Research Interests

Discrete subgroups of Lie groups, geometry of discrete groups, arithmetic and quasi-arithmetic lattices, hyperbolic geometry and topology, hyperbolic manifolds and orbifolds, hyperbolic reflection groups, Coxeter polytopes, arithmetic hyperbolic reflection groups, and reflective Lorentzian lattices.

Preprints/submitted papers

  1. On quasi-arithmeticity of hyperbolic gluings (with Dmitry Guschin and Andrei Vesnin)
    arXiv:2307.07000, 28 pp.
  2. Subspace stabilisers in hyperbolic lattices (with Misha Belolipetsky, Alexander Kolpakov & Leone Slavich)
    arXiv:2105.06897, 75 pp, under revision in a journal.

Published/accepted papers

  1. Quasi-arithmetic hyperbolic Coxeter prisms (with Khusrav Yorov)
    Proceedings of the American Mathematical Society (to appear).
  2. Geometric and arithmetic properties of Löbell polyhedra (with Sami Douba)
    Algebraic & Geometric Topology, 2025, Vol. 25, pp. 2281–2295.
  3. Thin hyperbolic reflection groups (with Alexander Kolpakov)
    Bulletin of the London Mathematical Society, 2025, Vol. 57, Issue 8, pp. 2498–2508.
  4. Infinitely many commensurability classes of compact Coxeter polyhedra in $\mathbb{H}^4$ and $\mathbb{H}^5$ (with Sami Douba & Jean Raimbault)
    Advances in Mathematics, 2024, Vol. 448, Paper No. 109705, 14 pp.
  5. Arithmetic trialitarian hyperbolic lattices are not LERF (with Leone Slavich and Hongbin Sun)
    International Mathematics Research Notices, 2024, Vol. 2024 (13), pp. 10081–10087.
  6. Kleinian sphere packings, reflection groups and arithmeticity (with Alex Kontorovich & Alexander Kolpakov)
    Mathematics of Computation, 2024, Vol. 93, no. 345, pp. 505–521.
  7. On volumes of hyperbolic right-angled polyhedra (with Stepan Alexandrov, Andrei Egorov & Andrei Vesnin)
    Sbornik: Mathematics, 2023, Vol. 214, no. 2, pp. 148–165.
  8. From geometry to arithmetic of compact hyperbolic Coxeter polytopes
    Transformation Groups, 2023, Vol. 28, no. 1, pp. 77–105.
  9. On faces of quasi-arithmetic Coxeter polytopes (with Alexander Kolpakov)
    International Mathematics Research Notices, 2021, Vol. 2021, Issue 4, pp. 3078–3096.
  10. On the classification of stably reflective hyperbolic $\mathbb{Z}[\sqrt{2}]$-lattices of rank $4$
    Doklady Mathematics, 2019, Vol. 99:3, pp. 241–244.
  11. Classification of (1,2)-reflective anisotropic hyperbolic lattices of rank 4
    Izvestiya Mathematics, 2019, Vol. 83:1, pp. 1–19.
  12. Vinberg’s Algorithm for Hyperbolic Lattices (with Alexander Perepechko)
    Mathematical Notes, 2018, Vol. 103:5, pp. 836–840.
  13. Reflective anisotropic hyperbolic lattices of rank 4
    Russian Mathematical Surveys, 2017, Vol. 1 (433), pp. 179–181

Conferences

  1. Graph-based Nearest Neighbor Search in Hyperbolic Spaces, ICLR 2022
    (with Liudmila Ostroumova-Prokhorenkova, Dima Baranchuk, Yura Demidovich, and Alexander Kolpakov)

Tools & Algorithms

  • SmaRBA: Small Ridges, Bounds and Applications, 2020
    N. Bogachev, SageMath worksheet, GitHub: SmaRBA.ipynb.
  • PLoF: Polytope’s lower-dimensional faces, 2020
    N. Bogachev and A. Kolpakov, SageMath worksheet, GitHub: PLoF
  • VinAl: Vinberg’s Algorithm, 2017-2022
    N. Bogachev and A. Perepechko: GitHub, DOI.

Unpublished notes

  • Preprint 2016: Reflective anisotropic hyperbolic lattices of rank 4, arXiv:1610.06148
  • Notes on determinantal point processes, 2015