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In [[mathematics]], a '''Cannon–Thurston map''' is any of a number of continuous group-equivariant maps between the boundaries of two [[hyperbolic metric space]]s extending a discrete isometric actions of the group on those spaces.

The notion originated from a seminal 1980s preprint of [[James W. Cannon|James Cannon]] and [[William Thurston]] "Group-invariant Peano curves" (eventually published in 2007) about fibered hyperbolic 3-manifolds.<ref name="CT07">{{cite journal |author1=James W. Cannon |author2=William P. Thurston| title=Group invariant Peano curves|journal=[[Geometry & Topology]]|volume=11|year=2007|issue=3|pages=1315–1356|doi=10.2140/gt.2007.11.1315|mr=2326947|doi-access=free}}</ref>

Cannon–Thurston maps provide many natural geometric examples of [[space-filling curves]].

==History==

The Cannon–Thurston map first appeared in a mid-1980s preprint of [[James W. Cannon]] and [[William Thurston]] called "Group-invariant Peano curves". The preprint remained unpublished until 2007,<ref name="CT07"/> but in the meantime had generated numerous follow-up works by other researchers.<ref>Darryl McCullough, MR2326947 (2008i:57016), [[Mathematical Reviews]], Review of: J. W. Cannon and W. P. Thurston, Group invariant Peano curves, Geom. Topol. '''11''' (2007), 1315–1355; 'This influential paper dates from the mid-1980's. Indeed, preprint versions are referenced in more than 30 published articles, going back as early as 1990.'</ref>

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]] <math>\tilde S</math> can be identified with the [[hyperbolic plane]] <math>\mathbb H^2</math>. Similarly, the universal cover of ''M'' can be identified with the [[hyperbolic space|hyperbolic 3-space]] <math>\mathbb H^3</math>. The inclusion <math> S\subseteq M</math> lifts to a <math>\pi_1(S)</math>-invariant inclusion <math>\tilde S=\mathbb H^2\subseteq \mathbb H^3=\tilde M</math>. This inclusion is highly distorted because the action of <math>\pi_1(S)</math> on 
<math>\mathbb H^3</math> is not [[geometrically finite group|geometrically finite]].

Nevertheless, Cannon and Thurston proved that this distorted inclusion  <math>\mathbb H^2\subseteq \mathbb H^3</math> extends to a continuous <math>\pi_1(S)</math>-equivariant map 
:<math>j:\mathbb S^1\to \mathbb S^2</math>, 
where <math>\mathbb S^1=\partial \mathbb H^2</math> and <math>\mathbb S^2=\partial \mathbb H^3</math>. 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 <math>j:\mathbb S^1\to \mathbb S^2</math>, via collapsing stable and unstable laminations of the monodromy [[Pseudo-Anosov map|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 <math>\mathbb S^2</math> having cardinality at most 2''g'', where ''g'' is the genus of&nbsp;''S''.

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–Thurston result.

==Cannon–Thurston maps and [[Kleinian group]]s==

===Kleinian representations of surface groups===

The original example of Cannon and Thurston can be thought of in terms of [[Kleinian group|Kleinian]] representations of the [[surface group]] <math>H=\pi_1(S)</math>. As a subgroup of <math>G=\pi_1(M)</math>, the group ''H'' acts on <math>\mathbb H^3=\tilde M</math> by isometries, and this action is [[properly discontinuous]]. Thus one gets a discrete representation <math> \rho:H\to \mathbb PSL(2,\mathbb C)=\operatorname{Isom}_+(\mathbb H^3)</math>.

The group <math>H=\pi_1(S)</math> also acts by isometries, properly discontinuously and co-compactly, on the universal cover <math>\mathbb H^2=\tilde S</math>, with the [[limit set]] <math>\Lambda H\subseteq \partial H^2=\mathbb S^1</math> being equal to <math>\mathbb S^1</math>. The Cannon–Thurston result can be interpreted as saying that these actions of ''H'' on <math>\mathbb H^2</math> and <math>\mathbb H^3</math> induce a continuous ''H''-equivariant map <math>j:\mathbb S^1\to \mathbb S^2</math>.

One can ask, given a hyperbolic surface ''S'' and a discrete representation <math> \rho:\pi_1(S)\to \mathbb PSL(2,\mathbb C)</math>, if there exists an induced continuous map <math>j:\Lambda H\to \mathbb S^2</math>.

For Kleinian representations of surface groups, the most general result in this direction is due to [[Mahan Mj]] (2014).<ref name=Mj14>{{cite journal|title=Cannon–Thurston maps for surface groups|author=Mahan Mj|journal=[[Annals of Mathematics]]|volume=179|issue=1|year=2014|pages=1–80|mr=3126566|doi=10.4007/annals.2014.179.1.1|arxiv=math/0607509|s2cid=119160004 }}</ref>
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 <math>\mathbb H^2=\tilde S</math> and <math>\Lambda \pi_1(S)=\mathbb S^1</math> (even if ''S'' has some cusps). In this setting Mj<ref name=Mj14/> proved the following theorem:

:Let ''S'' be a complete connected finite volume hyperbolic surface and let <math>H=\pi_1(S)</math>. Let <math> \rho: H\to \mathbb PSL(2,\mathbb C)</math> be a discrete faithful representation without accidental parabolics. Then <math>\rho</math> induces a continuous ''H''-equivariant map <math>j:\mathbb S^1\to \mathbb S^2</math>.

Here the "without accidental parabolics" assumption means that for <math>1\ne h\in H</math>, the element <math>\rho(h)</math> is a parabolic isometry of <math>\mathbb H^3</math> if and only if <math>h</math> is a parabolic isometry of <math>\mathbb H^2</math>.  One of important applications of this result is that in the above situation the limit set <math>\Lambda \rho(\pi_1(S))\subseteq \mathbb S^2</math> is locally connected.

This result of Mj was preceded by numerous other results in the same direction, such as [[Yair Minsky|Minsky]] (1994),<ref>{{cite journal|author=Yair Minsky|title=On rigidity, limit sets, and end invariants of hyperbolic 3-manifolds|journal=[[Journal of the American Mathematical Society]]|volume=7|year=1994|issue=3|pages=539–588|mr=1257060|doi=10.2307/2152785|jstor=2152785|url=https://www.ams.org/journals/jams/1994-07-03/S0894-0347-1994-1257060-3/S0894-0347-1994-1257060-3.pdf}}</ref> Alperin, Dicks and Porti (1999),<ref>{{cite journal |author1=Roger C. Alperin |author2=Warren Dicks |author3=Joan Porti|title=The boundary of the Gieseking tree in hyperbolic three-space|journal=[[Topology and Its Applications]]|volume=93|year=1999|issue=3 |pages=219–259|doi=10.1016/S0166-8641(97)00270-8 |mr=1688476|doi-access=free}}</ref> [[Curtis McMullen|McMullen]] (2001),<ref>{{cite journal|author=Curtis T. McMullen|title=Local connectivity, Kleinian groups and geodesics on the blowup of the torus|journal=[[Inventiones Mathematicae]]|volume=146|year=2001|issue=1|pages=35–91|doi=10.1007/PL00005809|bibcode=2001InMat.146...35M|mr=1859018|url=http://nrs.harvard.edu/urn-3:HUL.InstRepos:3637162|doi-access=free}}</ref> [[Brian Bowditch|Bowditch]] (2007)<ref>{{cite journal|author=Brian H. Bowditch|title=The Cannon–Thurston map for punctured-surface groups|journal=[[Mathematische Zeitschrift]]|volume=255|year=2007|pages=35–76|mr=2262721|doi=10.1007/s00209-006-0012-4|doi-access=free}}</ref> and (2013),<ref name=Bow13>{{cite book |author=Brian H. Bowditch |editor1=Craig D. Hodgson |editor2=William H. Jaco |editor3=Martin G. Scharlemann |editor4=Stephan Tillmann |title=Geometry and topology down under |publisher= Contemporary Mathematics, 597. American Mathematical Society |date=2013 |pages=65–138 |chapter=Stacks of hyperbolic spaces and ends of 3-manifolds|isbn=978-0-8218-8480-5 }}</ref> Miyachi (2002),<ref>Hideki Miyachi, ''Semiconjugacies between actions of topologically tame Kleinian groups'', 2002, preprint</ref> Souto (2006),<ref>{{cite web|author=Juan Souto|title=Cannon–Thurston maps for thick free groups|year=2006|work=Preprint|url=https://www.researchgate.net/publication/255606697}}</ref> [[Mahan Mj|Mj]] (2009),<ref>{{cite journal|author=Mahan Mj|title=Cannon–Thurston maps for pared manifolds of bounded geometry|journal=[[Geometry & Topology]]|volume=13|year=2009|pages=89–245|mr=2469517}}</ref> (2011),<ref>{{cite book|author=Mahan Mj|title=Actes du Séminaire de Théorie Spectrale et Géometrie. Volume 28. Année 2009–2010|chapter=Cannon–Thurston maps, i-bounded geometry and a theorem of McMullen|series=Seminar on Spectral Theory and Geometry, vol. 28|publisher=Univ. Grenoble I|year=2011}}</ref> and others.
In particular, Bowditch's 2013 paper<ref name=Bow13/> introduced the notion of a "stack" of Gromov-hyperbolic metric spaces and developed an alternative framework to that of Mj for proving various results about Cannon–Thurston maps.

===General Kleinian groups===

In a 2017 paper<ref name=Mj17>{{cite journal|title=Cannon–Thurston maps for Kleinian groups|author=Mahan Mj|journal=[[Forum of Mathematics|Forum of Mathematics, Pi]]|volume=5|year=2017|mr=3652816|doi=10.1017/fmp.2017.2|url=https://www.cambridge.org/core/services/aop-cambridge-core/content/view/15B4CAB9479FE0422A74D83AC8FED4D1/S2050508617000026a.pdf/cannonthurston_maps_for_kleinian_groups.pdf|doi-access=free}}</ref> Mj proved the existence of the Cannon–Thurston map in the following setting:

:Let <math>\rho:G\to \mathbb PSL(2,\mathbb C) </math> be a discrete faithful representation where ''G'' is a [[word-hyperbolic group]], and where <math>\rho(G)</math> contains no parabolic isometries of <math>\mathbb H^3</math>. Then <math>\rho</math> induces a continuous ''G''-equivariant map <math>j:\partial G\to \mathbb S^2</math>, where <math>\partial G</math> is the [[Gromov boundary]] of ''G'', and where the image of ''j'' is the [[limit set]] of ''G'' in <math>\mathbb S^2</math>.

Here "induces" means that the map <math>J: G\cup \partial G\to \mathbb H^3 \cup \mathbb S^2</math> is continuous, where <math>J|_{\partial G}=j</math> and <math>J(g)=gx_0, g\in G</math> (for some basepoint <math>x_0\in \mathbb H^3</math>). 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 lamination theorem|ending laminations]] of <math>\mathbb H^3/G</math>.

==Cannon–Thurston maps and [[word-hyperbolic group]]s==

===Existence and non-existence results===

Let ''G'' be a [[word-hyperbolic group]] and let ''H''&nbsp;≤&nbsp;''G'' be a subgroup such that ''H'' is also word-hyperbolic. If the inclusion ''i'':''H''&nbsp;→&nbsp;''G'' extends to a continuous map ''∂i'': ''∂H'' → ''∂G'' between their hyperbolic boundaries, the map  ''∂i'' is called a '''Cannon–Thurston map'''. Here "extends" means that the map between hyperbolic compactifications <math>\hat i: H\cup \partial H\to G\cup \partial G</math>, given by <math>\hat i|_H=i, \hat i|_{\partial H}=\partial i</math>, is continuous. In this setting, if the map ''∂i'' exists, it is unique and ''H''-equivariant, and the image ''∂i''(''∂H'') is equal to the [[limit set]] <math>\Lambda_{\partial G}(H)</math>.

If ''H''&nbsp;≤&nbsp;''G'' is [[Quasi-isometry#Definition|quasi-isometrically embedded]] (i.e. quasiconvex) subgroup, then the Cannon–Thurston map ''∂i'': ''∂H'' → ''∂G'' exists and is a [[topological embedding]].
However, it turns out that the Cannon–Thurston map exists in many other situations as well.

Mitra proved <ref>{{cite journal|mr=1604882|author=Mahan Mitra|title=Cannon–Thurston maps for hyperbolic group extensions|journal=[[Topology (journal)|Topology]]|volume=37|year=1998|issue=3|pages=527–538|doi=10.1016/S0040-9383(97)00036-0|doi-access=free}}</ref> that if ''G'' is word-hyperbolic and ''H''&nbsp;≤&nbsp;''G'' is a [[normal subgroup|normal]] word-hyperbolic subgroup, then the Cannon–Thurston map exists. (In this case if ''H'' and ''Q''&nbsp;=&nbsp;''G''/''H'' are infinite then ''H'' is not quasiconvex in ''G''.) The original Cannon–Thurston theorem about fibered hyperbolic 3-manifolds is a special case of this result.

If ''H''&nbsp;≤&nbsp;''G'' are two word-hyperbolic groups and ''H'' is normal in ''G'' then, by a result of Mosher,<ref>{{cite journal|mr=1443845|author=Lee Mosher|title=A hyperbolic-by-hyperbolic hyperbolic group|journal=[[Proceedings of the American Mathematical Society]]|volume=125|year=1997|issue=12|pages=3447–3455|doi=10.1090/S0002-9939-97-04249-4|url=https://www.ams.org/journals/proc/1997-125-12/S0002-9939-97-04249-4/S0002-9939-97-04249-4.pdf|doi-access=free}}</ref> the quotient group ''Q''&nbsp;=&nbsp;''G''/''H'' is also word-hyperbolic. In this setting Mitra also described the fibers of the map ''∂i'': ''∂H'' → ''∂G'' in terms of "algebraic ending laminations" on ''H'', parameterized by the boundary points ''z''&nbsp;∈&nbsp;''∂Q''.

In another paper<ref name=Mitra98>{{cite journal|author=Mahan Mitra, Mahan|title=Cannon–Thurston maps for trees of hyperbolic metric spaces.|journal=[[Journal of Differential Geometry]]|volume=48|year=1998|issue=1|pages=135–164|doi=10.4310/jdg/1214460609|mr=1622603|url=https://projecteuclid.org/download/pdf_1/euclid.jdg/1214460609|doi-access=free|arxiv=math/9609209}}</ref> Mitra considered the case where a word-hyperbolic group ''G'' splits as the [[graph of groups|fundamental group of a graph of groups]], where all vertex and edge groups are word-hyperbolic, and the edge-monomorphisms are [[Quasi-isometry#Definition|quasi-isometric embeddings]]. In this setting Mitra proved that for every vertex group <math>A_v</math>, for the inclusion map <math>i:A_v\to G</math> the Cannon–Thurston map <math>\partial i:\partial A_v\to \partial G</math> does exist.

By combining and iterating these constructions, Mitra produced<ref name=Mitra98/> examples of hyperbolic subgroups of hyperbolic groups ''H''&nbsp;≤&nbsp;''G'' where the [[subgroup distortion]] of ''H'' in ''G'' is an arbitrarily high tower of exponentials, and the Cannon–Thurston map <math>\partial i:\partial H\to \partial G</math> exists. Later Barker and Riley showed that one can arrange for ''H'' to have arbitrarily high [[primitive recursive]] distortion in ''G''.<ref>{{cite journal|mr=4077662 |author1=Owen Baker |author2=Timothy R. Riley|title=Cannon–Thurston maps, subgroup distortion, and hyperbolic hydra|journal=[[Groups, Geometry and Dynamics]]|volume=14|year=2020|issue=1|pages=255–282|doi=10.4171/ggd/543|arxiv=1209.0815|s2cid=119299936 }}</ref>

In a 2013 paper,<ref>{{cite journal |author1=Owen Baker |author2=Timothy R. Riley|title=Cannon–Thurston maps do not always exist|journal=[[Forum of Mathematics|Forum of Mathematics, Sigma]]|volume=1|year=2013|mr=3143716|doi=10.1017/fms.2013.4|url=https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2AD18B19161BECFB2ED7F797A6338227/S2050509413000042a.pdf/cannonthurston_maps_do_not_always_exist.pdf|doi-access=free}}</ref> Baker and Riley constructed the first example of a word-hyperbolic group ''G'' and a word-hyperbolic (in fact [[free group|free]]) subgroup ''H''&nbsp;≤&nbsp;''G'' such that the Cannon–Thurston map <math>\partial i:\partial H\to \partial G</math> does not exist.
Later Matsuda and Oguni generalized the Baker–Riley 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–Thurston map <math>\partial i:\partial H\to \partial G</math> does not exist.<ref name=MO>{{cite journal|mr=3176651 |author1=Yoshifumi Matsuda |author2=Shin-ichi Oguni |title=On Cannon–Thurston maps for relatively hyperbolic groups |journal=[[Journal of Group Theory]]|volume=17|year=2014|issue=1|pages=41–47|doi=10.1515/jgt-2013-0024|url=https://www.degruyter.com/downloadpdf/journals/jgth/17/1/article-p41.xml|arxiv=1206.5868|s2cid=119169019 }}</ref>

===Multiplicity of the Cannon–Thurston map===

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–Thurston map <math>\partial i:\partial H\to\partial G </math> 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–Thurston map <math>\partial i:\partial H\to\partial G </math> exists and is injective, then ''H'' is uasi-isometrically embedded in ''G''.<ref name=JKLO>{{cite journal|mr=3488025 |author1=Woojin Jeon |author2=Ilya Kapovich |author3=Christopher Leininger |author4=Ken'ichi Ohshika|title=Conical limit points and the Cannon–Thurston map|journal=[[Conformal Geometry and Dynamics]]|volume=20|year=2016|issue=4 |pages=58–80|doi=10.1090/ecgd/294 |doi-access=free|arxiv=1401.2638}}</ref>

It is known, for more general [[convergence group]]s reasons, that if ''H'' is a word-hyperbolic subgroup of a word-hyperbolic group ''G'' such that the Cannon–Thurston map <math>\partial i:\partial H\to\partial G </math> exists then for every [[convergence group#Conical limit points|conical limit point]] for ''H'' in <math>\partial G</math> has exactly one pre-image under <math>\partial i</math>.<ref>{{cite journal|author=Victor Gerasimov|title=Floyd maps for relatively hyperbolic groups|journal=[[Geometric and Functional Analysis]]|volume=22|year=2012|issue=5|pages=1361–1399|doi=10.1007/s00039-012-0175-6|mr=2989436|arxiv=1001.4482|s2cid=253648281 }}</ref> However, the converse fails: If  <math>\partial i:\partial H\to\partial G </math> exists and is non-injective, then there always exists a non-conical limit point of ''H'' in ''∂G'' with exactly one preimage under ''∂i''.<ref name=JKLO/>

It the context of the original Cannon–Thurston paper, and for many generalizations for the Kleinin representations <math>\rho:\pi_1(S)\to \mathbb PSL(2,\mathbb C),</math> the Cannon–Thurston map <math>j:\mathbb S^1\to \mathbb S^2</math> is known to be uniformly finite-to-one.<ref name=Mj17/> That means that for every point <math>p\in \mathbb S^2</math>, the full pre-image <math>j^{-1}(p)</math> is a finite set with cardinality bounded by a constant depending only on ''S''.<ref>{{cite book|author=Mahan Mj, Mahan|chapter=Cannon–Thurston maps|url=https://eta.impa.br/dl/116.pdf |title=Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures|pages=885–917|publisher=World Sci. Publ., Hackensack, NJ|year=2018|isbn= 978-981-3272-91-0}}</ref>

In general, it is known, as a consequence of the JSJ-decomposition theory for [[word-hyperbolic group]]s, that if <math>1\to H\to G\to Q\to 1</math> 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 group]]s and of a [[free group]].

If <math>H=\pi_1(S)</math> is the fundamental group of a closed hyperbolic surface ''S'', such hyperbolic extensions of ''H'' are described by the theory of "convex cocompact" subgroups of the [[Mapping class group of a surface|mapping class group]] Mod(''S''). Every subgroup Γ&nbsp;≤&nbsp;Mod(''S'') determines, via the Birman short exact sequence, an extension
:<math>1\to H\to E_\Gamma \to \Gamma\to 1</math>
Moreover, the group <math>E_\Gamma</math> is word-hyperbolic if and only if Γ&nbsp;≤&nbsp;Mod(''S'') is convex-cocompact.
In this case, by Mitra's general result, the Cannon–Thurston map ''∂i'':''∂H''&nbsp;→&nbsp;''∂E''<sub>Γ</sub> does exist. The fibers of the map ''∂i'' are described by a collection of ending laminations on ''S'' determined by&nbsp;Γ. This description implies that map ''∂i'' is uniformly finite-to-one.

If <math>\Gamma</math> is a convex-cocompact purely atoroidal subgroup of [[Out(Fn)|<math>\operatorname{Out}(F_n)</math>]] (where <math>n\ge 3</math>) then for the corresponding extension <math>1\to F_n\to E_\Gamma\to \Gamma\to 1</math> the group <math>E_\Gamma</math> is word-hyperbolic. In this setting Dowdall, Kapovich and Taylor proved<ref>{{cite journal|mr=3557464 |author1=Spencer Dowdall |author2=Ilya Kapovich |author3=Samuel J. Taylor|title=Cannon–Thurston maps for hyperbolic free group extensions|journal=[[Israel Journal of Mathematics]]|volume=216|year=2016|issue=2|pages=753–797|doi=10.1007/s11856-016-1426-2 |arxiv=1506.06974|s2cid=255427886 }}</ref> that the Cannon–Thurston map <math>\partial i: \partial F_n\to\partial E_\Gamma</math> is uniformly finite-to-one, with point preimages having cardinality <math>\le 2n</math>. This result was first proved by Kapovich and Lustig<ref>{{cite journal|mr=3335244|author=Ilya Kapovich and Martin Lustig|title=Cannon–Thurston fibers for iwip automorphisms of F<sub>N</sub>|journal=Journal of the London Mathematical Society|volume=91|year=2015|issue=1|pages=203–224|doi=10.1112/jlms/jdu069|arxiv=1207.3494|s2cid=30718832 }}</ref> under the extra assumption that <math>\Gamma</math> is infinite cyclic, that is, that <math>\Gamma</math> is generated by an atoroidal [[fully irreducible automorphism|fully irreducible]] element of <math>\operatorname{Out}(F_n)</math>.

Ghosh proved that for an arbitrary atoroidal <math>\phi\in\operatorname{Out}(F_n)</math> (without requiring <math>\Gamma=\langle \phi\rangle</math> to be convex cocompact) the Cannon–Thurston map <math>\partial i: \partial F_n\to\partial E_\Gamma</math> is uniformly finite-to-one, with a bound on the cardinality of point preimages depending only on ''n''.<ref>{{cite journal|author=Pritam Ghosh |title=Limits of conjugacy classes under iterates of hyperbolic elements of Out(<math>\mathbb{F}</math>)|
journal=[[Groups, Geometry and Dynamics]]|volume=14|year=2020|issue=1|pages=177–211|doi=10.4171/GGD/540|mr=4077660|s2cid=119295501 }}</ref> (However, Ghosh's result does not provide an explicit bound in terms of ''n'', and it is still unknown if the 2''n'' 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–Thurston map <math>\partial i: \partial H\to\partial G</math> exists, if the map  <math>\partial i</math> is finite-to-one. 
However, it is known that in this setting for every <math>p\in \Lambda_{\partial G} H</math> such that ''p'' is a [[convergence group|conical limit point]], the set <math>(\partial i)^{-1}(p)</math> has cardinality 1.

==Generalizations, applications and related results==

*As an application of the result about the existence of Cannon–Thurston maps for Kleinian surface group representations, Mj proved<ref name=Mj14/> that if <math>\Gamma\le \mathbb PSL(2,\mathbb C)</math> is a finitely generated Kleinian group such that the limit set <math>\Lambda\subseteq \partial \mathbb H^3</math> is connected, then <math>\Lambda</math> is locally connected.
*Leininger, Mj and Schleimer,<ref>{{cite journal|mr=2851869 |author1=Christopher J. Leininger |author2=Mahan Mj |author3=Saul Schleimer |title=The universal Cannon–Thurston map and the boundary of the curve complex |journal=[[Commentarii Mathematici Helvetici]]|volume=86|year=2011|issue=4|pages=769–816}}</ref> given a closed hyperbolic surface ''S'', constructed a 'universal' Cannon–Thurston map from a subset of <math>\partial \pi_1(S)=\mathbb S^1</math> to the boundary <math>\partial \mathcal C(S,z) </math> of the [[curve complex]] of ''S'' with one puncture, such that this map, in a precise sense, encodes all the Cannon–Thurston maps corresponding to arbitrary ending laminations on ''S''. As an application, they prove that <math>\partial \mathcal C(S,z) </math> is [[path-connected]] and locally path-connected.
*Leininger, Long and Reid<ref>{{cite journal |author1=Christopher J. Leininger |author2=Darren D. Long |author3=Alan W. Reid|title=Commensurators of finitely generated nonfree Kleinian groups|journal=[[Algebraic and Geometric Topology]]|volume=11|year=2011|issue=1|pages=605–624|doi=10.2140/agt.2011.11.605|mr=2783240|doi-access=free|arxiv=0908.2272}}</ref> used Cannon–Thurston maps to show that any finitely generated torsion-free nonfree Kleinian group  with limit set equal to <math>\mathbb S^2</math>, which is not a lattice and contains no parabolic elements, has discrete commensurator in <math>\mathbb PSL(2,\mathbb C)</math>.
*Jeon and Ohshika<ref>{{cite journal|mr=3570022 |author1=Woojin Jeon |author2=Ken'ichi Ohshika|title=Measurable rigidity for Kleinian groups|journal=[[Ergodic Theory and Dynamical Systems]]|volume=36|year=2016|issue=8|pages=2498–2511|doi=10.1017/etds.2015.15 |arxiv=1406.4594 |s2cid=119149073 }}</ref> used Cannon–Thurston maps to establish measurable rigidity for Kleinian groups.
*Inclusions of [[relatively hyperbolic group]]s 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–Thurston maps.<ref name=Mj14/><ref name=MjPal/><ref name=Pal/><ref name=MO/>
*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''&nbsp;→&nbsp;''Z'' (if such a map exists) is also referred to as a Cannon–Thurston map. Of particular interest in this setting is the case where ''G'' is word-hyperbolic and ''M''&nbsp;=&nbsp;''∂G'' is the hyperbolic boundary of ''G'', or where ''G'' is relatively hyperbolic and ''M''&nbsp;=&nbsp;''∂G'' is the Bowditch boundary of ''G''.<ref name=JKLO/>
* Mj and Pal<ref name=MjPal>{{cite journal|mr=2780738 |author1=Mahan Mj |author2=Abhijit Pal|title=Relative hyperbolicity, trees of spaces and Cannon–Thurston maps|journal=[[Geometriae Dedicata]]|volume=151|year=2011|pages=59–78|doi=10.1007/s10711-010-9519-2 |arxiv=0708.3578 |s2cid=7045852 }}</ref> obtained a generalization of Mitra's earlier result for graphs of groups to the [[relatively hyperbolic group|relatively hyperbolic]] context.
*Pal <ref name=Pal>{{cite journal|mr=2654898|author=Abhijitn Pal
|title=Relatively hyperbolic extensions of groups and Cannon–Thurston maps 
|journal=Proc. Indian Acad. Sci. Math. Sci. |volume=120|year=2010|issue=1|pages=57–68|doi=10.1007/s12044-010-0009-0
|s2cid=16597989
}}</ref> obtained a generalization of Mitra's earlier result, about the existence of the Cannon–Thurston map for short exact sequences of word-hyperbolic groups, to relatively hyperbolic contex.

*Mj and Rafi <ref>{{cite journal|mr=3797060 |author1=Mahan Mj |author2=Kasra Rafi|
title=Algebraic ending laminations and quasiconvexity|
journal=[[Algebraic and Geometric Topology]]|volume= 18|year=2018|issue=4|pages=1883–1916|doi=10.2140/agt.2018.18.1883 |arxiv=1506.08036 |s2cid=92985011 }}</ref> used the Cannon–Thurston map to study which subgroups are quasiconvex in extensions of free groups and surface groups by convex cocompact subgroups of <math>\operatorname{Out}t(F_n)</math> and of mapping class groups.

==References==
{{Reflist}}

==Further reading==
*{{cite book|author=Mahan Mj, Mahan|chapter=Cannon–Thurston maps|url=https://eta.impa.br/dl/116.pdf |title=Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures|pages=885–917|publisher=World Sci. Publ., Hackensack, NJ|year=2018|isbn= 978-981-3272-91-0}}

{{DEFAULTSORT:Cannon-Thurston map}}
[[Category:Group theory]]
[[Category:Dynamical systems]]
[[Category:Geometric topology]]
[[Category:Geometric group theory]]