[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/cannon-thurston-map-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki21\/cannon-thurston-map-wikipedia\/","headline":"Cannon\u2013Thurston map &#8211; Wikipedia","name":"Cannon\u2013Thurston map &#8211; Wikipedia","description":"before-content-x4 In mathematics, a Cannon\u2013Thurston map is any of a number of continuous group-equivariant maps between the boundaries of two","datePublished":"2020-05-04","dateModified":"2020-05-04","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki21\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fe5cccb77beeaf68832504920e99657492a7545b","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fe5cccb77beeaf68832504920e99657492a7545b","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki21\/cannon-thurston-map-wikipedia\/","wordCount":21707,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4In mathematics, a Cannon\u2013Thurston map is any of a number of continuous group-equivariant maps between the boundaries of two hyperbolic metric spaces extending a discrete isometric actions of the group on those spaces.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4The notion originated from a seminal 1980s preprint of James Cannon and William Thurston &#8220;Group-invariant Peano curves&#8221; (eventually published in 2007) about fibered hyperbolic 3-manifolds.[1]Cannon\u2013Thurston maps provide many natural geometric examples of space-filling curves.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsHistory[edit]Kleinian representations of surface groups[edit]General Kleinian groups[edit]Existence and non-existence results[edit]Multiplicity of the Cannon\u2013Thurston map[edit]Generalizations, applications and related results[edit]References[edit]Further reading[edit]History[edit]The Cannon\u2013Thurston map first appeared in a mid-1980s preprint of James W. Cannon and William Thurston called &#8220;Group-invariant Peano curves&#8221;. The preprint remained unpublished until 2007,[1] but in the meantime had generated numerous follow-up works by other researchers.[2]In their paper Cannon and Thurston considered the following situation. Let M be a closed hyperbolic 3-manifold that fibers over the circle with fiber S. Then S itself is a closed hyperbolic surface, and its universal cover S~{displaystyle {tilde {S}}} can be identified with the hyperbolic plane        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4H2{displaystyle mathbb {H} ^{2}}. Similarly, the universal cover of M can be identified with the hyperbolic 3-space H3{displaystyle mathbb {H} ^{3}}. The inclusion S\u2286M{displaystyle Ssubseteq M} lifts to a \u03c01(S){displaystyle pi _{1}(S)}-invariant inclusion S~=H2\u2286H3=M~{displaystyle {tilde {S}}=mathbb {H} ^{2}subseteq mathbb {H} ^{3}={tilde {M}}}. This inclusion is highly distorted because the action of \u03c01(S){displaystyle pi _{1}(S)} onH3{displaystyle mathbb {H} ^{3}} is not geometrically finite.Nevertheless, Cannon and Thurston proved that this distorted inclusion  H2\u2286H3{displaystyle mathbb {H} ^{2}subseteq mathbb {H} ^{3}} extends to a continuous \u03c01(S){displaystyle pi _{1}(S)}-equivariant mapj:S1\u2192S2{displaystyle j:mathbb {S} ^{1}to mathbb {S} ^{2}},where S1=\u2202H2{displaystyle mathbb {S} ^{1}=partial mathbb {H} ^{2}} and S2=\u2202H3{displaystyle mathbb {S} ^{2}=partial mathbb {H} ^{3}}. Moreover, in this case the map j is surjective, so that it provides a continuous onto function from the circle onto the 2-sphere, that is, a space-filling curve.Cannon and Thurston also explicitly described the map j:S1\u2192S2{displaystyle j:mathbb {S} ^{1}to mathbb {S} ^{2}}, via collapsing stable and unstable laminations of the monodromy pseudo-Anosov homeomorphism of S for this fibration of M. In particular, this description implies that the map j is uniformly finite-to-one, with the pre-image of every point of S2{displaystyle mathbb {S} ^{2}} having cardinality at most 2g, where g is the genus of\u00a0S.After the paper of Cannon and Thurston generated a large amount of follow-up work, with other researchers analyzing the existence or non-existence of analogs of the map j in various other set-ups motivated by the Cannon\u2013Thurston result.Kleinian representations of surface groups[edit]The original example of Cannon and Thurston can be thought of in terms of Kleinian representations of the surface group H=\u03c01(S){displaystyle H=pi _{1}(S)}. As a subgroup of G=\u03c01(M){displaystyle G=pi _{1}(M)}, the group H acts on H3=M~{displaystyle mathbb {H} ^{3}={tilde {M}}} by isometries, and this action is properly discontinuous. Thus one gets a discrete representation \u03c1:H\u2192PSL(2,C)=Isom+\u2061(H3){displaystyle rho :Hto mathbb {P} SL(2,mathbb {C} )=operatorname {Isom} _{+}(mathbb {H} ^{3})}.The group H=\u03c01(S){displaystyle H=pi _{1}(S)} also acts by isometries, properly discontinuously and co-compactly, on the universal cover H2=S~{displaystyle mathbb {H} ^{2}={tilde {S}}}, with the limit set \u039bH\u2286\u2202H2=S1{displaystyle Lambda Hsubseteq partial H^{2}=mathbb {S} ^{1}} being equal to S1{displaystyle mathbb {S} ^{1}}. The Cannon\u2013Thurston result can be interpreted as saying that these actions of H on H2{displaystyle mathbb {H} ^{2}} and H3{displaystyle mathbb {H} ^{3}} induce a continuous H-equivariant map j:S1\u2192S2{displaystyle j:mathbb {S} ^{1}to mathbb {S} ^{2}}.One can ask, given a hyperbolic surface S and a discrete representation \u03c1:\u03c01(S)\u2192PSL(2,C){displaystyle rho :pi _{1}(S)to mathbb {P} SL(2,mathbb {C} )}, if there exists an induced continuous map j:\u039bH\u2192S2{displaystyle j:Lambda Hto mathbb {S} ^{2}}.For Kleinian representations of surface groups, the most general result in this direction is due to Mahan Mj (2014).[3]Let S be a complete connected finite volume hyperbolic surface. Thus S is a surface without boundary, with a finite (possibly empty) set of cusps. Then one still has H2=S~{displaystyle mathbb {H} ^{2}={tilde {S}}} and \u039b\u03c01(S)=S1{displaystyle Lambda pi _{1}(S)=mathbb {S} ^{1}} (even if S has some cusps). In this setting Mj[3] proved the following theorem:Let S be a complete connected finite volume hyperbolic surface and let H=\u03c01(S){displaystyle H=pi _{1}(S)}. Let \u03c1:H\u2192PSL(2,C){displaystyle rho :Hto mathbb {P} SL(2,mathbb {C} )} be a discrete faithful representation without accidental parabolics. Then \u03c1{displaystyle rho } induces a continuous H-equivariant map j:S1\u2192S2{displaystyle j:mathbb {S} ^{1}to mathbb {S} ^{2}}.Here the &#8220;without accidental parabolics&#8221; assumption means that for 1\u2260h\u2208H{displaystyle 1neq hin H}, the element \u03c1(h){displaystyle rho (h)} is a parabolic isometry of H3{displaystyle mathbb {H} ^{3}} if and only if h{displaystyle h} is a parabolic isometry of H2{displaystyle mathbb {H} ^{2}}.  One of important applications of this result is that in the above situation the limit set \u039b\u03c1(\u03c01(S))\u2286S2{displaystyle Lambda rho (pi _{1}(S))subseteq mathbb {S} ^{2}} is locally connected.This result of Mj was preceded by numerous other results in the same direction, such as Minsky (1994),[4] Alperin, Dicks and Porti (1999),[5]McMullen (2001),[6]Bowditch (2007)[7] and (2013),[8] Miyachi (2002),[9] Souto (2006),[10]Mj (2009),[11] (2011),[12] and others.In particular, Bowditch&#8217;s 2013 paper[8] introduced the notion of a &#8220;stack&#8221; of Gromov-hyperbolic metric spaces and developed an alternative framework to that of Mj for proving various results about Cannon\u2013Thurston maps.General Kleinian groups[edit]In a 2017 paper[13] Mj proved the existence of the Cannon\u2013Thurston map in the following setting:Let \u03c1:G\u2192PSL(2,C){displaystyle rho :Gto mathbb {P} SL(2,mathbb {C} )} be a discrete faithful representation where G is a word-hyperbolic group, and where \u03c1(G){displaystyle rho (G)} contains no parabolic isometries of H3{displaystyle mathbb {H} ^{3}}. Then \u03c1{displaystyle rho } induces a continuous G-equivariant map j:\u2202G\u2192S2{displaystyle j:partial Gto mathbb {S} ^{2}}, where \u2202G{displaystyle partial G} is the Gromov boundary of G, and where the image of j is the limit set of G in S2{displaystyle mathbb {S} ^{2}}.Here &#8220;induces&#8221; means that the map J:G\u222a\u2202G\u2192H3\u222aS2{displaystyle J:Gcup partial Gto mathbb {H} ^{3}cup mathbb {S} ^{2}} is continuous, where J|\u2202G=j{displaystyle J|_{partial G}=j} and J(g)=gx0,g\u2208G{displaystyle J(g)=gx_{0},gin G} (for some basepoint x0\u2208H3{displaystyle x_{0}in mathbb {H} ^{3}}). In the same paper Mj obtains a more general version of this result, allowing G to contain parabolics, under some extra technical assumptions on G. He also provided a description of the fibers of j in terms of ending laminations of H3\/G{displaystyle mathbb {H} ^{3}\/G}.Existence and non-existence results[edit]Let G be a word-hyperbolic group and let H\u00a0\u2264\u00a0G be a subgroup such that H is also word-hyperbolic. If the inclusion i:H\u00a0\u2192\u00a0G extends to a continuous map \u2202i: \u2202H \u2192 \u2202G between their hyperbolic boundaries, the map  \u2202i is called a Cannon\u2013Thurston map. Here &#8220;extends&#8221; means that the map between hyperbolic compactifications i^:H\u222a\u2202H\u2192G\u222a\u2202G{displaystyle {hat {i}}:Hcup partial Hto Gcup partial G}, given by i^|H=i,i^|\u2202H=\u2202i{displaystyle {hat {i}}|_{H}=i,{hat {i}}|_{partial H}=partial i}, is continuous. In this setting, if the map \u2202i exists, it is unique and H-equivarinat, and the image \u2202i(\u2202H) is equal to the limit set \u039b\u2202G(H){displaystyle Lambda _{partial G}(H)}.If H\u00a0\u2264\u00a0G is quasi-isometrically embedded (i.e. quasiconvex) subgroup, then the Cannon\u2013Thurston map \u2202i: \u2202H \u2192 \u2202G exists and is a topological embedding.However, it turns out that the Cannon\u2013Thurston map exists in many other situations as well.Mitra proved [14] that if G is word-hyperbolic and H\u00a0\u2264\u00a0G is a normal word-hyperbolic subgroup, then the Cannon\u2013Thurston map exists. (In this case if H and Q\u00a0=\u00a0G\/H are infinite then H is not quasiconvex in G.) The original Cannon\u2013Thurston theorem about fibered hyperbolic 3-manifolds is a special case of this result.If H\u00a0\u2264\u00a0G are two word-hyperbolic groups and H is normal in G then, by a result of Mosher,[15] the quotient group Q\u00a0=\u00a0G\/H is also word-hyperbolic. In this setting Mitra also described the fibers of the map \u2202i: \u2202H \u2192 \u2202G in terms of &#8220;algebraic ending laminations&#8221; on H, parameterized by the boundary points z\u00a0\u2208\u00a0\u2202Q.In another paper[16] Mitra considered the case where a word-hyperbolic group G splits as the fundamental group of a graph of groups, where all vertex and edge groups are word-hyperbolic, and the edge-monomorphisms are quasi-isometric embeddings. In this setting Mitra proved that for every vertex group Av{displaystyle A_{v}}, for the inclusion map i:Av\u2192G{displaystyle i:A_{v}to G} the Cannon\u2013Thurston map \u2202i:\u2202Av\u2192\u2202G{displaystyle partial i:partial A_{v}to partial G} does exist.By combining and iterating these constructions, Mitra produced[16] examples of hyperbolic subgroups of hyperbolic groups H\u00a0\u2264\u00a0G where the subgroup distortion of H in G is an arbitrarily high tower of exponentials, and the Cannon\u2013Thurston map \u2202i:\u2202H\u2192\u2202G{displaystyle partial i:partial Hto partial G} exists. Later Barker and Riley showed that one can arrange for H to have arbitrarily high primitive recursive distortion in G.[17]In a 2013 paper,[18] Baker and Riley constructed the first example of a word-hyperbolic group G and a word-hyperbolic (in fact free) subgroup H\u00a0\u2264\u00a0G such that the Cannon\u2013Thurston map \u2202i:\u2202H\u2192\u2202G{displaystyle partial i:partial Hto partial G} does not exist.Later Matsuda and Oguni generalized the Baker\u2013Riley approach and showed that every non-elementary word-hyperbolic group H can be embedded in some word-hyperbolic group G in such a way that the Cannon\u2013Thurston map \u2202i:\u2202H\u2192\u2202G{displaystyle partial i:partial Hto partial G} does not exist.[19]Multiplicity of the Cannon\u2013Thurston map[edit]As noted above, if H is a quasi-isometrically embedded subgroup of a word-hyperbolic group G, then H is word-hyperbolic, and the Cannon\u2013Thurston map \u2202i:\u2202H\u2192\u2202G{displaystyle partial i:partial Hto partial G} exists and is injective. Moreover, it is known that the converse is also true: If H is a word-hyperbolic subgroup of a word-hyperbolic group G such that the Cannon\u2013Thurston map \u2202i:\u2202H\u2192\u2202G{displaystyle partial i:partial Hto partial G} exists and is injective, then H is uasi-isometrically embedded in G.[20]It is known, for more general convergence groups reasons, that if H is a word-hyperbolic subgroup of a word-hyperbolic group G such that the Cannon\u2013Thurston map \u2202i:\u2202H\u2192\u2202G{displaystyle partial i:partial Hto partial G} exists then for every concical limit point for H in \u2202G{displaystyle partial G} has exactly one pre-image under \u2202i{displaystyle partial i}.[21] However, the converse fails: If  \u2202i:\u2202H\u2192\u2202G{displaystyle partial i:partial Hto partial G} exists and is non-injective, then there always exists a non-conical limit point of H in \u2202G with exactly one preimage under \u2202i.[20]It the context of the original Cannon\u2013Thurston paper, and for many generalizations for the Kleinin representations \u03c1:\u03c01(S)\u2192PSL(2,C),{displaystyle rho :pi _{1}(S)to mathbb {P} SL(2,mathbb {C} ),} the Cannon\u2013Thurston map j:S1\u2192S2{displaystyle j:mathbb {S} ^{1}to mathbb {S} ^{2}} is known to be uniformly finite-to-one.[13] That means that for every point p\u2208S2{displaystyle pin mathbb {S} ^{2}}, the full pre-image j\u22121(p){displaystyle j^{-1}(p)} is a finite set with cardinality bounded by a constant depending only on S.[22]In general, it is known, as a consequence of the JSJ-decomposition theory for word-hyperbolic groups, that if 1\u2192H\u2192G\u2192Q\u21921{displaystyle 1to Hto Gto Qto 1} is a short exact sequence of three infinite torsion-free word-hyperbolic groups, then H is isomorphic to a free product of some closed surface groups and of a free group.If H=\u03c01(S){displaystyle H=pi _{1}(S)} is the fundamental group of a closed hyperbolic surface S, such hyperbolic extensions of H are described by the theory of &#8220;convex cocompact&#8221; subgroups of the mapping class group Mod(S). Every subgroup \u0393\u00a0\u2264\u00a0Mod(S) determines, via the Birman short exact sequence, an extension1\u2192H\u2192E\u0393\u2192\u0393\u21921{displaystyle 1to Hto E_{Gamma }to Gamma to 1}Moreover, the group E\u0393{displaystyle E_{Gamma }} is word-hyperbolic if and only if \u0393\u00a0\u2264\u00a0Mod(S) is convex-cocompact.In this case, by Mitra&#8217;s general result, the Cannon\u2013Thurston map \u2202i:\u2202H\u00a0\u2192\u00a0\u2202E\u0393 does exist. The fibers of the map \u2202i are described by a collection of ending laminations on S determined by\u00a0\u0393. This description implies that map \u2202i is uniformly finite-to-one.If \u0393{displaystyle Gamma } is a convex-cocompact purely atoroidal subgroup of Out\u2061(Fn){displaystyle operatorname {Out} (F_{n})} (where n\u22653{displaystyle ngeq 3}) then for the corresponding extension 1\u2192Fn\u2192E\u0393\u2192\u0393\u21921{displaystyle 1to F_{n}to E_{Gamma }to Gamma to 1} the group E\u0393{displaystyle E_{Gamma }} is word-hyperbolic. In this setting Dowdall, Kapovich and Taylor proved[23] that the Cannon\u2013Thurston map \u2202i:\u2202Fn\u2192\u2202E\u0393{displaystyle partial i:partial F_{n}to partial E_{Gamma }} is uniformly finite-to-one, with point preimages having cardinality \u22642n{displaystyle leq 2n}. This result was first proved by Kapovich and Lustig[24] under the extra assumption that \u0393{displaystyle Gamma } is infinite cyclic, that is, that \u0393{displaystyle Gamma } is generated by an autoroidal fully irreducible element of Out\u2061(Fn){displaystyle operatorname {Out} (F_{n})}.Ghosh proved that for an arbitrary atoroidal \u03d5\u2208Out\u2061(Fn){displaystyle phi in operatorname {Out} (F_{n})} (without requiring \u0393=\u27e8\u03d5\u27e9{displaystyle Gamma =langle phi rangle } to be convex cocompact) the Cannon\u2013Thurston map \u2202i:\u2202Fn\u2192\u2202E\u0393{displaystyle partial i:partial F_{n}to partial E_{Gamma }} is uniformly finite-to-one, with a bound on the cardinality of point preimages depending only on n.[25] (However, Ghosh&#8217;s result does not provide an explicit bound in terms of n, and it is still unknown if the 2n bound always holds in this case.)It remains unknown, whenever H is a word-hyperbolic subgroup of a word-hyperbolic group G such that the Cannon\u2013Thurston map \u2202i:\u2202H\u2192\u2202G{displaystyle partial i:partial Hto partial G} exists, if the map  \u2202i{displaystyle partial i} is finite-to-one.However, it is known that in this setting for every p\u2208\u039b\u2202GH{displaystyle pin Lambda _{partial G}H} such that p is a conical limit point, the set (\u2202i)\u22121(p){displaystyle (partial i)^{-1}(p)} has cardinality 1.Generalizations, applications and related results[edit]As an application of the result about the existence of Cannon\u2013Thurston maps for Kleinian surface group representations, Mj proved[3] that if \u0393\u2264PSL(2,C){displaystyle Gamma leq mathbb {P} SL(2,mathbb {C} )} is a finitely generated Kleinian group such that the limit set \u039b\u2286\u2202H3{displaystyle Lambda subseteq partial mathbb {H} ^{3}} is connected, then \u039b{displaystyle Lambda } is locally connected.Leininger, Mj and Schleimer,[26] given a closed hyperbolic surface S, constructed a &#8216;universal&#8217; Cannon\u2013Thurston map from a subset of \u2202\u03c01(S)=S1{displaystyle partial pi _{1}(S)=mathbb {S} ^{1}} to the boundary \u2202C(S,z){displaystyle partial {mathcal {C}}(S,z)} of the curve complex of S with one puncture, such that this map, in a precise sense, encodes all the Cannon\u2013Thurston maps corresponding to arbitrary ending laminations on S. As an application, they prove that \u2202C(S,z){displaystyle partial {mathcal {C}}(S,z)} is path-connected and locally path-connected.Leininger, Long and Reid[27] used Cannon\u2013Thurston maps to show that any finitely generated torsion-free nonfree Kleinian group  with limit set equal to S2{displaystyle mathbb {S} ^{2}}, which is not a lattice and contains no parabolic elements, has discrete commensurator in PSL(2,C){displaystyle mathbb {P} SL(2,mathbb {C} )}.Jeon and Ohshika[28] used Cannon\u2013Thurston maps to establish measurable rigidity for Kleinian groups.Inclusions of relatively hyperbolic groups as subgroups of other relatively hyperbolic groups in many instances also induce equivariant continuous maps between their Bowditch boundaries; such maps are also referred to as Cannon\u2013Thurston maps.[3][29][30][19]More generally, if G is a group acting as a discrete convergence group on two metrizable compacta M and Z, a continuous G-equivariant map M\u00a0\u2192\u00a0Z (if such a map exists) is also referred to as a Cannon\u2013Thurston map. Of particular interest in this setting is the case where G is word-hyperbolic and M\u00a0=\u00a0\u2202G is the hyperbolic boundary of G, or where G is relatively hyperbolic and M\u00a0=\u00a0\u2202G is the Bowditch boundary of G.[20]Mj and Pal[29] obtained a generalization of Mitra&#8217;s earlier result for graphs of groups to the relatively hyperbolic context.Pal [30] obtained a generalization of Mitra&#8217;s earlier result, about the existence of the Cannon\u2013Thurston map for short exact sequences of word-hyperbolic groups, to relatively hyperbolic contex.Mj and Rafi [31] used the Cannon\u2013Thurston map to study which subgroups are quasiconvex in extensions of free groups and surface groups by convex cocompact subgroups of Out\u2061t(Fn){displaystyle operatorname {Out} t(F_{n})} and of mapping class groups.References[edit]^ a b James W. Cannon; William P. Thurston (2007). &#8220;Group invariant Peano curves&#8221;. Geometry &#038; Topology. 11 (3): 1315\u20131356. doi:10.2140\/gt.2007.11.1315. MR\u00a02326947.^ Darryl McCullough, MR2326947 (2008i:57016), Mathematical Reviews, Review of: J. W. Cannon and W. P. Thurston, Group invariant Peano curves, Geom. Topol. 11 (2007), 1315\u20131355; &#8216;This influential paper dates from the mid-1980&#8217;s. Indeed, preprint versions are referenced in more than 30 published articles, going back as early as 1990.&#8217;^ a b c d Mahan Mj (2014). &#8220;Cannon\u2013Thurston maps for surface groups&#8221;. Annals of Mathematics. 179 (1): 1\u201380. arXiv:math\/0607509. doi:10.4007\/annals.2014.179.1.1. MR\u00a03126566. S2CID\u00a0119160004.^ Yair Minsky (1994). &#8220;On rigidity, limit sets, and end invariants of hyperbolic 3-manifolds&#8221; (PDF). Journal of the American Mathematical Society. 7 (3): 539\u2013588. doi:10.2307\/2152785. JSTOR\u00a02152785. MR\u00a01257060.^ Roger C. Alperin; Warren Dicks; Joan Porti (1999). &#8220;The boundary of the Gieseking tree in hyperbolic three-space&#8221;. Topology and Its Applications. 93 (3): 219\u2013259. doi:10.1016\/S0166-8641(97)00270-8. MR\u00a01688476.^ Curtis T. McMullen (2001). &#8220;Local connectivity, Kleinian groups and geodesics on the blowup of the torus&#8221;. Inventiones Mathematicae. 146 (1): 35\u201391. Bibcode:2001InMat.146&#8230;35M. doi:10.1007\/PL00005809. MR\u00a01859018.^ Brian H. Bowditch (2007). &#8220;The Cannon\u2013Thurston map for punctured-surface groups&#8221;. Mathematische Zeitschrift. 255: 35\u201376. doi:10.1007\/s00209-006-0012-4. MR\u00a02262721.^ a b Brian H. Bowditch (2013). &#8220;Stacks of hyperbolic spaces and ends of 3-manifolds&#8221;.  In Craig D. Hodgson; William H. Jaco; Martin G. Scharlemann; Stephan Tillmann (eds.). Geometry and topology down under. Contemporary Mathematics, 597. American Mathematical Society. pp.\u00a065\u2013138. 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Sci. 120 (1): 57\u201368. doi:10.1007\/s12044-010-0009-0. MR\u00a02654898. S2CID\u00a016597989.^ Mahan Mj; Kasra Rafi (2018). &#8220;Algebraic ending laminations and quasiconvexity&#8221;. Algebraic and Geometric Topology. 18 (4): 1883\u20131916. arXiv:1506.08036. doi:10.2140\/agt.2018.18.1883. MR\u00a03797060. S2CID\u00a092985011.Further reading[edit]     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/cannon-thurston-map-wikipedia\/#breadcrumbitem","name":"Cannon\u2013Thurston map &#8211; Wikipedia"}}]}]