Criterion for Cannon’s Conjecture

@article{Markovi2012CriterionFC,
  title={Criterion for Cannon’s Conjecture},
  author={Vladimir Markovi{\'c}},
  journal={Geometric and Functional Analysis},
  year={2012},
  volume={23},
  pages={1035-1061}
}
  • V. Marković
  • Published 25 May 2012
  • Mathematics
  • Geometric and Functional Analysis
The Cannon Conjecture from the geometric group theory asserts that a word hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to a Kleinian group. We prove the following Criterion for Cannon’s Conjecture: a hyperbolic group G (that acts effectively on its boundary) whose boundary is homeomorphic to the 2-sphere is isomorphic to a Kleinian group if and only if every two points in the boundary of G are separated by a quasi… 
Characterizing candidates for Cannon's Conjecture from Geometric Measure Theory
. We show that recent work of Song implies that torsion-free hyperbolic groups with Gromov boundary S 2 are realized as fundamental groups of closed 3-manifolds of constant negative curvature if and
Thurston’s Vision and the Virtual Fibering Theorem for 3-Manifolds
The vision and results of William Thurston (1946–2012) have shaped the theory of 3-dimensional manifolds for the last four decades. The high point was Perelman’s proof of Thurston’s Geometrization
Virtual properties of 3-manifolds dedicated to the memory of Bill Thurston
We will discuss the proof of Waldhausen’s conjecture that compact aspherical 3-manifolds are virtually Haken, as well as Thurston’s conjecture that hyperbolic 3manifolds are virtually fibered. The
Hyperbolic groups with planar boundaries
We prove that the class of convex-cocompact Kleinian groups is quasi-isometrically rigid. We also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually
Hyperbolic groups with planar boundaries
We prove that the class of convex-cocompact Kleinian groups is quasi-isometrically rigid. We also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually
Detecting sphere boundaries of hyperbolic groups
We show that a one-ended simply connected at infinity hyperbolic group $G$ with enough codimension-1 surface subgroups has $\partial G \cong \mathbb{S}^2$. Combined with a result of Markovic, our
Fuchsian groups, circularly ordered groups and dense invariant laminations on the circle
We propose a program to study groups acting faithfully on S 1 in terms of numbers of pairwise transverse dense invariant laminations. We give some examples of groups that admit a small number of
Essential surfaces in graph pairs
  • H. Wilton
  • Mathematics
    Journal of the American Mathematical Society
  • 2018
A well-known question of Gromov asks whether every one-ended hyperbolic group Γ \Gamma has a surface subgroup. We give a positive answer when Γ \Gamma is the fundamental group of
Homotopy and Homology at Infinity and at the Boundary
In this paper we study the relationship between the homology and homotopy of a space at infinity and at its boundary. Firstly, we prove that if a locally connected, connected, δ-hyperbolic space that
Geometric Group Theory
Geometry and topology Metric spaces Differential geometry Hyperbolic space Groups and their actions Median spaces and spaces with measured walls Finitely generated and finitely presented groups
...
...

References

SHOWING 1-10 OF 28 REFERENCES
A boundary criterion for cubulation
We give a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a ${\rm CAT}(0)$ cube complex. We describe applications towards lattices and
Hyperbolic Manifolds and Discrete Groups
Preface.-Three-dimensional Topology.-Thurston Norm.-Geometry of the Hyperbolic Space.-Kleinian Groups.-Teichmuller Theory of Riemann Surfaces.-Introduction to the Orbifold Theory.-Complex Projective
The virtual Haken conjecture
We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are
RESEARCH ANNOUNCEMENT: THE STRUCTURE OF GROUPS WITH A QUASICONVEX HIERARCHY
Let $G$ be a word-hyperbolic group with a quasiconvex hierarchy. We show that $G$ has a finite index subgroup $G'$ that embeds as a quasiconvex subgroup of a right-angled Artin group. It follows
Packing subgroups in relatively hyperbolic groups
Our main result establishes the bounded packing of relatively quasiconvex subgroups of a relatively hyperbolic group, under mild hypotheses. As an application, we prove that relatively quasiconvex
Convergence groups are Fuchsian groups
A group of homeomorphisms of the circle satisfying the "convergence property" is shown to be the restriction of a discrete group of Mobius transformations of the unit disk. This completes the proof
Homeomorphic conjugates of Fuchsian groups.
By a Fuchsian group we mean a discrete subgroup of M. It may contain also orientation reversing elements. Usually a Fuchsian group is thought to act on I) but this is not a problem since the action
Surface subgroups from homology
Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H2(G; ℚ) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the
WIDTHS OF SUBGROUPS
We say that the width of an infinite subgroup H in G is n if there exists a collection of n essentially distinct conjugates of H such that the intersection of any two elements of the collection is
Hyperbolic surface subgroups of one‐ended doubles of free groups
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 has a hyperbolic surface subgroup if (1) the free
...
...