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Nikolay Bogachev

Publications and source records attributed to Nikolay Bogachev.

12 recordsLinked to original sources
arXiv2025

Random walks on cocompact Fuchsian and Kleinian groups

The question of the singularity at infinity of the hitting measure of random walks has a long history, originating from the work of Furstenberg in the 1960s. In 2011, Kaimanovich and Le Prince conjectured that the hitting measure of any finitely supported random walk on a discrete subgroup $\Gamma$ of $\mathrm{SL}_N(\mathbb R)$ is singular at infinity with respect to the Lebesgue measure. Using algebraic and geometric convergence and hyperbolic Dehn filling, we prove the singularity conjecture for certain measures on ``most'' cocompact Fuchsian and Kleinian groups.

math.DS↗
arXiv2024

Quasi-arithmetic hyperbolic Coxeter prisms

In 1974, Kaplinskaja classified all simplicial straight hyperbolic Coxeter prisms. In this paper, we determine precisely which of these prisms are properly quasi-arithmetic or arithmetic. We also present some observations regarding commensurability classes and systoles of the associated orbifolds.

math.GT↗
arXiv2024

Kleinian sphere packings, reflection groups, and arithmeticity

In this paper we study crystallographic sphere packings and Kleinian sphere packings, introduced first by Kontorovich and Nakamura in 2017 and then studied further by Kapovich and Kontorovich in 2021. In particular, we solve the problem of existence of crystallographic sphere packings in certain higher dimensions posed by Kontorovich and Nakamura. In addition, we present a geometric doubling procedure allowing to obtain sphere packings from some Coxeter polyhedra without isolated roots, and study "properly integral" packings (that is, ones which are integral but not superintegral). Our techniques rely extensively on computations with Lorentzian quadratic forms, their orthogonal groups, and associated higher-dimensional hyperbolic polyhedra.

math.GT↗
arXiv2024

Arithmetic trialitarian hyperbolic lattices are not LERF

A group is LERF (locally extended residually finite) if all its finitely generated subgroups are separable. We prove that the trialitarian arithmetic lattices in $\mathbf{PSO}_{7,1}(\mathbb{R})$ are not LERF. This result, together with previous work by the third author, implies that all arithmetic lattices in $\mathbf{PO}_{n,1}(\mathbb{R})$, $n>3$, are not LERF.

math.GR↗
arXiv2024

Thin hyperbolic reflection groups

We study a family of Zariski dense finitely generated discrete subgroups of $\mathrm{Isom}(\mathbb{H}^d)$, $d \geqslant 2$, defined by the following property: any group in this family contains at least one reflection in a hyperplane. As an application we obtain a general description of all thin hyperbolic reflection groups. In particular, we show that the Vinberg algorithm applied to a non-reflective Lorentzian lattice gives rise to an infinite sequence of thin reflection subgroups in $\mathrm{Isom}(\mathbb{H}^d)$, for any $d \geqslant 2$. Moreover, every such group is a subgroup of a group produced by the Vinberg algorithm applied to a Lorentzian lattice independently on the latter being reflective. As a consequence, all thin hyperbolic reflection groups are enumerable.

math.GR↗
arXiv2023

On quasi-arithmeticity of hyperbolic gluings

We study a more general version of the gluings of hyperbolic orbifolds in the spirit of Gromov and Piatetski-Shapiro, where the gluing pieces, called the building blocks, are no longer assumed to be arithmetic or incommensurable. We prove that if such a general hyperbolic gluing along a common finite-volume totally geodesic hypersurface is quasi-arithmetic (this is a broader notion than that of arithmeticity) then each building block must be quasi-arithmetic as well and, moreover, with the same ambient group and adjoint trace field. We also show that there exist arithmetic gluings whose building blocks are incommensurable even despite the reflection with respect to the lift of the gluing locus commensurates the fundamental group of the gluing. On the other hand, we provide an example of nonarithmetic but quasi-arithmetic orbifolds such that a specific gluing of such an orbifold with itself along the boundary gives rise to an arithmetic hyperbolic orbifold. We illustrate the above results in the setting of reflection groups and hyperbolic Coxeter polyhedra and apply them to rule out the (quasi-)arithmeticity of a family of ideal hyperbolic right-angled $3$-polyhedra, namely, certain ``twisted'' ideal right-angled antiprisms, which play an important role in low-dimensional geometry and topology.

math.GT↗
arXiv2023

Geometric and arithmetic properties of Löbell polyhedra

The Löbell polyhedra form an infinite family of compact right-angled hyperbolic polyhedra in dimension $3$. We observe, through both elementary and more conceptual means, that the ``systoles'' of the Löbell polyhedra approach $0$, so that these polyhedra give rise to particularly straightforward examples of closed hyperbolic $3$-manifolds with arbitrarily small systole, and constitute an infinite family even up to commensurability. By computing number theoretic invariants of these polyhedra, we refine the latter result, and also determine precisely which of the Löbell polyhedra are quasi-arithmetic.

math.GT↗
arXiv2022

From geometry to arithmetic of compact hyperbolic Coxeter polytopes

We establish some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and show that these constraints can be a very useful tool for the classification problem of reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.

math.GT↗
arXiv2022

On volumes of hyperbolic right-angled polyhedra

In this paper we obtain new upper bounds on volumes of right-angled polyhedra in hyperbolic space $\mathbb{H}^3$ in three different cases: for ideal polyhedra with all vertices on the ideal hyperbolic boundary, for compact polytopes with only finite (or usual) vertices, and for finite volume polyhedra with vertices of both types.

math.GT↗
arXiv2021

Subspace stabilisers in hyperbolic lattices

This paper shows that immersed totally geodesic $m$-dimensional suborbifolds of $n$-dimensional arithmetic hyperbolic orbifolds correspond to finite subgroups of the commensurator whenever $m \geqslant \frac{n-1}{2}$. We call such totally geodesic suborbifolds finite centraliser subspaces (or fc-subspaces) and use them to formulate an arithmeticity criterion for hyperbolic lattices. We show that a hyperbolic orbifold $M$ is arithmetic if and only if it has infinitely many fc-subspaces, and exhibit examples of non-arithmetic orbifolds that contain non-fc subspaces of codimension one. We provide an algebraic characterization of totally geodesically immersed suborbifolds of arithmetic hyperbolic orbifolds by analysing Vinberg's commensurability invariants. This allows us to construct examples with the property that the adjoint trace field of the geodesic suborbifold properly contains the adjoint trace field of the orbifold. The case of special interest is that of exceptional trialitarian $7$-dimensional orbifolds. We show that every such orbifold contains a totally geodesic arithmetic hyperbolic $3$-orbifold of exceptional type. Finally, we study arithmetic properties of orbifolds that descend to their totally geodesic suborbifolds, proving that all suborbifolds in a (quasi-)arithmetic orbifold are (quasi-)arithmetic.

math.GT↗
arXiv2020

On faces of quasi-arithmetic Coxeter polytopes

We prove that each lower-dimensional face of a quasi-arithmetic Coxeter polytope, which happens to be itself a Coxeter polytope, is also quasi-arithmetic. We also provide a sufficient condition for a codimension $1$ face to be actually arithmetic, as well as a few computed examples.

math.GT↗