Finite index subgroups of fully residually free groups

TL;DR

This paper develops graph-theoretic criteria to determine finite index subgroups in fully residually free groups, providing effective methods and extending classical theorems to this class of groups.

Contribution

It introduces a new criterion for finite index subgroups in fully residually free groups and proves an analogue of Greenberg-Stallings Theorem for these groups.

Findings

Criterion for finite index subgroups established

Effective checking procedure developed

Analogue of Greenberg-Stallings Theorem proved

Abstract

Using graph-theoretic techniques for f.g. subgroups of $F^{\mathbb{Z}[t]}$ we provide a criterion for a f.g. subgroup of a f.g. fully residually free group to be of finite index. Moreover, we show that this criterion can be checked effectively. Also we obtain an analogue of Greenberg-Stallings Theorem for f.g. fully residually free groups, and prove that a f.g. non-abelian subgroup of a f.g. fully residually free group is of finite index in its commensurator.