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Commentarii Mathematici Helvetici

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Volume 96, Issue 1, 2021, pp. 47–63
DOI: 10.4171/CMH/506

Published online: 2021-03-12

Irreducibility of a free group endomorphism is a mapping torus invariant

Jean Pierre Mutanguha[1]

(1) University of Arkansas, Fayetteville, USA

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.

Keywords: Irreducible, atoroidal, mapping torus, group invariant

Mutanguha Jean Pierre: Irreducibility of a free group endomorphism is a mapping torus invariant. Comment. Math. Helv. 96 (2021), 47-63. doi: 10.4171/CMH/506