Failure of the finitely generated intersection property for ascending HNN extensions of free groups

TL;DR

This paper demonstrates that ascending HNN extensions of free groups do not satisfy the finitely generated intersection property, providing new insights into their algebraic structure and conditions for failure.

Contribution

It proves the failure of FGIP for a broad class of ascending HNN extensions of free groups and offers a sufficient condition for this failure in relative hyperbolic groups.

Findings

FGIP fails for ascending HNN extensions of free groups

Provides a condition for FGIP failure in relative hyperbolic groups

Applies results to free-by-cyclic groups of exponential growth

Abstract

The main result in this paper is the failure of the finitely generated intersection property (FGIP) of ascending HNN extensions of non-cyclic finite rank free groups. This class of group consists of free-by-cyclic groups and properly ascending HNN extensions of free groups. We also give a sufficient condition for the failure of the FGIP in the context of relative hyperbolicity, we apply this to free-by-cyclic groups of exponential growth.