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Tree Lattices and the Neumann-Zagier conjecture

  • Joshua Evan Greene
  • Ahmad Issa
  • Duncan McCoy
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Abstract

A tree with integer weights on its vertices presents an integral lattice called a tree lattice. We prove that if a tree lattice is positive definite, then one of the vertex elements is irreducible, i.e. determines a facet of its Voronoi polyhedron. As a corollary, we settle a 1985 conjecture of Neumann and Zagier: if a tree lattice is isometric to the lattice of integer points in Euclidean space, then the tree contains a vertex of weight one.

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Cite

BibTeXRIS
BibTeX
@misc{greene2026tree,
  title = {{Tree Lattices and the Neumann-Zagier conjecture}},
  author = {Joshua Evan Greene and Ahmad Issa and Duncan McCoy},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27095},
  eprint = {2610.27095},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27095},
}
RIS
TY  - UNPB
TI  - Tree Lattices and the Neumann-Zagier conjecture
AU  - Joshua Evan Greene
AU  - Ahmad Issa
AU  - Duncan McCoy
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.27095
UR  - https://arxiv.org/abs/2610.27095
AB  - A tree with integer weights on its vertices presents an integral lattice called a tree lattice. We prove that if a tree lattice is positive definite, then one of the vertex elements is irreducible, i.e. determines a facet of its Voronoi polyhedron. As a corollary, we settle a 1985 conjecture of Neumann and Zagier: if a tree lattice is isometric to the lattice of integer points in Euclidean space, then the tree contains a vertex of weight one.
ER  -