Related papers: Discrete RP groups with a parabolic generator

In this paper we give necessary and sufficient conditions for discreteness of a group generated by a hyperbolic element and an elliptic one of odd order. This completes the classification of discrete groups with non-$\pi$-loxodromic…

Let $G$ be a finitely generated group of isometries of $\HH^m$, hyperbolic $m$-space, for some positive integer $m$. %or equivalently elements of $PSL(2,\CC)$. The discreteness problem is to determine whether or not $G$ is discrete. Even in…

We give a complete list of orbifolds uniformised by discrete non-elementary two-generator subgroups of PSL(2,C) without invariant plane whose generators and their commutator have real traces.

We describe all real points of the parameter space of two-generator Kleinian groups with a parabolic generator, that is, we describe a certain two-dimensional slice through this space. In order to do this we gather together known…

Let $K$ be a non-archimedean local field with residue field of characteristic $p$. We give necessary and sufficient conditions for a two-generator subgroup $G$ of ${\rm PSL_2}(K)$ to be discrete, where either $K=\mathbb{Q}_p$ or $G$…

The Gehring-Martin-Tan inequality for 2-generator subgroups of PSL(2,C) is one of the best known discreteness conditions. A Kleinian group $G$ is called a Gehring-Martin-Tan group if the equality holds for the group $G$. We give a method…

We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of $PSL(2,{\bold C})$. We then examine certain two-generator groups which arise as extremals in various geometric problems in…

In this paper we will consider the 2-fold symmetric complex hyperbolic triangle groups generated by three complex reflections through angle 2pi/p with p no smaller than 2. We will mainly concentrate on the groups where some elements are…

Let $\Gamma$ be a subgroup of $PSL(2,R)$ generated by three parabolic transformations. The main goal of this paper is to present an algorithm to determine whether or not $\Gamma$ is discrete. Historically discreteness algorithms have been…

We study and classify the purely parabolic discrete subgroups of $PSL(3,\Bbb{C})$. This includes all discrete subgroups of the Heisenberg group ${\rm Heis}(3,\Bbb{C})$. While for $PSL(2,\Bbb{C})$ every purely parabolic subgroup is Abelian…

The discreteness problem for finitely generated subgroups of $PSL(2,\mathbb{R})$ and $PSL(2,\mathbb{C})$ is a long-standing open problem. In this paper we consider whether or not this problem is decidable by an algorithm. Our main result is…

The problem of determining whether or not a non-elementary subgroup of $PSL(2,\CC)$ is discrete is a long standing one. The importance of two generator subgroups comes from J{\o}rgensen's inequality which has as a corollary the fact that a…

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…

Let SL(2, $\mathbb H$) be the group of $2 \times 2$ quaternionic matrices $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with quaternionic determinant $\det A=|ad-aca^{-1} b|=1$. This group acts by the orientation-preserving isometries of…

We consider non-elementary representations of two generator free groups in $PSL(2,\mathbb{C})$, not necessarily discrete or free, $G = < A, B >$. A word in $A$ and $B$, $W(A,B)$, is a palindrome if it reads the same forwards and backwards.…

In this paper, we give a complete description of the representations of all upper triangular complex Kleinian subgroups of PSL(3,C). In https://doi.org/10.1007/s00574-021-00254-9 we show that any solvable group is virtually triangularizable…

In earlier work we introduced geometrically natural probability measures on the group of all M\"obius transformations in order to study "random" groups of M\"obius transformations, random surfaces, and in particular random two-generator…

We follow the dual approach to Coxeter systems and show for Weyl groups a criterium which decides whether a set of reflections is generating the group depending on the root and the coroot lattice. Further we study special generating sets…

We prove that the groups PSL_n(q) are (2,3)-generated for n=9,10 or 11 and all q. Actually, we find out explicit generators x_n and y_n of respective orders 2 and 3, for the groups SL_n(q).

Discrete subgroups of SL(2,R) are well understood, and classified by the geometry of the corresponding hyperbolic surfaces. Discrete subgroups of higher-rank semisimple Lie groups, such as SL(n,R) for n>2, remain more mysterious. While…