This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of its associated Deligne complex. We prove that an Artin group is weakly hyperbolic relative to its finite (or spherical) type parabolic subgroups if and only if its Deligne complex is a Gromov hyperbolic space. For a 2-dimensional Artin group the Deligne complex is Gromov hyperbolic preci...

A closed, and say orientable, Riemannian 3–manifold (M, ρ) is hyperbolic if the metric ρ has constant sectional curvature κρ = −1. Equivalently, there is a discrete and torsion free group Γ of isometries of hyperbolic 3–space H3 such that the manifolds (M, ρ) and H3/Γ are isometric. It is well-known that the fundamental group π1(M) of every closed 3–manifold which admits a hyperbolic metric is ...

The action of Möbius transformations with real coefficients preserves the hyperbolic metric in the upper half-plane model of the hyperbolic plane. The modular group is an interesting group of hyperbolic isometries generated by two Möbius transformations, namely, an order-two element, g2 HzL = -1 ê z, and an element of infinite order, g• HzL = z + 1. Viewing the action of the group elements on a...

Gromov asked whether every one-ended word-hyperbolic group contains a hyperbolic surface group. We prove that every one-ended double of a free group contains a hyperbolic surface group if the free group has rank two, or every generator is used the same number of times in the list of amalgamating words. Our method is based on formulating a stronger conjecture on tilings of closed surfaces combin...

The retrosection theorem says that any hyperbolic or Riemann surface can be uniformized by a Schottky group. We generalize this theorem to the case of hyperbolic 2-orbifolds by giving necessary and sufficient conditions for a hyperbolic 2-orbifold, in terms of its signature, to admit a uniformization by a Kleinian group which is a finite extension of a Schottky group. Equivalent^, the condition...

In 1987, Gromov conjectured that for every non-elementary hyperbolic group G there is an n = n(G) such that the quotient group G/Gn is infinite. The article confirms this conjecture. In addition, a description of finite subgroups of G/Gn is given, it is proven that the word and conjugacy problem are solvable in G/Gn and that ⋂∞ k=1G k = {1}. The proofs heavily depend upon prior authors’ results...