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.
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The preprint remained unpublished until 2007,[1] but in the meantime had generated numerous follow-up works by other researchers.
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.
, 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.
The original example of Cannon and Thurston can be thought of in terms of Kleinian representations of the surface group
also acts by isometries, properly discontinuously and co-compactly, on the universal cover
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.
In this setting Mj[3] proved the following theorem: Here the "without accidental parabolics" assumption means that for
One of important applications of this result is that in the above situation the limit set
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.
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
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.
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
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
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
[20] It the context of the original Cannon–Thurston paper, and for many generalizations for the Kleinin representations
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 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
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.
In this setting Dowdall, Kapovich and Taylor proved[23] that the Cannon–Thurston map
This result was first proved by Kapovich and Lustig[24] under the extra assumption that