Mahan Mj
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.
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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.
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 …
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.
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。
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.
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。
(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 …
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。
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.
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.
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。
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…
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。
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.
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。
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。
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 study Thurston's Lipschitz and curve metrics, as well as the arc metric on the Teichmueller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston's stretch maps and prove the following: (1) On the Teichmueller space of the torus with one boundary component, the Lipschitz and the curve metrics coincide and define a geodesic metric on this space. (2) On the same space, the arc and the curve metrics coincide when the length o…
Let $S_{1}$ and $S_{2}$ be connected orientable surfaces of genus $g_{1}, g_{2} \geq 3$, $n_{1},n_{2} \geq 0$ punctures, and empty boundary. Let also $φ: \mathcal{HT}(S_{1}) \rightarrow \mathcal{HT}(S_{2})$ be an edge-preserving alternating map between their Hatcher-Thurston graphs. We prove that $g_{1} \leq g_{2}$ and that there is also a multicurve of cardinality $g_{2} - g_{1}$ contained in every element of the image. We also prove that if $n_{1} = 0$ and $g_{1} = g_{2}$, then the map $\widetildeφ$ obtained by filling the punctures of $S_{2}$, is induced by a homeomorphism of $S_{1}$.
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.
Finite and infinite degree Thurston maps with a small postsingular set
本文研究拓扑球面上除单个本性奇点外处处有定义的Thurston映射，针对具有四个后临界值的广泛情形建立了W. Thurston刻画定理的类似物，并通过分析一维Teichmüller空间上的拉回映射推导出相应Hurwitz类的性质。
Wolf gave a homeomorphism from the Teichmüller space to the space of quadratic differentials on a closed Riemann surface by using harmonic maps. Moreover, using harmonic maps rays, he gave a compactification of the Teichmüller space and show that it coincides with the Thurston compactification. In this paper, we extend the harmonic maps compactification to the Teichmüller spaces of punctured Riemann surfaces, and show that it still coincides with the Thurston compactification.
In the Teichmüller space of a hyperbolic surface of finite type, we construct geodesic lines for Thurston's asymmetric metric having the property that when they are traversed in the reverse direction, they are also geodesic lines (up to reparametrization). The lines we construct are special stretch lines in the sense of Thurston. They are directed by complete geodesic laminations that are not chain-recurrent, and they have a nice description in terms of Fenchel-Nielsen coordinates. At the basis of the construction are certain maps with controlled Lipschitz constants between right-angled hyperb…
Towards Nielsen-Thurston classification for surfaces of infinite type: well-tempered homeomorphisms
该研究针对无穷型曲面，引入良态映射类并证明其满足Nielsen-Thurston型分类定理，给出曲面的典范不变子曲面分解，首次回归为周期或平移。
Thurston's Theorem: Entropy in Dimension One
本文现代化并补全 Thurston 关于自由群外自同构遍历 traintrack 代表元拓扑熵与弱 Perron 数等价性的证明，尤其补上遍历性证明。
A theory of transversely oriented spun-normal immersed surfaces in ideally triangulated 3--manifolds is developed in this paper, including linear functionals determining the boundary curves, Euler characteristic and homology class of these immersions. This is used to develop and implement an algorithm to compute the unit ball of the Thurston norm for cusped hyperbolic 3--manifolds of finite volume. As an application of independent interest, we give an upper bound on the minimal entropy of pseudo-Anosov maps of surfaces with number of cusps bounded linearly in genus.
Associated to a Thurston map $f: S^2 \to S^2$ with postcritical set $P$ are several different invariants obtained via pullback: a relation on the set of free homotopy classes of curves in $S^2- P$, a linear operator on the free $\R$-module generated by these homotopy classes of curves, a virtual endomorphism on the pure mapping class group, an analytic self-map of an associated Teichmueller space, and an analytic self-correspondence on an associated moduli space. Viewing all of these objects as invariants of $f$, we investigate harmonious relationships between their properties.
We obtain an ordering of closed aspherical 4-manifolds that carry a non-hyperbolic Thurston geometry. As application, we derive that the Kodaira dimension of geometric 4-manifolds is monotone with respect to the existence of maps of non-zero degree.
In this paper, we develop a way to extract information about a random walk associated with a typical Thurston's construction. We first observe that a typical Thurston's construction entails a free group of rank 2. We also present a proof of the spectral theorem for random walks associated with Thurston's construction that have finite second moment with respect to the Teichmüller metric. Its general case was remarked by Dahmani and Horbez. Finally, under a hypothesis not involving moment conditions, we prove that random walks eventually become pseudo-Anosov. As an application, we first discuss …
Thurston related $\mathbb{C}{\rm P}^1$-structures (complex projective structures) and equivariant pleated surfaces in the hyperbolic-three space $\mathbb{H}^3$, in order to give a parameterization of the deformation space of $\mathbb{C}{\rm P}^1$-structures. In this note, we summarize Thurston's parametrization of $\mathbb{C}{\rm P}^1$-structures, based on Kamishima-Tan and Kulkani-Pinkall. We, in addition, give independent proofs for the following well-known theorems on $\mathbb{C}{\rm P}^1$-structures by means of pleated surfaces given by the parameterization. (1) Goldman's Theorem on $\math…
A Thurston map is a branched covering map from $§^2$ to $§^2$ with a finite postcritical set. We associate a natural Gromov hyperbolic graph $\G=\G(f,\mathcal C)$ with an expanding Thurston map $f$ and a Jordan curve $\mathcal C$ on $§^2$ containing $\post(f)$. The boundary at infinity of $\G$ with associated visual metrics can be identified with $§^2$ equipped with the visual metric induced by the expanding Thurston map $f$. We define asymptotic upper curvature of an expanding Thurston map $f$ to be the asymptotic upper curvature of the associated Gromov hyperbolic graph, and establish a conn…
Let $R$ be a compact, connected, orientable surface of genus $g$ with $n$ boundary components with $g \geq 2$, $n \geq 0$. Let $\mathcal{N}(R)$ be the nonseparating curve graph, $\mathcal{C}(R)$ be the curve graph and $\mathcal{HT}(R)$ be the Hatcher-Thurston graph of $R$. We prove that if $λ: \mathcal{N}(R) \rightarrow\mathcal{N}(R)$ is an edge-preserving map, then $λ$ is induced by a homeomorphism of $R$. We prove that if $θ: \mathcal{C}(R) \rightarrow \mathcal{C}(R)$ is an edge-preserving map, then $θ$ is induced by a homeomorphism of $R$. We prove that if $R$ is closed and $τ: \mathcal{HT}…
We establish basic geometric and topological properties of Thurston's Master Teapot and the Thurston set for superattracting unimodal self-maps of intervals. In particular, the Master Teapot is connected, contains the unit cylinder, and its intersection with a set $\mathbb{D} \times \{c\}$ grows monotonically with $c$. We show that the Thurston set described above is not equal to the Thurston set for postcritically finite tent maps, and we provide an arithmetic explanation for why certain gaps appear in plots of finite approximations of the Thurston set.
Thurston obtained a classification of individual surface homeomorphisms via the dynamics of the corresponding mapping class elements on Teichmüller space. In this paper we present certain extended versions of this, first, to random products of homeomorphisms and second, to holomorphic self-maps of Teichmüller spaces.
Let $M^3$ be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, $F_{best}$, of a \emph{harmonic} map $f: M^3 \to S^1$ with Morse-type singularities delivers the Thurston norm $χ_-([F_{best}])$ of its homology class $[F_{best}] \in H_2(M^3; \Z)$. In particular, for a map $f$ with connected fibers and any well-positioned oriented surface $Σ\subset M$ in the homology class of a fiber, we show that the Thurston number $χ_-(Σ)$ satisfies an inequality $$χ_-(Σ) \geq χ_-(F_{best}) - ρ^\circ(Σ, f)\cdot Var_{χ_-}(f).$$ Here the variation $Var_{χ_-}(f)$ i…
Infinite chains of perfect fits for expanding Thurston maps
该研究证明多项式配对中完美拟合无限链不存在的结论是全纯现象，构造了含周期临界轨道的组合扩张Thurston映射，其层叠结构存在无限多完美拟合无限链，且无法由扩张有理映射实现。
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