ℤ_p-folded surfaces and the ℤ_p-Thurston norm

@misc{hobkirk2026ℤpfolded,
  title = {{$\mathbbZ_p$-folded surfaces and the $\mathbbZ_p$-Thurston norm}},
  author = {William Hobkirk},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09422},
  eprint = {2610.09422},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09422},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - ℤ_p-folded surfaces and the ℤ_p-Thurston norm
AU  - William Hobkirk
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09422
UR  - https://arxiv.org/abs/2610.09422
AB  - ℤ_p-folded surfaces are a class of oriented folded surfaces with a fundamental H_2(ℤ/pℤ) class. In this paper we study the ℤ_p-folded surfaces that embed in $3$-manifolds. This perspective naturally leads to the definition of the ℤ_p-Thurston norm, which generalises the ℤ_2-Thurston norm of Jaco, Rubinstein and Tillmann and adapts Turaev's definition of θ. We define canonical ℤ_p-folded surfaces, which connect the theory of ℤ_p-folded surfaces to the study of one-vertex $3$-manifold triangulations, and we relate their Euler characteristic to the ℤ_p-Thurston norm. We generalise a classical result of Bredon and Wood by proving that the maximum Euler characteristic of a non-separating non-empty ℤ_3-folded surface embedded in L(3n,1) is 2−n for n∈ℤ^>0, determining the ℤ_3-Thurston norm in these cases. We conclude by demonstrating the motivating application of this theory to the study of generalised triangulations by proving that any generalised triangulation of the lens space L(3n,1) must have at least n tetrahedra. This final result is a preview of joint work with Spreer in which we determine the exact complexity of L(3n,1) using more sophisticated techniques.
ER  -