arXiv research

Study thin hyperbolic reflection groups and their properties.

A hyperbolic reflection group is a discrete group generated by reflections in the faces of an $n$-dimensional hyperbolic polyhedron. This survey article is dedicated to the study of arithmetic hyperbolic reflection groups with an emphasis on the results that were obtained in the last ten years and on the open problems.

Constructs hyperbolic reflection groups with 3D limit sets.

A hyperbolic 3-simplex reflection group is a Coxeter group arising as a lattice in the isometry group of hyperbolic 3-space, with fundamental domain a geodesic simplex (possibly with some ideal vertices). The classification of these groups is known, and there are exactly 9 cocompact examples, and 23 non-cocompact examp…

Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing ground fields of arithmetic hyperbolic reflection groups are defined, and good bounds of their degrees (over Q) are obtained. For example, degree of the ground field of any arithmetic hyperbolic reflection group in dimension at…

We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.

This paper continues arXiv.org:math.AG/0609256, arXiv:0708.3991 and arXiv:0710.0162 . Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimension at least 3 are defined, and explicit bounds of their degrees (over …

New families of hyperbolic polyhedra yield infinitely many unique reflection groups.

A hyperbolic lattice is called \textit{$(1{,}2)$-reflective} if its automorphism group is generated by $1$- and $2$-reflections up to finite index. In this paper we prove that the fundamental polyhedron of a $\mathbb{Q}$-arithmetic cocompact reflection group in the three-dimensional Lobachevsky space contains an edge s…

This paper is a follow-up to our joint paper with I. Agol, P. Storm and K. Whyte "Finiteness of arithmetic hyperbolic reflection groups". The main purpose is to investigate the effective side of the method developed there and its possible application to the problem of classification of arithmetic hyperbolic reflection …

The paper improves Vinberg's algorithm for arithmetic hyperbolic lattices.

Geometric constraints help classify hyperbolic polytopes.

New groups found in hyperbolic space with infinite fields of definition.

A discrete subgroup of the group of isometries of the hyperbolic space is called reflective if up to a finite index it is generated by reflections in hyperplanes. The main result of this paper is a complete classification of the reflective (and quasi-reflective) subgroups among the Bianchi groups and their extensions.

Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.

This paper continues arXiv.org:math.AG/0609256 and arXiv:0708.3991 Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimensions at least 4 are defined, and good explicit bounds of their degrees (over Q) are obtain…

A hyperbolic lattice is called \textit{$1.2$-reflective} if the subgroup of its automorphism group generated by all $1$- and $2$-reflections is of finite index. The main result of this article is a complete classification of $1.2$-reflective maximal anisotropic lattices of rank $4$.

An integral hyperbolic lattice is called reflective if its automorphism group is generated by reflections, up to finite index. Since 1981, it is known that their number is essentially finite. We show that K3 surfaces over C with reflective Picard lattices can be characterized in terms of compositions of their self-corr…

Study on hyperbolic polyhedra and their volume, proving finiteness of arithmetic groups.

Two groups with specific limit sets in hyperbolic spaces are identified.

We compute the equivariant $K$-homology of the classifying space for proper actions, for compact 3-dimensional hyperbolic reflection groups. This coincides with the topological $K$-theory of the reduced $C^\ast$-algebra associated to the group, via the Baum-Connes conjecture. We show that, for any such reflection group…

The study determines discreteness of complex hyperbolic triangle groups.

We classify the 3-dimensional hyperbolic polyhedral orbifolds that contain no embedded essential 2-suborbifolds, up to decomposition along embedded hyperbolic triangle orbifolds (turnovers). We give a necessary condition for a 3-dimensional hyperbolic polyhedral orbifold to contain an immersed (singular) hyperbolic tur…

Characterizes Coxeter groups with convex cocompact representations in projective space.

Picard modular groups are shown to be generated by complex reflections.

We present several formulas for the traces of elements in complex hyperbolic triangle groups generated by complex reflections. The space of such groups of fixed signature is of real dimension one. We parameterise this space by a real invariant alpha of triangles in the complex hyperbolic plane. The main result of the p…

Identifies Anosov representations of hyperbolic triangle groups in SL(3,R).

Defines fundamental racks for braid spaces of complex reflection groups.

The reflection length of an element of a Coxeter group is the minimal number of conjugates of the standard generators whose product is equal to that element. In this paper we prove the conjecture of McCammond and Petersen that reflection length is unbounded in any non-affine Coxeter group. Among the tools used, the con…

We compute the Cheeger constants of a collection of hyperbolic surfaces corresponding to maximal non-compact arithmetic Fuchsian groups, and to subgroups which are the rotation subgroup of maximal reflection groups. The Cheeger constants are geometric quantities, but relate to the smallest eigenvalues of Maass cusp for…

The transition constant was introduced in our 1981 paper and denoted as N(14). It is equal to the maximal degree of the ground fields of V-arithmetic connected edge graphs with 4 vertices and of the minimality 14. This constant is fundamental since if the degree of the ground field of an arithmetic hyperbolic reflectio…

The paper defines parabolic subgroups for complex braid groups and proves they form a lattice.

In this paper, it is shown that a Fuchsian group, acting on the upper half-plane model for $\mathbb{H}^2$, admits a Ford domain which is also a Dirichlet domain, for some center, if and only if it is an index 2 subgroup of a reflection group. This is used to exhibit an example of a maximal arithmetic hyperbolic reflect…

The notion of limit roots of a Coxeter group W was recently introduced (see arXiv:1112.5415 and arXiv:1303.6710): they are the accumulation points of directions of roots of a root system for W. In the case where the root system lives in a Lorentzian space W admits a faithful representation as a discrete reflection grou…

Study geometric properties of a complex hyperbolic group action.

Minimal covolume group found in hyperbolic 3-space.

For every dimension d, there is an infinite family of convex co-compact reflection groups of isometries of hyperbolic d-space --- the superideal (simplicial and cubical) reflection groups --- with the property that a random group at any density less than a half (or in the few relators model) contains quasiconvex subgro…

Following the previous work of Nikulin and Agol, Belolipetsky, Storm, and Whyte it is known that there exist only finitely many (totally real) number fields that can serve as fields of definition of arithmetic hyperbolic reflection groups. We prove a new bound on the degree $n_k$ of these fields in dimension 3: $n_k$ d…

Study general hyperbolic gluings, proving quasi-arithmeticity of building blocks.

Cube complexes allow hyperbolic groups to have Anosov representations.

In this note we prove that a complex hyperbolic triangle group of type (m,m,infinity), i.e. a group of isometries of the complex hyperbolic plane, generated by complex reflections in three complex geodesics meeting at angles Pi/m, Pi/m and 0, is not discrete if the product of the three generators is regular elliptic.

We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bic…

New trick builds hyperbolic manifolds from compact ones, proving some don't virtually fiber.

For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the property that all interior angles between incident faces are integral submultiples of Pi, there is a naturally associated Coxeter group generated by reflections in the faces. Furthermore, this Coxeter group is a lattice inside the isometry group …

We generalize arc coordinates for maximal representations on a pair of pants.

In this paper we consider ultra-parallel complex hyperbolic triangle groups of type $[m_1,m_2,0]$, i.e. groups of isometries of the complex hyperbolic plane, generated by complex reflections in three ultra-parallel complex geodesics two of which intersect on the boundary. We prove some discreteness and non-discreteness…

We provide a concrete criterion to determine whether or not two given elements of PU(2,1) can be written as products of real reflections, with one reflection in common. As an application, we show that the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$ with $d=1,2,3,7,11$ are generated by real reflections up to ind…

Following Vinberg, we find the criterions for a subgroup generated by reflections $Γ\subset \SL^{\pm}(n+1,\mathbb{R})$ and its finite-index subgroups to be definable over $\mathbb{A}$ where $\mathbb{A}$ is an integrally closed Noetherian ring in the field $\mathbb{R}$. We apply the criterions for groups generated by re…