Related papers: New complex- and quaternion-hyperbolic reflection …

We study the recently discovered isomorphisms between hyperbolic Weyl groups and unfamiliar modular groups. These modular groups are defined over integer domains in normed division algebras, and we focus on the cases involving quaternions…

In this paper we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps. In particular, a few classical results of Steinberg and Deligne & Lusztig on complex representations of finite…

This work builds on the foundation laid by Gordon and Wilson in the study of isometry groups of solvmanifolds, i.e. Riemannian manifolds admitting a transitive solvable group of isometries. We restrict ourselves to a natural class of…

We prove an asymptotic analog of the classical Hurewicz theorem on mappings which lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces…

We give examples of hyperbolic groups which contain subgroups that are of type $\mathscr{F}_{3}$ but not of type $\mathscr{F}_{4}$. These groups are obtained by Dehn filling starting from a non-uniform lattice in ${\rm PO}(8,1)$ which was…

When the maximal isometry group of a four-dimensional spacetime acts simply transitively, such a Ricci-flat metric is uniquely determined to be the Petrov solution. This isometry group is almost abelian; that is, its Lie algebra contains an…

For an even, integral hyperbolic lattice $L$, the symmetry group of $L$ is the quotient of the group of isometries of $L$ by the Weyl subgroup of $(-2)$-reflections. Following Nikulin, the exceptional lattice of $L$ is defined as the…

The new local group LB1 introduced previously will be studied and reviewed in detail, depicting its unique nature that makes it a new group in fundamental physics. It will be made clear that even though most of its elements are Lorentz…

We give necessary and sufficient conditions for a linear reflection group in the sense of Vinberg to be Zariski-dense in the ambient projective general linear group. As an application, we show that every irreducible right-angled Coxeter…

Minkowski space is the local model of 3 dimensionnal flat spacetimes. Recent progress in the description of globally hyperbolic flat spacetimes showed strong link between Lorentzian geometry and Teichm{\"u}ller space. We notice that…

We prove that there are only finitely many isometry classes of even lattices $L$ of signature $(2,n)$ for which the space of cusp forms of weight $1+n/2$ for the Weil representation of the discriminant group of $L$ is trivial. We compute…

We give variants of lifting construction, which define new classes of modular forms on the Siegel upper half-space of complex dimension 3 with respect to the full paramodular groups (defining moduli of Abelian surfaces with arbitrary…

This is the announcement and partly the review of our results about classification of Lorentzian Kac-Moody algebras of the rank three. One of our results gives the classification of Lorentzian Kac-Moody algebras with denominator identity…

We give uniform formulas for the number of full reflection factorizations of a parabolic quasi-Coxeter element in a Weyl group or complex reflection group, generalizing the formula for the genus-0 Hurwitz numbers. This paper is the…

This paper studies groups of maximal size acting harmonically on a finite graph. Our main result states that these maximal graph groups are exactly the finite quotients of the modular group $\Gamma=\left<x,y \ | \ x^2=y^3=1\right>$ of size…

Let G be a connected complex simple Lie group with maximal compact subgroup U. Let g be the Lie algebra of G, and X = G/U be the associated Riemannian globally symmetric space of type IV. We have constructed three types of arithmetic…

We derive explicit isomorphisms between certain congruence subgroups of the Siegel modular group, the Hermitian modular group over an arbitrary imaginary-quadratic number field and the modular group over the Hurwitz quaternions of degree 2…

In the framework of the study of homogeneous Lorentzian three-manifolds, we consider here the only class of examples which admit a four-dimensional group of isometries but are neither Lorentzian Bianchi-Cartan-Vranceanu spaces nor plane…

We survey the existing parts of a classification of finite groups generated by orthogonal transformations in a finite-dimensional Euclidean space whose fixed point subspace has codimension one or two and extend it to a complete…

The classification of reflective modular forms is an important problem in the theory of automorphic forms on orthogonal groups. In this paper, we develop an approach based on the theory of Jacobi forms to give a full classification of…