François Guéritaud
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.
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We show that a veering triangulation $τ$ specifies a face $σ$ of the Thurston norm ball of a closed three-manifold, and computes the Thurston norm in the cone over $σ$. Further, we show that $τ$ collates exactly the taut surfaces representing classes in the cone over $σ$ up to isotopy. The analysis includes nonlayered veering triangulations and nonfibered faces.
From veering triangulations to convergence actions and back again
本文证明有限体积尖点双曲三维流形的基本群在转向二维球面上的作用是几何有限的收敛作用，进而得到等变同胚及 Cannon-Thurston 映射，并实现改进的近似绘制算法。
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.
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 …
Pullbacks of metric bundles and Cannon-Thurston maps
本文改进了度量丛拉回与Cannon-Thurston映射相关论证，并补充附录阐明flaring条件在度量丛与典范度量图丛间的等价性。
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.
Landry, Minsky and Taylor [LMT] introduced two polynomial invariants of veering triangulations -- the taut polynomial and the veering polynomial. We give algorithms to compute these invariants. In their definition [LMT] use only the upper track of the veering triangulation, while we consider both the upper and the lower track. We prove that the lower and the upper taut polynomials are equal. However, we show that there are veering triangulations whose lower and upper veering polynomials are different. [LMT] related the Teichmüller polynomial of a fibred face of the Thurston norm ball with the …
We study the connections between subsurface projections in curve and arc complexes in fibered 3-manifolds and Agol's veering triangulation. The main theme is that large-distance subsurfaces in fibers are associated to large simplicial regions in the veering triangulation, and this correspondence holds uniformly for all fibers in a given fibered face of the Thurston norm.
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 …
Finiteness of Cannon--Thurston fibers
本文证明在大多数已知设定中，Cannon-Thurston映射是均匀有限到一的，回答了Swarup的问题并扩展了先前结果。
Cannon-Thurston maps and measures
本文给出 Gadre 等人关于闭双曲 3-流形纤维化于圆周时，由 Gibbs 平衡态在理想边界与圆周上诱导的测度类经 Cannon-Thurston 映射前推后相互奇异这一结果的简短证明。
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.
(Non-)existence of Cannon-Thurston maps for Morse boundaries
本文研究正规子群在Morse边界上Cannon-Thurston映射的存在性问题，证明其与双曲情形不同，可同时出现存在与不存在的现象。
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 …
Let $M$ be a closed hyperbolic 3-manifold with a fibered face $σ$ of the unit ball of the Thurston norm on $H_2(M)$. If $M$ satisfies a certain condition related to Agol's veering triangulations, we construct a taut branched surface in $M$ spanning $σ$. This partially answers a 1985 question of Oertel, and extends an earlier partial answer due to Mosher.
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.
Cannon--Thurston maps for Anosov foliations
研究Anosov叶面的universal circles，证明其存在Cannon--Thurston型映射，揭示流形基本群的伪Anosov动力学。
Every pseudo-Anosov mapping class $φ$ defines an associated veering triangulation $τ_φ$ of a punctured mapping torus. We show that generically, $τ_φ$ is not geometric. Here, the word "generic" can be taken either with respect to random walks in mapping class groups or with respect to counting geodesics in moduli space. Tools in the proof include Teichmüller theory, the Ending Lamination Theorem, study of the Thurston norm, and rigorous computation.
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.
Landing rays and ray Cannon-Thurston maps
本文构造了不存在Cannon-Thurston映射的双曲群对，将锥形极限点性质转化为Cannon-Thurston映射存在性判据，还给出了Baker-Riley反例的几何证明。
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.
Mutations and faces of the Thurston norm ball dynamically represented by multiple distinct flows
本文利用 veering 三角剖分定义组合突变，证明 Thurston 范数球的非纤维面可由相差一个 veering 突变的两个不同 pseudo-Anosov 流动态表示。
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…
Arbitrarily large veering triangulations with a vanishing taut polynomial
本文构造了一列四面体数目趋于无穷的倾斜三角剖分，其紧多项式消失，从而编码横截于环面的非圆阿诺索夫流。
Quasi-geodesics in the Cannon-Thurston metric
研究闭纤维3 - 流形上由伪阿诺索夫单值化纤维化产生的坎农 - 瑟斯顿度量，描述该度量中特定的拟测地线族，并用于获取典型测地线统计及度量类奇点几何准则。
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.
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…
Cohomology fractals, Cannon-Thurston maps, and the geodesic flow
本文通过光线追踪生成双曲三维流形的上同调分形图像，证明其与Cannon-Thurston映射的对应关系，并揭示其极限为无穷远球面上的分布。
We introduce a polynomial invariant $V_τ\in \mathbb{Z}[H_1(M)/\text{torsion}]$ associated to a veering triangulation $τ$ of a $3$-manifold $M$. In the special case where the triangulation is layered, i.e. comes from a fibration, $V_τ$ recovers the Teichmüller polynomial of the fibered faces canonically associated to $τ$. Via Dehn filling, this gives a combinatorial description of the Teichmüller polynomial for any hyperbolic fibered $3$-manifold. For a general veering triangulation $τ$, we show that the surfaces carried by $τ$ determine a cone in homology that is dual to its cone of positive c…
Non-existence of Cannon-Thurston maps for hierarchically hyperbolic groups
研究分层双曲群双曲正规子群的 Cannon - Thurston 映射不存在性，涵盖曲面映射类群正规子群等，还证明了自由循环群子群相关结果及 CAT(0)群可视边界与相对分层双曲边界同胚。
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…
Singularity of Cannon-Thurston maps
研究封闭纤维化双曲3-流形中坎农-瑟斯顿映射，利用测地线性质证明圆上多种自然测度经此映射到球面上会变奇异，通过对比双曲测地线靠近纤维时间比例得出结论，还对特定圆测度证明了有效结果。
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…
Cannon-Thurston Maps, Harmonic Maps, and Minimal Surfaces in Fibered Hyperbolic 3-Manifolds
本文研究闭曲面伪Anosov同胚映射环的纤维同伦类，将Cannon-Thurston边界映射与调和映射及最小面积极小曲面相联系，给出面积曲率界并分析调和能量函数。
The taut polynomial and the Alexander polynomial
本文证明转向三角剖分的绷紧多项式等于底层流形的扭结亚历山大多项式，并建立其与非扭结亚历山大多项式及 Dehn 填充后流形亚历山大多项式的关系，推广了 McMullen 定理并给出可定向纤维类存在的充要条件。
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 $Λ_H$
Limits of limit sets in rank-one symmetric spaces
本文研究一秩对称空间等距群几何有限子群序列的极限集连续性，证明在强收敛下极限集按豪斯多夫距离连续变化，并给出弱保型条件下 Cannon-Thurston 映射的一致收敛性。
Surjectivity of the Cannon--Thurston map in metric (graph) bundles
研究度量（图）丛中 Cannon--Thurston 映射的满射性，在纤维与非初等双曲群一致拟等距或纤维是一端双曲度量空间这两种情形下证明该映射满射，推广了相关定理并回答了问题。
Embeddings of trees of hyperbolic metric spaces and Cannon--Thurston maps
研究超几何群自由积的Cannon-Thurston映射存在性条件
We characterise completely when limit sets, as parametrised by Cannon-Thurston maps, move discontinuously for a sequence of algebraically convergent quasi-Fuchsian groups.
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.
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.
Exotic codimension one Anosov flows
本文在闭双曲三维流形上的某些圆丛中构造了外斯阿诺夫流，构造了Verjovsky猜想的反例，并利用伪外斯阿诺夫拟测地流相关的Cannon-Thurston映射，得到了某些四维流形上无穷多个轨道不等价的外斯阿诺夫流。
To each once-punctured-torus bundle, $T_ϕ$, over the circle with pseudo-Anosov monodromy $ϕ$, there are associated two tessellations of the complex plane: one, $Δ(ϕ)$, is (the projection from $\infty$ of) the triangulation of a horosphere at $\infty$ induced by the canonical decomposition into ideal tetrahedra, and the other, $CW(ϕ)$, is a fractal tessellation given by the Cannon-Thurston map of the fiber group switching back and forth between gray and white each time it passes through $\infty$. In this paper, we study the relation between $Δ(ϕ)$ and $CW(ϕ)$.
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.
Taut polynomials from finite quotients of fibered hyperbolic 3-manifold groups
本文证明，当单值映射为完全穿孔时，纤维化双曲3-流形群的有限商群能检测Thurston范数球的纤维化面的taut多项式，并发展了扭曲多变量Alexander多项式的profinite不变性的一般框架。
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