Nikolay Bogachev
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.
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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.
Several families of incommensurable noncompact hyperbolic Coxeter polytopes
研究对141个有八个面的有限体积双曲考克斯特五维多面体分类，利用特定方法在4至9维构造无限多个两两不可公度的非紧考克斯特多面体，可公度类数量至少随体积指数增长。
A convex polytope $P$ in the real projective space with reflections in the facets of $P$ is a Coxeter polytope if the reflections generate a subgroup $Γ$ of the group of projective transformations so that the $Γ$-translates of the interior of $P$ are mutually disjoint. It follows from work of Vinberg that if $P$ is a Coxeter polytope, then the interior $Ω$ of the $Γ$-orbit of $P$ is convex and $Γ$ acts properly discontinuously on $Ω$. A Coxeter polytope $P$ is $2$-perfect if $P \smallsetminus Ω$ consists of only some vertices of $P$. In this paper, we describe the deformation spaces of $2$-per…
In this paper we study $\times_0$-products of Lannér diagrams. We prove that every $\times_0$-product of at least four Lannér diagrams with at least one diagram of order $\ge 3$ is superhyperbolic. As a corollary, we obtain that known classifications exhaust all compact hyperbolic Coxeter polytopes that are combinatorially equivalent to products of simplices. We also consider compact hyperbolic Coxeter polytopes whose every Lannér subdiagram has order $2$. The second result of this paper slightly improves recent Burcroff's upper bound on the dimension of such polytopes to $12$.
In this paper, we obtain a complete classification of compact hyperbolic Coxeter five-dimensional polytopes with nine facets.
Finite-volume hyperbolic Coxeter 4-dimensional polytopes with 7 facets
本文对具有7个面的有限体积双曲Coxeter 4维多胞体进行了完整分类，共获得331种分类结果。
In this paper, we obtain the complete classification for compact hyperbolic Coxeter four-dimensional polytopes with eight facets.
We construct infinite series of non-simple ideal hyperbolic Coxeter 4-polytopes whose growth rates are Perron numbers. This infinite series is the first example of such a non-compact infinite polytopal series.
We complete the classification of compact hyperbolic Coxeter $d$-polytopes with $d+4$ facets for $d=4$ and $5$. By previous work of Felikson and Tumarkin, the only remaining dimension where new polytopes may arise is $d=6$. We derive a new method for generating the combinatorial type of these polytopes via the classification of point set order types. In dimensions $4$ and $5$, there are $348$ and $51$ polytopes, respectively, yielding many new examples for further study. We furthermore provide new upper bounds on the dimension $d$ of compact hyperbolic Coxeter polytopes with $d+k$ facets for $…
In this paper, we compute the covolume of the group of units of the quadratic form f_d^n(x) = x_1^2 + x_2^2 + . . . + x_n^2 - d x_{n+1}^2 with d an odd, positive, square-free integer. Mcleod has determined the hyperbolic Coxeter fundamental domain of the reflection subgroup of the group of units of the quadratic form f_3^n. We apply our covolume formula to compute the volumes of these hyperbolic Coxeter polytopes.
Using a variant of an unpublished argument due to Agol, we show that an irreducible right-angled Coxeter group on ${n \geq 3}$ vertices embeds as a thin subgroup of a uniform arithmetic lattice in an indefinite orthogonal group $\mathrm{O}(p, q)$ for some $p, q \geq 1$ satisfying $p+q=n$.
By gluing together the sides of eight copies of an all-right angled hyperbolic 6-dimensional polytope, two orientable hyperbolic 6-manifolds with Euler characteristic -1 are constructed. They are the first known examples of orientable hyperbolic 6-manifolds having the smallest possible volume.
We observe that a large part of the volume of a hyperbolic polyhedron is taken by a tubular neighbourhood of its boundary, and use this to give a new proof for the finiteness of arithmetic maximal reflection groups following a recent work with M. Fraczyk and S. Hurtado. We also study in more depth the case of polygons in the hyperbolic plane.
Quasi-arithmetic hyperbolic Coxeter prisms
本文精确确定了Kaplinskaja在1974年分类的单纯直双曲Coxeter棱柱中哪些是真正拟算术或算术的，并提出了关于相关配边可换类和收缩的观察。
The smoothness of the real projective deformation spaces of orderable Coxeter 3-polytopes
本文证明，对于可排序且正规类型的Coxeter三维多胞形，若其底层组合多胞形非多边形上的锥，则其形变空间为光滑流形，通过分析到实现空间的自然映射得出。
Projective reflection groups of finite covolume
该研究证明了有限体积的Coxeter多面体与Vinberg域的关系，指出其在反射群中的不变性条件。
Finiteness of integral representations on 2-perfect truncation polytopes
研究二完美截断多面体的整数表示有限性，证明其变形空间仅含有限多个整数表示，扩展到一般不可约、大、二完美截断多面体。
In this paper, we construct a right-angled 5-polytope P of finite volume such that all the right-angled Coxeter groups with Fuchsian ends obtained from P are locally rigid.
A theorem of Tits - Vinberg allows to build an action of a Coxeter group $Γ$ on a properly convex open set $Ω$ of the real projective space, thanks to the data $P$ of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically finite. We describe an hypothesis that make those conditions necessary. Under this hypothesis, we describe the Zariski closure of $Γ$, find the maximal $Γ$-invariant convex, when there is a unique $Γ$-invariant convex, when the convex $Ω$ is strictly convex, when we can find a $Γ$-…
A Coxeter $n$-orbifold is an $n$-dimensional orbifold based on a polytope with silvered boundary facets. Each pair of adjacent facets meet on a ridge of some order $m$, whose neighborhood is locally modeled on ${\mathbb R}^n$ modulo the dihedral group of order $2m$ generated by two reflections. For $n \geq 3$, we study the deformation space of real projective structures on a compact Coxeter $n$-orbifold $Q$ admitting a hyperbolic structure. Let $e_+(Q)$ be the number of ridges of order $\geq 3$. A neighborhood of the hyperbolic structure in the deformation space is a cell of dimension $e_+(Q) …
In his seminal 1951 paper "Extreme forms" Coxeter \cite{cox51} observed that for $n \ge 9$ one can add vectors to the perfect lattice $\sfA_9$ so that the resulting perfect lattice, called $\sfA_9^2$ by Coxeter, has exactly the same set of minimal vectors. An inhomogeneous analog of the notion of perfect lattice is that of a lattice with a perfect Delaunay polytope: the vertices of a perfect Delaunay polytope are the analogs of minimal vectors in a perfect lattice. We find a new infinite series $P(n,s)$ for $s\geq 2$ and $n+1\geq 4s$ of $n$-dimensional perfect Delaunay polytopes. A remarkable …
We determine the three hyperbolic 5-orbifolds of smallest volume among compact arithmetic orbifolds, and we identify their fundamental groups with hyperbolic Coxeter groups. This gives two different ways to compute the volume of these orbifolds.
We study arithmetic properties of the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups whose dihedral angles are of the form $\fracπ{m}$ for $m=2,3,4,5,6$ and show that the growth rates are always Perron numbers.
For small $n$, the known compact hyperbolic $n$-orbifolds of minimal volume are intimately related to Coxeter groups of smallest rank. For $n=2$ and $3$, these Coxeter groups are given by the triangle group $[7,3]$ and the tetrahedral group $[3,5,3]$, and they are also distinguished by the fact that they have minimal growth rate among all cocompact hyperbolic Coxeter groups in $\hbox{Isom}\mathbb H^n$, respectively. In this work, we consider the cocompact Coxeter simplex group $G_4$ with Coxeter symbol $[5,3,3,3]$ in $\hbox{Isom}\mathbb H^4$ and the cocompact Coxeter prism group $G_5$ based on…
The minimal covolume right-angled Coxeter group in hyperbolic 3-space
本文证明由特定直角三角双锥体面反射生成的直角Coxeter群在双曲3空间中具有最小余体积，该群为算术群，余体积等于卡塔兰常数。
Character varieties of a transitioning Coxeter 4-orbifold
本文研究 Kerckhoff 和 Storm 的直角 Coxeter 群在双曲、Anti-de Sitter 与半管几何中的特征簇及其坍缩奇点，并利用四维直角尖点群的刚性性质进行分析。
We show that the 1-cusped quotient of the hyperbolic space $\mathbb{H}^3$ by the tetrahedral Coxeter group $Γ_*=[5,3,6]$ has minimal volume among all non-arithmetic cusped hyperbolic 3-orbifolds, and as such it is uniquely determined. Furthermore, the lattice $Γ_*$ is incommensurable to any Gromov-Piatetski-Shapiro type lattice. Our methods have their origin in the work of C. Adams. We extend considerably this approach via the geometry of the underlying horoball configuration induced by a cusp.
A group of isometries of a hyperbolic $n$-space is called a reflection group if it is generated by reflections in hyperbolic hyperplanes. Vinberg gave a semi-algorithm for finding a maximal reflection sublattice in a given arithmetic subgroup of $O(n,1)$ of the simplest type. We provide an effective termination condition for Vinberg's semi-algorithm with which it becomes an algorithm for finding maximal reflection sublattices. The main new ingredient of the proof is an upper bound for the number of faces of an arithmetic hyperbolic Coxeter polyhedron in terms of its volume.
Multitriangulations, and more generally subword complexes, yield a large family of simplicial complexes that are homeomorphic to spheres. Until now, all attempts to prove or disprove that they can be realized as convex polytopes faced major obstacles. In this article, we lay out the foundations of a framework -- built upon notions from algebraic combinatorics and discrete geometry -- that allows a deeper understanding of geometric realizations of subword complexes of Coxeter groups. Namely, we describe explicitly a family of chirotopes that encapsulate the necessary information to obtain geome…
On quasi-arithmeticity of hyperbolic gluings
研究广义双曲 orbifold 粘合的拟算术性，证明拟算术粘合要求构建块也拟算术且共享环境群与迹域，并据此排除了一族理想直角反棱柱的拟算术性。
The smallest hyperbolic 6-manifolds
通过将一个全直角Coxeter多胞形的拷贝粘在一起，构造了若干个具有欧拉示性数-1的开双曲6-流形.它们是已知的第一个具有最小可能体积的双曲6-流形的例子。
We determine the adjoint trace field of gluings of general hyperbolic manifolds. This provides a new method to prove the nonarithmeticity of gluings, which can be applied to the classical construction of Gromov and Piatetski-Shapiro (and generalizations) as well as certain gluings of pieces of commensurable arithmetic manifolds. As an application we give many new examples of nonarithmetic gluings and prove that the unique nonarithmetic Coxeter 5-simplex is not commensurable to any gluing of arithmetic pieces.
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