Let N^h be a hyperbolic 3-manifold of bounded geometry corresponding to a hyperbolic structure on a pared manifold (M,P). Further, suppose that (\partial{M} - P) is incompressible, i.e. the boundary of M is incompressible away from cusps. Further, suppose that M_{gf} is a geometrically finite hyperbolic structure on (M,P). Then there is a Cannon- Thurston map from the limit set of M_{gf} to that of N^h. Further, the limit set of N^h is locally connected. This answers in part a question attributed to Thurston.
In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian surface groups. In this paper we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon-Thurston map is not one-to-one. In particular, the Cannon-Thurston map is finite-to-one. This completes the proof of the conjectural picture of Cannon-Thurston maps for surface groups.
This is an expository paper. We prove the Cannon-Thurston property for bounded geometry surface groups with or without punctures. We prove three theorems, due to Cannon-Thurston, Minsky and Bowditch. The proofs are culled out of earlier work of the author.
Using the Birmanexact sequence for pure mapping class groups, we construct a universal Cannon--Thurston map onto the boundary of a curve complex for a surface with punctures we call surviving curve complex. Along the way we prove hyperbolicity of this complex and identify its boundary as a space of laminations. As a corollary we obtain a universal Cannon--Thurston map to the boundary of the ordinary curve complex, extending earlier work of the second author with Mj and Schleimer.
Any hyperbolic surface bundle over the circle gives rise to a continuous surjection from the circle to the sphere, by work of Cannon and Thurston. We prove that the order in which this surjection fills out the sphere is dictated by a natural triangulation of the surface bundle (introduced by Agol) when all singularities of the invariant foliations are at punctures of the fiber.
We introduce the notion of manifolds of amalgamation geometry and its generalization, split geometry. We show that the limit set of any surface group of split geometry is locally connected, by constructing a natural Cannon-Thurston map.
The notion of i-bounded geometry generalises simultaneously bounded geometry and the geometry of punctured torus Kleinian groups. We show that the limit set of a surface Kleinian group of i-bounded geometry is locally connected by constructing a natural Cannon-Thurston map. This is an exposition of a special case of the main result of arXiv:math/0607509.
We prove the existence of Cannon-Thurston maps for simply and doubly degenerate surface Kleinian groups. As a consequence we prove that connected limit sets of finitely generated Kleinian groups are locally connected.
Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the quasi-isometrically embedded condition. Let $v$ be a vertex of $T$. Let $({X_v},d_v)$ denote the hyperbolic metric space corresponding to $v$. Then $i : X_v \rightarrow X$ extends continuously to a map $\hat{i} : \widehat{X_v} \rightarrow \widehat{X}$. This generalizes a Theorem of Cannon and Thurston. The techniques are used to give a new proof of a result of Minsky: Thurston's ending lamination conjecture for certain Kleinian groups. Applications to graphs of hyperbolic groups and local connectivity of limit sets of Kleinian …
We show that Cannon-Thurston maps exist for degenerate free groups without parabolics, i.e. for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon-Thurston maps for surface groups, we show that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon-Thurston maps for degenerate free groups without parabolics correspond to end-points of leaves of an ending lamination in the Masur domain, whenever a point has more than …
In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian punctured surface groups without accidental parabolics. In this note we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon-Thurston map is not one-to-one. This extends earlier work done for closed surface groups.
Mahan Mitra (Mj) proved Cannon--Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group. We prove that Cannon--Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic CAT(0) groups with isolated flats with respect to the visual boundaries. We also show Cannon--Thurston maps do not exist for infinite infinite-index normal CAT(0) subgroups with isolated flats in non-hyperbolic CAT(0) groups with isolated flats. We obtain a structure theorem for the normal subgroups in these settings and show that outer automorphism groups of hyperbolic groups have …
Baker and Riley proved that a free group of rank 3 can be contained in a hyperbolic group as a subgroup for which the Cannon-Thurston map is not well-defined. By using their result, we show that the phenomenon occurs for not only a free group of rank 3 but also every non-elementary hyperbolic group. In fact it is shown that a similar phenomenon occurs for every non-elementary relatively hyperbolic group.
For a Coxeter group $W$ we have an associating bi-linear form $B$ on a real vector space. We assume that $B$ has the signature $(n-1,1)$. In this case we have the Cannon-Thurston map for $W$, that is, a $W$-equivariant continuous surjection from the Gromov boundary of $W$ to the limit set of $W$. We focus on the case where Coxeter groups contain affine special subgroups.
We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalises a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces.
In genus two and higher, the fundamental group of a closed surface acts naturally on the curve complex of the surface with one puncture. Combining ideas from previous work of Kent--Leininger--Schleimer and Mitra, we construct a universal Cannon--Thurston map from a subset of the circle at infinity for the closed surface group onto the boundary of the curve complex of the once-punctured surface. Using the techniques we have developed, we also show that the boundary of this curve complex is locally path-connected.
Let $1\to (K,K_1)\to (G,N_G(K_1))\to(Q,Q_1)\to 1$ be a short exact sequence of pairs of finitely generated groups with $K$ strongly hyperbolic relative to proper subgroup $K_1$. Assuming that for all $g\in G$ there exists $k\in K$ such that $gK_1g^{-1}=kK_1k^{-1}$, we prove that there exists a quasi-isometric section $s\colon Q \to G$. Further we prove that if $G$ is strongly hyperbolic relative to the normalizer subgroup $N_G(K_1)$ and weakly hyperbolic relative to $K_1$, then there exists a Cannon-Thurston map for the inclusion $i\colonΓ_K\to Γ_G$.
We give an overview of the theory of Cannon-Thurston maps which forms one of the links between the complex analytic and hyperbolic geometric study of Kleinian groups. We also briefly sketch connections to hyperbolic subgroups of hyperbolic groups and end with some open questions.
For a Coxeter group $W$ we have an associating bi-linear form $B$ on suitable real vector space. We assume that $B$ has the signature $(n-1,1)$ and all the bi-linear form associating rank $n' (\ge 3)$ Coxeter subgroups generated by subsets of $S$ has the signature $(n',0)$ or $(n'-1,1)$. Under these assumptions, we see that there exists the Cannon-Thurston map for $W$, that is, the $W$-equivariant continuous surjection from the Gromov boundary of $W$ to the limit set of $W$. To see this we construct an isometric action of $W$ on an ellipsoid with the Hilbert metric. As a consequence, we see th…
Let $G$ be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space $Z$ so that there exists a continuous $G$-equivariant map $i:\partial G\to Z$, which we call a \emph{Cannon-Thurston map}. We obtain two characterzations (a dynamical one and a geometric one) of conical limit points in $Z$ in terms of their pre-images under the Cannon-Thurston map $i$. As an application we prove, under the extra assumption that the action of $G$ on $Z$ has no accidental parabolics, that if the map $i$ is not injective then there exists a non-conical limit point $z\in Z…
We show that for a strongly convergent sequence of purely loxodromic finitely generated Kleinian groups with incompressible ends, Cannon-Thurston maps, viewed as maps from a fixed base limit set to the Riemann sphere, converge uniformly. For algebraically convergent sequences we show that there exist examples where even pointwise convergence of Cannon-Thurston maps fails.
This paper gives a detailed analysis of the Cannon--Thurston maps associated to a general class of hyperbolic free group extensions. Let $F_N$ denote a free groups of finite rank $N\ge 3$ and consider a \emph{convex cocompact} subgroup $Γ\le Out(F_N)$, i.e. one for which the orbit map from $Γ$ into the free factor complex of $F_N$ is a quasi-isometric embedding. The subgroup $Γ$ determines an extension $E_Γ$ of $F_N$, and the main theorem of Dowdall--Taylor \cite{DT1} states that in this situation $E_Γ$ is hyperbolic if and only if $Γ$ is purely atoroidal. Here, we give an explicit geometric d…
For any atoroidal iwip $ϕ\in Out(F_N)$ the mapping torus group $G_ϕ=F_N\rtimes_ϕ<t>e$ is hyperbolic, and the embedding $ι: F_N \overset{\lhd}{\longrightarrow} G_ϕ$ induces a continuous, $F_N$-equivariant and surjective {\em Cannon-Thurston map} $\hat ι: \partial F_N \to \partial G_ϕ$. We prove that for any $ϕ$ as above, the map $\hat ι$ is finite-to-one and that the preimage of every point of $\partial G_ϕ$ has cardinality $\le 2N$. We also prove that every point $S\in \partial G_ϕ$ with $\ge 3$ preimages in $\partial F_N$ has the form $(wt^m)^\infty$ where $w\in F_N, m\ne 0$, and that there a…
When 1 -> H -> G -> Q -> 1 is a short exact sequence of three infinite, word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point z in the Gromov boundary of Q an "ending lamination" on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a de…
Given a hyperbolic subgroup $H$ of a hyperbolic group $G$ for which a Cannon-Thurston map $\hat i:\partial H \ra \partial G$ exists, we study the limit set $Λ_H$ of $H$ with respect to its action on $\partial G$. We prove that the set of conical limit points is exactly the subset of $Λ_H$ consisting of the points to which the Cannon-Thurston map $\hat i$ injects. Moreover, we show that when $H$ is not quasi-convex in $G$, there exists a non-conical limit point in $Λ_H$
We prove the existence of Cannon-Thurston maps for Kleinian groups corresponding to pared manifolds whose boundary is incompressible away from cusps. We also describe the structure of these maps in terms of ending laminations.
We characterise completely when limit sets, as parametrised by Cannon-Thurston maps, move discontinuously for a sequence of algebraically convergent quasi-Fuchsian groups.
We give a generalization of Thurston's Bounded Image Theorem for skinning maps, which applies to pared 3-manifolds with incompressible boundary that are not necessarily acylindrical. Along the way we study properties of divergent sequences in the deformation space of such a manifold, establishing the existence of compact cores satisfying a certain notion of uniform geometry.
In this note we discuss the behavior of the Gromov boundaries and limit sets for the surface subgroups of the mapping class group with accidental parabolics constructed by the author and A. Reid in earlier work. Specifically, we show that generically there are no Cannon--Thurston maps from the Gromov boundary to Thurston's boundary of Teichmuller space.
We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometric limit is geometrically finite, the maps of limit sets converge pointwise but not uniformly.
Given a closed hyperbolic 3-manifold M with a quasigeodesic flow we construct a π_1-equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal P to the lifted flow on H^3 has a natural compactification as a closed disc that inherits a π_1 action. The embedding of P in H^3 extends continuously to the compactification and the restriction to the boundary is a surjective π_1-equivariant map from S^1 to S^2_\infty. This generalizes the result of Cannon and Thurston for fibered hyperbolic 3-manifolds.
