Irreducibility of a Free Group Endomorphism is a Mapping Torus Invariant

DOI10.4171/CMH/506arXiv1910.04285MaRDI QIDQ6326904

Publication date: 9 October 2019

Abstract: We prove that the property of a free group endomorphism being irreducible is a group invariant of the ascending HNN extension it defines. This answers a question posed by Dowdall-Kapovich-Leininger. We further prove that being irreducible and atoroidal is a commensurability invariant. The invariance follows from an algebraic characterization of ascending HNN extensions that determines exactly when their defining endomorphisms are irreducible and atoroidal; specifically, we show that the endomorphism is irreducible and atoroidal if and only if the ascending HNN extension has no infinite index subgroups that are ascending HNN extensions.

Mathematics Subject Classification ID

Geometric group theory (20F65) Free nonabelian groups (20E05) Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations (20E06) Hyperbolic groups and nonpositively curved groups (20F67)

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