkable class of solvable groups without center, and then we do considerations on their Hirsch length. Among other results, restrictions on the Schur multiplier will be discussed.