Virtually abelian subgroups of $IA_n(Z/3)$ are abelian

TL;DR

The paper proves that any virtually abelian subgroup of $IA_n(Z/3)$ is actually abelian, clarifying the structure of subgroups in this context.

Contribution

It establishes that within $IA_n(Z/3)$, virtual abelianness implies actual abelianness, providing a key structural insight.

Findings

Virtually abelian subgroups of $IA_n(Z/3)$ are abelian.

The result simplifies the analysis of subgroup properties in $IA_n(Z/3)$.

Supports the use of finite index subgroups to study subgroup properties.

Abstract

When studying subgroups of $Out(F_n)$, one often replaces a given subgroup $H$ with one of its finite index subgroups $H_0$ so that virtual properties of $H$ become actual properties of $H_0$. In many cases, the finite index subgroup is $H_0 = H \cap IA_n(Z/3)$. For which properties is this a good choice? Our main theorem states that being abelian is such a property. Namely, every virtually abelian subgroup of $IA_n(Z/3)$ is abelian.