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. Quasi-arithmetic hyperbolic Coxeter prisms (with Khusrav Yorov)
    arXiv:2312.17193, pdf
  2. On quasi-arithmeticity of hyperbolic hybrids (with Dmitry Guschin and Andrei Vesnin)
    arXiv:2307.07000
  3. Thin hyperbolic reflection groups (with Alexander Kolpakov)
    arXiv:2112.14642
  4. Subspace stabilisers in hyperbolic lattices (with Misha Belolipetsky, Alexander Kolpakov & Leone Slavich)
    arXiv:2105.06897

Published/accepted papers

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