Torsion growth in finitely presented pro-$p$ groups

TL;DR

This paper investigates torsion growth in finitely presented pro-$p$ groups, showing it can be arbitrarily fast in general, but is limited to polynomial growth in $ ext{Z}_p$-analytic groups, highlighting different behaviors in these classes.

Contribution

It demonstrates the potential for unbounded torsion growth in finitely presented pro-$p$ groups and contrasts this with the polynomial bounds in $ ext{Z}_p$-analytic groups.

Findings

Torsion in abelianizations can grow arbitrarily fast in finitely presented pro-$p$ groups.

In $ ext{Z}_p$-analytic groups, torsion growth is at most polynomial.

The result highlights a fundamental difference between general pro-$p$ groups and $ ext{Z}_p$-analytic groups.

Abstract

We prove that torsion in the abelianizations of open normal subgroups in finitely presented pro-$p$ groups can grow arbitrarily fast. By way of contrast in $\mathbb Z_p$- analytic groups the torsion growth is at most polynomial.