Cannon–Thurston map
In mathematics, the notion of 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. 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 The preprint remained unpublished until 2007, but in the meantime had generated numerous follow-up works by other researchers. 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 formula_1 can be identified with the hyperbolic plane formula_2. Similarly, the universal cover of "M" can be identified with the hyperbolic 3-space formula_3. The inclusion formula_4 lifts to a formula_5-invariant inclusion formula_6. This inclusion is highly distorted because  the action of formula_5 on formula_3 is not geometrically finite. Nevertheless, Cannon and Thurston proved that this distorted inclusion formula_9 extends to a continuous formula_5-equivariant map where formula_12 and formula_13. 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 formula_11, 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 formula_15 having cardinality at most 2"g", 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. Kleinian representations of surface groups. The original example of Cannon and Thurston can be thought of in terms of Kleinian representations of the surface group formula_16. As a subgroup  of formula_17, the group "H" acts on formula_18 by isometries, and this action is properly discontinuous. Thus one gets a discrete representation formula_19. The group formula_16 also acts by isometries, properly discontinuously and co-compactly, on the universal cover formula_21, with the limit set formula_22 being equal to formula_23. The Cannon–Thurston result can be interpreted as saying that these actions of "H" on formula_2 and formula_3 induce a continuous "H"-equivariant map formula_11. One can ask, given a hyperbolic surface "S" and a discrete representation formula_27, if there exists an induced continuous map formula_28. For Kleinian representations of surface groups , the most general result in this direction is due to Mahan Mj (2014). 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 formula_21 and formula_30 (even if "S" has some cusps). In this setting Mj proved the following theorem: Here the "without accidental parabolics" assumption means that for formula_35, the element formula_36 is a parabolic isometry of formula_3 if and only if formula_38 is a parabolic isometry of formula_2. One of important applications of this result is that in the  above situation the limit set formula_40 is locally connected. This result of Mj was preceded by numerous other results in the same direction, such as Minsky (1994), Alperin, Dicks and Porti (1999), McMullen (2001), Bowditch (2007) and (2013), Miyachi (2002), Souto (2006), Mj (2009), (2011), and others. In particular, Bowditch's 2013 paper 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 Mj proved the existence of the Cannon–Thurston map in the following  setting: Here "induces" means that the map formula_48 is continuous, where formula_49 and formula_50 (for some basepoint formula_51). 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 formula_52. Cannon–Thurston maps and word-hyperbolic groups. Existence and non-existence results. 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 formula_53, given by formula_54, is continuous. In this setting, if the map "∂i" exists, it is unique and "H"-equivarinat, and the image "∂i"("∂H") is equal to the limit set formula_55. 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 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, 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 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 formula_56, for the inclusion map formula_57 the Cannon–Thurston map formula_58 does exist. By combining and iterating these constructions, Mitra produced 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 formula_59 exists. Later Barker and Riley showed that one can arrange for "H" to have arbitrarily high primitive recursive distortion in "G". In a 2013 paper, 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 formula_59 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 formula_59 does not exist. 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 formula_62 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 formula_62 exists and is injective, then "H  is uasi-isometrically embedded in G". 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 formula_62 exists then for every concical limit point for "H" in formula_46 has exactly one pre-image under formula_66. However, the converse fails: If formula_62 exists and is non-injective, then there always exists a non-conical limit point of "H" in "∂G" with exactly one preimage under "∂i". It the context of the original Cannon–Thurston paper, and for many generalizations for the Kleinin  representations formula_68 the Cannon–Thurston map formula_11 is known to be uniformly finite-to-one. That means that for every point formula_70, the full pre-image formula_71 is a finite set with cardinality bounded by a constant depending only on "S". In general, it is known, as a consequence of the JSJ-decomposition theory for word-hyperbolic groups, that if formula_72 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 formula_16 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 formula_75 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" → "∂E" 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 formula_76 is a convex-cocompact purely atoroidal subgroup of formula_77 (where formula_78) then for the corresponding extension formula_79 the group formula_75 is word-hyperbolic. In this setting Dowdall, Kapovich and Taylor proved that the Cannon–Thurston map formula_81 is uniformly finite-to-one, with point preimages having cardinality formula_82. This result was first proved by Kapovich and Lustig under the extra assumption that formula_76 is infinite cyclic, that is, that formula_76 is generated by an autoroidal fully irreducible element of formula_77. Ghosh proved that for an arbitrary atoroidal formula_86 (without requiring formula_87 to be convex cocompact) the  Cannon–Thurston map formula_81 is uniformly finite-to-one, with a bound on the cardinality of point preimages depending only on "n". (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 formula_89 exists, if the map formula_66 is finite-to-one. However, it is known that in this setting for every formula_91 such that "p" is a conical limit  point, the set formula_92 has cardinality 1.