Thurston norm, polytopes and splitting complexity

Abstract

Let G be a finitely generated torsion-free group satisfying the Strong Atiyah Conjecture. Assuming that b₁⁽²⁾(G)=0, we associate to each surjective integral character ϕ the quantity b₁⁽²⁾(ker ϕ). We prove that the resulting function extends to a seminorm on H¹(G;ℝ) and is the thickness function of a polytope in H₁(G;ℝ). More generally, assuming that G is of type FP_k(ℚ) and has b_k⁽²⁾(G)=0, we prove that the function assigning b_k⁽²⁾(ker ϕ) to each surjective integral character ϕ is likewise the thickness function of a polytope. This confirms a conjecture of Friedl, Lück, and Tillmann. For torsion-free virtually (finitely generated free)-by-cyclic groups, we relate the resulting invariant to the L²-Betti complexity of a character, thereby confirming a conjecture of Gardam and Kielak. As an application, we establish the existence of an algorithm for computing the Bieri--Neumann--Strebel invariant of free-by-cyclic groups and discuss connections with the isomorphism problem for such groups.

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BibTeX RIS

BibTeX

@misc{jaikinzapirain2026thurston,
  title = {{Thurston norm, polytopes and splitting complexity}},
  author = {Andrei Jaikin-Zapirain and Monika Kudlinska and Pablo Sánchez-Peralta},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2606.31774},
  eprint = {2606.31774},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.31774},
}

RIS

TY  - UNPB
TI  - Thurston norm, polytopes and splitting complexity
AU  - Andrei Jaikin-Zapirain
AU  - Monika Kudlinska
AU  - Pablo Sánchez-Peralta
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2606.31774
UR  - https://arxiv.org/abs/2606.31774
AB  - Let G be a finitely generated torsion-free group satisfying the Strong Atiyah Conjecture. Assuming that b_1^(2)(G)=0, we associate to each surjective integral character ϕ the quantity b_1^(2)(ker ϕ). We prove that the resulting function extends to a seminorm on H^1(G;ℝ) and is the thickness function of a polytope in H_1(G;ℝ). More generally, assuming that G is of type FP_k(ℚ) and has b_k^(2)(G)=0, we prove that the function assigning b_k^(2)(ker ϕ) to each surjective integral character ϕ is likewise the thickness function of a polytope. This confirms a conjecture of Friedl, Lück, and Tillmann. For torsion-free virtually (finitely generated free)-by-cyclic groups, we relate the resulting invariant to the L^2-Betti complexity of a character, thereby confirming a conjecture of Gardam and Kielak. As an application, we establish the existence of an algorithm for computing the Bieri--Neumann--Strebel invariant of free-by-cyclic groups and discuss connections with the isomorphism problem for such groups.
ER  -

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