arXiv subjects

Nikolay Gusevskii

Publications and source records attributed to Nikolay Gusevskii.

6 recordsLinked to original sources

arXiv2024

Moduli of triples of points in quaternionic hyperbolic geometry

In this work, we describe the moduli of triples of points in quaternionic projective space which define uniquely the congruence classes of such triples relative to the action of the isometry group of quaternionic hyperbolic space ${\rm H}^n_{\mathbb{Q}}$. To solve this problem, we introduce some basic invariants of triples of points in quaternionic hyperbolic geometry. In particular, we define quaternionic analogues of the Goldman invariants for mixed configurations of points introduced by him in complex hyperbolic geometry.

math.DG↗

arXiv2023

On bisectors in quaternionic hyperbolic space

In this paper, we study a problem related to geometry of bisectors in quaternionic hyperbolic geometry. We develop some of the basic theory of bisectors in quaternionic hyperbolic space $H^n_Q$. In particular, we show that quaternionic bisectors enjoy various decompositions by totally geodesic submanifolds of $H^n_Q$. In contrast to complex hyperbolic geometry, where bisectors admit only two types of decomposition (described by Mostow and Goldman), we show that in the quaternionic case geometry of bisectors is more rich. The main purpose of the paper is to describe an infinite family of different decompositions of bisectors in $H^n_Q$ by totally geodesic submanifolds of $H^n_Q$ isometric to complex hyperbolic space $H^n_C$ which we call the fan decompositions. Also, we derive a formula for the orthogonal projection onto totally geodesic submanifolds in $H^n_Q$ isometric to $H^n_C$. Using this, we introduce a new class of hypersurfaces in $H^n_Q$, which we call complex hyperbolic packs in $H^n_Q$. We hope that the complex hyperbolic packs will be useful for constructing fundamental polyhedra for discrete groups of isometries of quaternionic hyperbolic space.

math.DG↗

arXiv2013

A note on trace fields of complex hyperbolic groups

We show that if $Γ$ is an irreducible subgroup of ${\rm SU}(2,1)$, then $Γ$ contains a loxodromic element $A$. If $A$ has eigenvalues $λ_1 = λe^{iφ},$ $λ_2 = e^{-2iφ}$, $λ_3 = λ^{-1}e^{iφ}$, we prove that $Γ$ is conjugate in ${\rm SU}(2,1)$ to a subgroup of ${\rm SU}(2,1,\mathbb{Q}(Γ,λ)),$ where $\mathbb{Q}(Γ, λ)$ is the field generated by the trace field $\mathbb{Q}(Γ)$ of $Γ$ and $λ$. It follows from this that if $Γ$ is an irreducible subgroup of ${\rm SU}(2,1)$ such that the trace field $\mathbb{Q}(Γ)$ is real, then $Γ$ is conjugate in ${\rm SU}(2,1)$ to a subgroup of ${\rm SO}(2,1)$. As a geometric application of the above, we get that if $G$ is an irreducible discrete subgroup of ${\rm PU}(2,1)$, then $G$ is an $\mathbb{R}$-Fuchsian subgroup of ${\rm PU}(2,1)$ if and only if the invariant trace field $k(G)$ of $G$ is real.

math.DG↗

arXiv2011

Complex Hyperbolic Structures on Disc Bundles over Surfaces

We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyperbolic disc bundles M->Σ that: admit both real and complex hyperbolic structures; satisfy the equality 2(χ+e)=3τ; satisfy the inequality χ/2 PU(2,1) with fractional Toledo invariant; where χ is the Euler characteristic of Σ, e denotes the Euler number of M, and τ stands for the Toledo invariant of M. To get a satisfactory explanation of the equality 2(χ+e)=3τ, we conjecture that there exists a holomorphic section in all our examples. In order to reduce the amount of calculations, we systematically explore coordinate-free methods.

math.GT↗

arXiv2009

On the moduli space of quadruples of points in the boundary of complex hyperbolic space

We consider the space $\mathcal M$ of ordered quadruples of distinct points in the boundary of complex hyperbolic $n$-space, $\ch{n},$ up to its holomorphic isometry group ${\rm PU}(n,1).$ One of the important problems in complex hyperbolic geometry is to construct and describe a moduli space for $\mathcal M$. For $n=2$, this problem was considered by Falbel, Parker, and Platis. The main purpose of this paper is to construct a moduli space for $\mathcal M $ for any dimension $n \geq 1$. The major innovation in our paper is the use of the Gram matrix instead of a standard position of points.

math.GT↗

arXiv2007

Complex Hyperbolic Structures on Disc Bundles over Surfaces. II. Example of a Trivial Bundle

This article is based on the methods developed in [AGG]. We construct a complex hyperbolic structure on a trivial disc bundle over a closed orientable surface $Σ$ (of genus 2) thus solving a long standing problem in complex hyperbolic geometry (see [Gol1, p. 583] and [Sch, p. 14]). This example answers also [Eli, Open Question 8.1] if a trivial circle bundle over a closed surface of genus >1 admits a holomorphically fillable contact structure. The constructed example M satisfies the relation $2(χ+e)=3τ$ which is necessary for the existence of a holomorphic section of the bundle, where $χ=χΣ$ stands for the Euler characteristic of $Σ$, e=eM, for the Euler number of the bundle, and $τ$, for the Toledo invariant. (The relation is also valid for the series of examples constructed in [AGG].) Open question: Does there exist a holomorphic section of the bundle M?

math.GT↗