We introduce the co-surface graph CS of a finitely generated free group F and use it to study the geometry of hyperbolic group extensions of F. Among other things, we show that the Gromov boundary of the co-surface graph is equivariantly homeomorphic to the space of free arational F–trees and use this to prove that a finitely generated subgroup of Out(F) quasi-isometrically embeds into the co-s...

We define a parameterization for an n-holed tori based on the hyperbolic polygon. We model the domain using a manifold with 2n+ 2 charts, and linear fractional transformations for transition functions. We embed the manifold using standard spline techniques to produce a surface. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling, Curve, Surface, Solid, and Objec...

We show how the real lines on a del Pezzo surface of degree 1 can be split into two species, elliptic and hyperbolic, via certain distinguished, intrinsically defined, $${\text {Pin}}^-$$ -structure locus surface. prove that this splitting is invariant under automorphisms deformations surface, difference between total numbers hyperbolic always equal to 16.

We prove that all hierarchically hyperbolic groups have nite asymptotic dimension. One application of this result is to obtain the sharpest known bound on the asymptotic dimension of the mapping class group of a nite type surface: improving the bound from exponential to at most quadratic in the complexity of the surface. We also apply the main result to various other hierarchically hyperbolic g...

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...

We prove that the 82 link complement is the minimal volume orientable hyperbolic manifold with 4 cusps. Its volume is twice of the volume V8 of the ideal regular octahedron, i.e. 7.32... = 2V8. The proof relies on Agol’s argument used to determine the minimal volume hyperbolic 3-manifolds with 2 cusps. We also need to estimate the volume of a hyperbolic 3-manifold with totally geodesic boundary...

Let F ′,F be any two closed orientable surfaces of genus g′ > g ≥ 1, and f : F → F be any pseudo-Anosov map. Then we can “extend” f to be a pseudoAnosov map f ′ : F ′ → F ′ so that there is a fiber preserving degree one map M(F ′, f ′) → M(F, f ) between the hyperbolic surface bundles. Moreover the extension f ′ can be chosen so that the surface bundlesM(F ′, f ′) andM(F, f ) have the same firs...

Hyperbolic polynomials have their origins in partial diierential equations. We show in this paper that they have applications in interior point methods for convex programming. Each homogeneous hyperbolic polynomial p has an associated open and convex cone called its hyperbolicity cone. We give an explicit representation of this cone in terms of polynomial inequalities. The function F (x) = ? lo...