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We define a class of groups constructed from rings equipped with an involution. We show that under suitable conditions, these groups are either algebraic or arithmetic, including as special cases the orientation-preserving isometry group of…

We show that all spin groups of non-definite, quinary quadratic forms over a field with characteristic 0 can be represented as 2 by 2 matrices with entries in an associated quaternion algebra. Over local and global fields, we further study…

The goal of this paper is to demonstrate the use of techniques from hyperbolic geometry to compute generating sets of certain subgroups of $SL^+(2,\mathbb{C})$; specifically, $SO^+(Q,\mathbb{Z})$ for $Q$ some integral quadratic form of…

Special orthogonal matrices with rational elements form the group SO(n,Q), where Q is the field of rational numbers. A theorem describing the structure of an arbitrary matrix from this group is proved. This theorem yields an algorithm for…

Let $K$ be a $\mathbb{Q}$-Clifford algebra associated to an $(n-1)$-ary positive definite quadratic form and let $\mathcal{O}$ be a maximal order in $K$. A Clifford-Bianchi group is a group of the form $\operatorname{SL}_2(\mathcal{O})$…

Let O be the ring of S-integers in a number field k. We prove that if the group of units O^* is infinite then every matrix in $\Gamma$ = SL_2(O) is a product of at most 9 elementary matrices. This completes a long line of research in this…

We study a generalization of the Fuchsian triangle groups to the hyperbolic 3-space, namely, the groups generated by half-turns in three hyperbolic lines. The role of the hyperbolic triangles is now played by the right-angled hexagons. This…

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $\mathbb{K}[x,y]$ the polynomial ring. The group $\text{SL}_{2}\left(\mathbb{K}[x,y]\right)$ of all matrices with determinant equal to $1$ over $\mathbb{K}[x,y]$…

Using the rings of Lipschitz and Hurwitz integers $\mathbb{H}(\mathbb{Z})$ and $\mathbb{H}ur(\mathbb{Z})$ in the quaternion division algebra $\mathbb{H}$, we define several Kleinian discrete subgroups of $PSL(2,\mathbb{H})$

Embeddings of word structures into matrix semigroups provide a natural bridge between combinatorics on words and linear algebra. However, low-dimensional matrix semigroups impose strong structural restrictions on possible embeddings.…

Let $\mathcal{O}$ be the ring of integers of a number field, and let $n\geq 3$. This paper studies bi-interpretability of the ring of integers $\mathbb{Z}$ with the special linear group $\text{SL}_n(\mathcal{O})$, the general linear group…

The present article determines the indices of the principal congruence subgroups of the Bianchi groups $B_d$, $SL(2,\mathcal O)$ and elementary matrix group $E$ and extends Wohlfahrt's Theorem to $B_d$, $SL(2,\mathcal O)$ and $E$, where…

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 study the orbit of $\mathbb{R}$ under the Bianchi group $\operatorname{PSL}_2(\mathcal{O}_K)$, where $K$ is an imaginary quadratic field. The orbit, called a Schmidt arrangement $\mathcal{S}_K$, is a geometric realisation, as an…

We classify the polycyclic totally ordered simple dimension groups, i.e. dimension groups given by a dense embedding of n-dimensional lattice into the real line. Our method is based on the geometry of simple geodesics on the hyperbolic…

In this paper we use techniques from convex projective geometry to produce many new examples of thin subgroups of lattices in special linear groups that are isomorphic to the fundamental groups of finite volume hyperbolic manifolds. More…

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group…

We consider non-elementary Kleinian groups \Gamma, without invariant plane, generated by an elliptic and a hyperbolic element with their axes lying in one plane. We find presentations and a complete list of orbifolds uniformized by such…

Uhlenbeck proved that a set of simple elements generates the group of rational loops in GL(n,C) that satisfy the U(n)-reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality…

Upper and lower bounds are given for the maximum Euclidean curvature among faces in Bianchi's fundamental polyhedron for $PSL_2(O)$ in the upper-half space model of hyperbolic space, where $O$ is an imaginary quadratic ring of integers with…