Linear Representations of Ascending Free-Group Mapping Tori

Definitions and mathematical statement

Let $n\in{\mathbb{Z}}$ with $n\ge 2$, and let $F_n$ be the free group of rank $n$. Let $\phi:F_n\to F_n$ be an injective endomorphism that is proper, understood here to mean $\phi(F_n)\subsetneq F_n$, and irreducible in the sense intended by the supplied source. The record does not define irreducibility, so this hypothesis remains unresolved; in particular, no choice between conventions concerning invariant free factors, free-factor systems, or iterates is imposed here.

Define the mapping-torus group, equivalently the ascending HNN extension associated to $\phi$, by \[ G_\phi=\left\langle F_n,t\ \middle|\ t^{-1}gt=\phi(g)\text{ for every }g\in F_n\right\rangle. \] Also define \[ \operatorname{SL}_2({\mathbb{C}})=\{A\in M_2({\mathbb{C}}):\det A=1\}. \] An embedding means an injective group homomorphism; no discreteness condition on its image is specified.

The supplied question, read universally, asks: for every $n\ge2$ and every proper injective endomorphism $\phi:F_n\to F_n$ satisfying the source's intended irreducibility hypothesis, does there exist an injective homomorphism \[ \rho:G_\phi\longrightarrow\operatorname{SL}_2({\mathbb{C}})? \] A fully self-contained formulation requires clarification of irreducibility. The interpretation of proper as nonsurjective should also be checked against the cited report.

Short English statement

Does every mapping-torus group of an irreducible, injective, nonsurjective endomorphism of a finite-rank free group of rank at least two admit a faithful representation in $\operatorname{SL}_2({\mathbb{C}})$? The source does not define irreducibility, so the exact class of endomorphisms remains to be clarified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001513$, title "Linear Representations of Ascending Free-Group Mapping Tori," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified. The record gives proposed year 2010.
• Supplied source citation: Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2011), Oberwolfach Reports, pp. 1761--1825, DOI 10.4171/owr/2010/30. https://doi.org/10.4171/owr/2010/30. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labeled "Calegari Dunfield": https://arxiv.org/abs/math/0412136.
• Supplied literature source labeled "original OWR report": https://doi.org/10.4171/owr/2010/30.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the embedding question remains unresolved for irreducible proper injective free-group endomorphisms, while Calegari and Dunfield only show that irreducibility cannot be omitted. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is a supplied verification assertion, not verification performed for this draft.