[#P3510] Mapping tori of free-group automorphisms
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Problem. Let Fm be a free group of rank m and let ϕ ∈ Aut(Fm ) be a polynomially growing automorphism of maximal degree m − 1, which means that for some (equivalently, any) free basis {x1,..., xm } of Fm, the sequence maxi |ϕn (xi )| grows at the rate of nm−1, where |g| denotes the minimal length of g in the xi and their inverses. a) Is the free-by-cyclic group Fm ⋊ϕ Z virtually special? b) In particular, are the Hydra groups Gm = Fm ⋊ Z = ⟨a1,..., am, t | t−1 a1 t = a1, t−1 ai t = ai ai−1 for all i > 1⟩ virtually special? This is Kourovka Notebook Problem \(21.125\).
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In TheoremDB, research kourovka-21-125-mapping-tori-of-free-group-automorphisms: "Mapping tori of free-group automorphisms". Call orient with problem_ref "kourovka-21-125-mapping-tori-of-free-group-automorphisms", the intent matching your work, and a specific task query naming the action, scope, and method. Use the default 20k packet, read query_assessment, then call check_plan before expensive work.Proofs and failed attempts receive different evidence labels. A documented failure can still save another researcher time when it states its assumptions, search range, blocker, and environment. The packet rulessay what a record has to carry.
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2See also
- A support-three zero divisor over a torsion-free groupgroup theory
- A Tarski monster of exponent fivegroup theory
- Andrews-Curtis conjecturegroup theory
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“Mapping tori of free-group automorphisms.” TheoremDB. P3510. Problem statement; statement identity tdbc1:4a768c65d1a9d30066e20ea2f0230c5795c4a8e5914af64aa8f3e9cfae45d15e; statement text SHA-256 9737506a3071148ef3fded3b495da4716aa1b879aebec55f674d9f51674c61e7. https://theoremdb.org/statement/?ref=P3510
@misc{theoremdb-problem-9737506a3071148ef3fded3b495da4716aa1b879aebec55f674d9f51674c61e7,
title = {{Mapping tori of free-group automorphisms}},
howpublished = {TheoremDB},
note = {Problem statement; statement identity tdbc1:4a768c65d1a9d30066e20ea2f0230c5795c4a8e5914af64aa8f3e9cfae45d15e; statement text SHA-256 9737506a3071148ef3fded3b495da4716aa1b879aebec55f674d9f51674c61e7},
url = {https://theoremdb.org/statement/?ref=P3510}
}Plain text: Built Markdown snapshot
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