In mathematics, a Cannon–Thurston 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.

The notion originated from a seminal 1980s preprint of James Cannon and William Thurston "Group-invariant Peano curves" (eventually published in 2007) about fibered hyperbolic 3-manifolds.[1]

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

History

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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 The preprint remained unpublished until 2007,[1] but in the meantime had generated numerous follow-up works by other researchers.[2]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   can be identified with the hyperbolic plane  . Similarly, the universal cover of M can be identified with the hyperbolic 3-space  . The inclusion   lifts to a  -invariant inclusion  . This inclusion is highly distorted because the action of   on   is not geometrically finite.

Nevertheless, Cannon and Thurston proved that this distorted inclusion   extends to a continuous  -equivariant map

 ,

where   and  . 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  , 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   having cardinality at most 2g, where g is the genus of 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 groups

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Kleinian representations of surface groups

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The original example of Cannon and Thurston can be thought of in terms of Kleinian representations of the surface group  . As a subgroup of  , the group H acts on   by isometries, and this action is properly discontinuous. Thus one gets a discrete representation  .

The group   also acts by isometries, properly discontinuously and co-compactly, on the universal cover  , with the limit set   being equal to  . The Cannon–Thurston result can be interpreted as saying that these actions of H on   and   induce a continuous H-equivariant map  .

One can ask, given a hyperbolic surface S and a discrete representation  , if there exists an induced continuous map  .

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   and   (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  . Let   be a discrete faithful representation without accidental parabolics. Then   induces a continuous H-equivariant map  .

Here the "without accidental parabolics" assumption means that for  , the element   is a parabolic isometry of   if and only if   is a parabolic isometry of  . One of important applications of this result is that in the above situation the limit set   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's 2013 paper[8] 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

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In a 2017 paper[13] Mj proved the existence of the Cannon–Thurston map in the following setting:

Let   be a discrete faithful representation where G is a word-hyperbolic group, and where   contains no parabolic isometries of  . Then   induces a continuous G-equivariant map  , where   is the Gromov boundary of G, and where the image of j is the limit set of G in  .

Here "induces" means that the map   is continuous, where   and   (for some basepoint  ). 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  .

Cannon–Thurston maps and word-hyperbolic groups

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Existence and non-existence results

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Let G be a word-hyperbolic group and let H ≤ G be a subgroup such that H is also word-hyperbolic. If the inclusion i:H → 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  , given by  , 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  .

If H ≤ G is 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 [14] that if G is word-hyperbolic and H ≤ G is a normal word-hyperbolic subgroup, then the Cannon–Thurston map exists. (In this case if H and Q = 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 ≤ G are two word-hyperbolic groups and H is normal in G then, by a result of Mosher,[15] the quotient group Q = 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 ∈ ∂Q.

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  , for the inclusion map   the Cannon–Thurston map   does exist.

By combining and iterating these constructions, Mitra produced[16] examples of hyperbolic subgroups of hyperbolic groups H ≤ G where the subgroup distortion of H in G is an arbitrarily high tower of exponentials, and the Cannon–Thurston map   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 ≤ G such that the Cannon–Thurston map   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   does not exist.[19]

Multiplicity of the Cannon–Thurston map

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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   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   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–Thurston map   exists then for every conical limit point for H in   has exactly one pre-image under  .[21] However, the converse fails: If   exists and is non-injective, then there always exists a non-conical limit point of H in ∂G with exactly one preimage under ∂i.[20]

It the context of the original Cannon–Thurston paper, and for many generalizations for the Kleinin representations   the Cannon–Thurston map   is known to be uniformly finite-to-one.[13] That means that for every point  , the full pre-image   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   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   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 Mod(S). Every subgroup Γ ≤ Mod(S) determines, via the Birman short exact sequence, an extension

 

Moreover, the group   is word-hyperbolic if and only if Γ ≤ Mod(S) is convex-cocompact. In this case, by Mitra's general result, the Cannon–Thurston map ∂i:∂H → ∂EΓ does exist. The fibers of the map ∂i are described by a collection of ending laminations on S determined by Γ. This description implies that map ∂i is uniformly finite-to-one.

If   is a convex-cocompact purely atoroidal subgroup of   (where  ) then for the corresponding extension   the group   is word-hyperbolic. In this setting Dowdall, Kapovich and Taylor proved[23] that the Cannon–Thurston map   is uniformly finite-to-one, with point preimages having cardinality  . This result was first proved by Kapovich and Lustig[24] under the extra assumption that   is infinite cyclic, that is, that   is generated by an atoroidal fully irreducible element of  .

Ghosh proved that for an arbitrary atoroidal   (without requiring   to be convex cocompact) the Cannon–Thurston map   is uniformly finite-to-one, with a bound on the cardinality of point preimages depending only on n.[25] (However, Ghosh'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–Thurston map   exists, if the map   is finite-to-one. However, it is known that in this setting for every   such that p is a conical limit point, the set   has cardinality 1.

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  • As an application of the result about the existence of Cannon–Thurston maps for Kleinian surface group representations, Mj proved[3] that if   is a finitely generated Kleinian group such that the limit set   is connected, then   is locally connected.
  • Leininger, Mj and Schleimer,[26] given a closed hyperbolic surface S, constructed a 'universal' Cannon–Thurston map from a subset of   to the boundary   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   is path-connected and locally path-connected.
  • Leininger, Long and Reid[27] used Cannon–Thurston maps to show that any finitely generated torsion-free nonfree Kleinian group with limit set equal to  , which is not a lattice and contains no parabolic elements, has discrete commensurator in  .
  • Jeon and Ohshika[28] used Cannon–Thurston 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–Thurston 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 → 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 = ∂G is the hyperbolic boundary of G, or where G is relatively hyperbolic and M = ∂G is the Bowditch boundary of G.[20]
  • Mj and Pal[29] obtained a generalization of Mitra's earlier result for graphs of groups to the relatively hyperbolic context.
  • Pal [30] 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 [31] used the Cannon–Thurston map to study which subgroups are quasiconvex in extensions of free groups and surface groups by convex cocompact subgroups of   and of mapping class groups.

References

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  1. ^ a b James W. Cannon; William P. Thurston (2007). "Group invariant Peano curves". Geometry & Topology. 11 (3): 1315–1356. doi:10.2140/gt.2007.11.1315. MR 2326947.
  2. ^ 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.'
  3. ^ a b c d Mahan Mj (2014). "Cannon–Thurston maps for surface groups". Annals of Mathematics. 179 (1): 1–80. arXiv:math/0607509. doi:10.4007/annals.2014.179.1.1. MR 3126566. S2CID 119160004.
  4. ^ Yair Minsky (1994). "On rigidity, limit sets, and end invariants of hyperbolic 3-manifolds" (PDF). Journal of the American Mathematical Society. 7 (3): 539–588. doi:10.2307/2152785. JSTOR 2152785. MR 1257060.
  5. ^ Roger C. Alperin; Warren Dicks; Joan Porti (1999). "The boundary of the Gieseking tree in hyperbolic three-space". Topology and Its Applications. 93 (3): 219–259. doi:10.1016/S0166-8641(97)00270-8. MR 1688476.
  6. ^ Curtis T. McMullen (2001). "Local connectivity, Kleinian groups and geodesics on the blowup of the torus". Inventiones Mathematicae. 146 (1): 35–91. Bibcode:2001InMat.146...35M. doi:10.1007/PL00005809. MR 1859018.
  7. ^ Brian H. Bowditch (2007). "The Cannon–Thurston map for punctured-surface groups". Mathematische Zeitschrift. 255: 35–76. doi:10.1007/s00209-006-0012-4. MR 2262721.
  8. ^ a b Brian H. Bowditch (2013). "Stacks of hyperbolic spaces and ends of 3-manifolds". 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. 65–138. ISBN 978-0-8218-8480-5.
  9. ^ Hideki Miyachi, Semiconjugacies between actions of topologically tame Kleinian groups, 2002, preprint
  10. ^ Juan Souto (2006). "Cannon–Thurston maps for thick free groups". Preprint.
  11. ^ Mahan Mj (2009). "Cannon–Thurston maps for pared manifolds of bounded geometry". Geometry & Topology. 13: 89–245. MR 2469517.
  12. ^ Mahan Mj (2011). "Cannon–Thurston maps, i-bounded geometry and a theorem of McMullen". Actes du Séminaire de Théorie Spectrale et Géometrie. Volume 28. Année 2009–2010. Seminar on Spectral Theory and Geometry, vol. 28. Univ. Grenoble I.
  13. ^ a b Mahan Mj (2017). "Cannon–Thurston maps for Kleinian groups" (PDF). Forum of Mathematics, Pi. 5. doi:10.1017/fmp.2017.2. MR 3652816.
  14. ^ Mahan Mitra (1998). "Cannon–Thurston maps for hyperbolic group extensions". Topology. 37 (3): 527–538. doi:10.1016/S0040-9383(97)00036-0. MR 1604882.
  15. ^ Lee Mosher (1997). "A hyperbolic-by-hyperbolic hyperbolic group" (PDF). Proceedings of the American Mathematical Society. 125 (12): 3447–3455. doi:10.1090/S0002-9939-97-04249-4. MR 1443845.
  16. ^ a b Mahan Mitra, Mahan (1998). "Cannon–Thurston maps for trees of hyperbolic metric spaces". Journal of Differential Geometry. 48 (1): 135–164. arXiv:math/9609209. doi:10.4310/jdg/1214460609. MR 1622603.
  17. ^ Owen Baker; Timothy R. Riley (2020). "Cannon–Thurston maps, subgroup distortion, and hyperbolic hydra". Groups, Geometry and Dynamics. 14 (1): 255–282. arXiv:1209.0815. doi:10.4171/ggd/543. MR 4077662. S2CID 119299936.
  18. ^ Owen Baker; Timothy R. Riley (2013). "Cannon–Thurston maps do not always exist" (PDF). Forum of Mathematics, Sigma. 1. doi:10.1017/fms.2013.4. MR 3143716.
  19. ^ a b Yoshifumi Matsuda; Shin-ichi Oguni (2014). "On Cannon–Thurston maps for relatively hyperbolic groups". Journal of Group Theory. 17 (1): 41–47. arXiv:1206.5868. doi:10.1515/jgt-2013-0024. MR 3176651. S2CID 119169019.
  20. ^ a b c Woojin Jeon; Ilya Kapovich; Christopher Leininger; Ken'ichi Ohshika (2016). "Conical limit points and the Cannon–Thurston map". Conformal Geometry and Dynamics. 20 (4): 58–80. arXiv:1401.2638. doi:10.1090/ecgd/294. MR 3488025.
  21. ^ Victor Gerasimov (2012). "Floyd maps for relatively hyperbolic groups". Geometric and Functional Analysis. 22 (5): 1361–1399. arXiv:1001.4482. doi:10.1007/s00039-012-0175-6. MR 2989436. S2CID 253648281.
  22. ^ Mahan Mj, Mahan (2018). "Cannon–Thurston maps". Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures (PDF). World Sci. Publ., Hackensack, NJ. pp. 885–917. ISBN 978-981-3272-91-0.
  23. ^ Spencer Dowdall; Ilya Kapovich; Samuel J. Taylor (2016). "Cannon–Thurston maps for hyperbolic free group extensions". Israel Journal of Mathematics. 216 (2): 753–797. arXiv:1506.06974. doi:10.1007/s11856-016-1426-2. MR 3557464. S2CID 255427886.
  24. ^ Ilya Kapovich and Martin Lustig (2015). "Cannon–Thurston fibers for iwip automorphisms of FN". Journal of the London Mathematical Society. 91 (1): 203–224. arXiv:1207.3494. doi:10.1112/jlms/jdu069. MR 3335244. S2CID 30718832.
  25. ^ Pritam Ghosh (2020). "Limits of conjugacy classes under iterates of hyperbolic elements of Out( )". Groups, Geometry and Dynamics. 14 (1): 177–211. doi:10.4171/GGD/540. MR 4077660. S2CID 119295501.
  26. ^ Christopher J. Leininger; Mahan Mj; Saul Schleimer (2011). "The universal Cannon–Thurston map and the boundary of the curve complex". Commentarii Mathematici Helvetici. 86 (4): 769–816. MR 2851869.
  27. ^ Christopher J. Leininger; Darren D. Long; Alan W. Reid (2011). "Commensurators of finitely generated nonfree Kleinian groups". Algebraic and Geometric Topology. 11 (1): 605–624. arXiv:0908.2272. doi:10.2140/agt.2011.11.605. MR 2783240.
  28. ^ Woojin Jeon; Ken'ichi Ohshika (2016). "Measurable rigidity for Kleinian groups". Ergodic Theory and Dynamical Systems. 36 (8): 2498–2511. arXiv:1406.4594. doi:10.1017/etds.2015.15. MR 3570022. S2CID 119149073.
  29. ^ a b Mahan Mj; Abhijit Pal (2011). "Relative hyperbolicity, trees of spaces and Cannon–Thurston maps". Geometriae Dedicata. 151: 59–78. arXiv:0708.3578. doi:10.1007/s10711-010-9519-2. MR 2780738. S2CID 7045852.
  30. ^ a b Abhijitn Pal (2010). "Relatively hyperbolic extensions of groups and Cannon–Thurston maps". Proc. Indian Acad. Sci. Math. Sci. 120 (1): 57–68. doi:10.1007/s12044-010-0009-0. MR 2654898. S2CID 16597989.
  31. ^ Mahan Mj; Kasra Rafi (2018). "Algebraic ending laminations and quasiconvexity". Algebraic and Geometric Topology. 18 (4): 1883–1916. arXiv:1506.08036. doi:10.2140/agt.2018.18.1883. MR 3797060. S2CID 92985011.

Further reading

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