Related papers: Pascal's Triangles in Abelian and Hyperbolic Group…

Let $G = \langle A,B \rangle$ be a non-elementary two generator subgroup of the isometry group of $\mathbb{H}^2$, the hyperbolic plane. If $G$ is discrete and free and geometrically finite, its quotient is a pair of pants and in prior work…

We continue the study of the geometry of infinite geodesics in first passage percolation (FPP) on Gromov-hyperbolic groups G, initiated by Benjamini-Tessera and developed further by the authors. It was shown earlier by the authors that,…

We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring $\mathbb{Z} G$ of a finite nilpotent group $G$, this provided the rational group algebra $\mathbb{Q} G$…

We classify groups G such that the unit group U(ZG) is hypercentral. In the second part we classify groups G whose modular group algebra has hyperbolic unit group V(KG).

We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, all of them rather involved or else relying on sophisticated number theoretical…

We study the problem of finding generators for the fundamental group G of a space of the following sort: one removes a family of complex hyperplanes from n dimensional complex vector space, or n dimensional complex hyperbolic space, or the…

We construct certain subgroups of hyperbolic triangle groups which we call "congruence" subgroups. These groups include the classical congruence subgroups of SL_2(ZZ), Hecke triangle groups, and 19 families of arithmetic triangle groups…

It is known that the bounded Geodesic Length Problem in free metabelian groups is NP-complete (in particular, the Geodesic Problem is NP-hard). We construct a 2-approximation polynomial time deterministic algorithm for the Geodesic Problem.…

Algebraic hyperbolicity serves as a bridge between differential geometry and algebraic geometry. Generally, it is difficult to show that a given projective variety is algebraically hyperbolic. However, it was established recently that a…

Identifying parallel sides of a collection of Euclidean polygons yields a flat surface with cone points of angles multiples of 2 pi, naturally a compact Riemann surface but also an algebraic curve, and a hyperbolic surface. In general two…

Let $G$ be a simply connected and simple algebraic group defined and split over a finite prime field $\mathbb{F}_p$ of $p$ elements. In this paper, using an $\mathbb{F}_p$-linear map splitting Frobenius endomorphism on a hyperalgebra…

We construct cross sections for the geodesic flow on the orbifolds $\Gamma\backslash H$ which are tailor-made for the requirements of transfer operator approaches to Maass cusp forms and Selberg zeta functions. Here, $H$ denotes the…

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…

For any finite group G and integer i, let $\mathcal{H}^i(G)$ be the set of all the isomorphism classes of the Galois cohomology groups $\hat{H}^i(K/k,E_K)$, where K/k runs over all the unramified G-extension of number fields and E_K denotes…

The set of axes of hyperbolic elements in a Fuchsian group depends on the commensurability class of the group. In fact, it has been conjectured that it determines the commensurability class and this has been verified in for groups of the…

The subgroup pattern of a finite groups $G$ is the table of marks of $G$ together with a list of representatives of the conjugacy classes of subgroups of $G$. In this article we present an algorithm for the computation of the subgroup…

We say that a group G is a cube group if it is generated by a set S of involutions such that the corresponding Cayley graph Cay(G,S) is isomorphic to a cube. Equivalently, G is a cube group if it acts on a cube such that the action is…

A triple space is a homogeneous space $G/H$ where $G=G_0\times G_0\times G_0$ is a threefold product group and $H\simeq G_0$ the diagonal subgroup of $G$. This paper concerns the geometry of the triple spaces with $G_0=\SL(2,\R)$,…

Given a binary quasigroup $G$ of order $n$, a $d$-iterated quasigroup $G[d]$ is the $(d+1)$-ary quasigroup equal to the $d$-times composition of $G$ with itself. The Cayley table of every $d$-ary quasigroup is a $d$-dimensional latin…

We generalize the group law of curves of degree three by chords and tangents to the Jacobi variety of a hyperelliptic curve. In the case of genus 2 we accomplish the construction by a cubic parabola. We derive explicit rational formulas for…