On subgroup separability of free-by-cyclic and deficiency 1 groups

TL;DR

This paper investigates the conditions under which free-by-cyclic groups with polynomial monodromy are subgroup separable, and demonstrates that most random deficiency 1 groups are not subgroup separable, highlighting key structural properties.

Contribution

It characterizes subgroup separability in free-by-cyclic groups with polynomial monodromy and shows that random deficiency 1 groups generally lack this property.

Findings

Free-by-cyclic groups with polynomial monodromy are subgroup separable iff virtually F_n x Z.

Most random deficiency 1 groups are not subgroup separable with positive probability.

Provides a criterion linking group structure to subgroup separability.

Abstract

We show that a free-by-cyclic group with a polynomially growing monodromy is subgroup separable exactly when it is virtually $F_n \times \mathbb{Z}$. We also prove that random deficiency 1 groups are not subgroup separable with positive asymptotic probability.