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Algebraic fiberings of right-angled Coxeter groups on bipartite cone graphs

  • Kasia Jankiewicz
  • Shayan Miri
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Abstract

We study systems of moves and states in the sense of [JNW21] on certain bipartite graphs, and the associated homomorphisms from the commutator subgroups of right-angled Coxeter groups to ℤ. We give examples of right-angled Coxeter groups whose graphs do not satisfy the legal system criterion, but contain finite index right-angled Coxeter subgroups whose graphs admit legal systems. We also provide a family of examples of graphs without legal systems, where no sequence of the considered graph operations yields a graph with a legal system. Moreover, we show that for a bipartite graph and an orbit of non-empty states under the bipartite system of moves, the associated homomorphism from the commutator subgroup of the right-angled Coxeter group to ℤ has finitely generated kernel if and only if the system is legal.

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Cite

BibTeXRIS
BibTeX
@misc{jankiewicz2026algebraic,
  title = {{Algebraic fiberings of right-angled Coxeter groups on bipartite cone graphs}},
  author = {Kasia Jankiewicz and Shayan Miri},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37641},
  eprint = {2610.37641},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37641},
}
RIS
TY  - UNPB
TI  - Algebraic fiberings of right-angled Coxeter groups on bipartite cone graphs
AU  - Kasia Jankiewicz
AU  - Shayan Miri
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37641
UR  - https://arxiv.org/abs/2610.37641
AB  - We study systems of moves and states in the sense of [JNW21] on certain bipartite graphs, and the associated homomorphisms from the commutator subgroups of right-angled Coxeter groups to ℤ. We give examples of right-angled Coxeter groups whose graphs do not satisfy the legal system criterion, but contain finite index right-angled Coxeter subgroups whose graphs admit legal systems. We also provide a family of examples of graphs without legal systems, where no sequence of the considered graph operations yields a graph with a legal system. Moreover, we show that for a bipartite graph and an orbit of non-empty states under the bipartite system of moves, the associated homomorphism from the commutator subgroup of the right-angled Coxeter group to ℤ has finitely generated kernel if and only if the system is legal.
ER  -