Cannon–Thurston map – Wikipedia

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[edit]

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 The preprint remained unpublished until 2007,^[1] but in the meantime had generated numerous follow-up works by other researchers.^[2]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

${displaystyle {tilde {S}}}$

can be identified with the hyperbolic plane

${displaystyle mathbb {H} ^{2}}$

. Similarly, the universal cover of M can be identified with the hyperbolic 3-space

${displaystyle mathbb {H} ^{3}}$

. The inclusion

${displaystyle Ssubseteq M}$

lifts to a

${displaystyle pi _{1}(S)}$

-invariant inclusion

${displaystyle {tilde {S}}=mathbb {H} ^{2}subseteq mathbb {H} ^{3}={tilde {M}}}$

. This inclusion is highly distorted because the action of

${displaystyle pi _{1}(S)}$

on

${displaystyle mathbb {H} ^{3}}$

is not geometrically finite.

Nevertheless, Cannon and Thurston proved that this distorted inclusion

${displaystyle mathbb {H} ^{2}subseteq mathbb {H} ^{3}}$

extends to a continuous

${displaystyle pi _{1}(S)}$

-equivariant map

${displaystyle j:mathbb {S} ^{1}to mathbb {S} ^{2}}$

,

where

${displaystyle mathbb {S} ^{1}=partial mathbb {H} ^{2}}$

and

${displaystyle mathbb {S} ^{2}=partial mathbb {H} ^{3}}$

. 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

${displaystyle j:mathbb {S} ^{1}to mathbb {S} ^{2}}$

, 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

${displaystyle mathbb {S} ^{2}}$

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.

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

${displaystyle H=pi _{1}(S)}$

. As a subgroup of

${displaystyle G=pi _{1}(M)}$

, the group H acts on

${displaystyle mathbb {H} ^{3}={tilde {M}}}$

by isometries, and this action is properly discontinuous. Thus one gets a discrete representation

${displaystyle rho :Hto mathbb {P} SL(2,mathbb {C} )=operatorname {Isom} _{+}(mathbb {H} ^{3})}$

.

The group

${displaystyle H=pi _{1}(S)}$

also acts by isometries, properly discontinuously and co-compactly, on the universal cover

${displaystyle mathbb {H} ^{2}={tilde {S}}}$

, with the limit set

${displaystyle Lambda Hsubseteq partial H^{2}=mathbb {S} ^{1}}$

being equal to

${displaystyle mathbb {S} ^{1}}$

. The Cannon–Thurston result can be interpreted as saying that these actions of H on

${displaystyle mathbb {H} ^{2}}$

and

${displaystyle mathbb {H} ^{3}}$

induce a continuous H-equivariant map

${displaystyle j:mathbb {S} ^{1}to mathbb {S} ^{2}}$

.

One can ask, given a hyperbolic surface S and a discrete representation

${displaystyle rho :pi _{1}(S)to mathbb {P} SL(2,mathbb {C} )}$

, if there exists an induced continuous map

${displaystyle j:Lambda Hto mathbb {S} ^{2}}$

.

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

${displaystyle mathbb {H} ^{2}={tilde {S}}}$

and

${displaystyle Lambda pi _{1}(S)=mathbb {S} ^{1}}$

(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

${displaystyle H=pi _{1}(S)}$

. Let

${displaystyle rho :Hto mathbb {P} SL(2,mathbb {C} )}$

be a discrete faithful representation without accidental parabolics. Then

${displaystyle rho }$

induces a continuous H-equivariant map

${displaystyle j:mathbb {S} ^{1}to mathbb {S} ^{2}}$

.

Here the “without accidental parabolics” assumption means that for

${displaystyle 1neq hin H}$

, the element

${displaystyle rho (h)}$

is a parabolic isometry of

${displaystyle mathbb {H} ^{3}}$

if and only if

${displaystyle h}$

is a parabolic isometry of

${displaystyle mathbb {H} ^{2}}$

. One of important applications of this result is that in the above situation the limit set

${displaystyle Lambda rho (pi _{1}(S))subseteq mathbb {S} ^{2}}$

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’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[edit]

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

Let

${displaystyle rho :Gto mathbb {P} SL(2,mathbb {C} )}$

be a discrete faithful representation where G is a word-hyperbolic group, and where

${displaystyle rho (G)}$

contains no parabolic isometries of

${displaystyle mathbb {H} ^{3}}$

. Then

${displaystyle rho }$

induces a continuous G-equivariant map

${displaystyle j:partial Gto mathbb {S} ^{2}}$

, where

${displaystyle partial G}$

is the Gromov boundary of G, and where the image of j is the limit set of G in

${displaystyle mathbb {S} ^{2}}$

.

Here “induces” means that the map

${displaystyle J:Gcup partial Gto mathbb {H} ^{3}cup mathbb {S} ^{2}}$

is continuous, where

${displaystyle J|_{partial G}=j}$

and

${displaystyle J(g)=gx_{0},gin G}$

(for some basepoint

${displaystyle x_{0}in mathbb {H} ^{3}}$

). 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

${displaystyle mathbb {H} ^{3}/G}$

.

Existence and non-existence results[edit]

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

${displaystyle {hat {i}}:Hcup partial Hto Gcup partial G}$

, given by

${displaystyle {hat {i}}|_{H}=i,{hat {i}}|_{partial H}=partial i}$

, 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

${displaystyle Lambda _{partial G}(H)}$

.

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

${displaystyle A_{v}}$

, for the inclusion map

${displaystyle i:A_{v}to G}$

the Cannon–Thurston map

${displaystyle partial i:partial A_{v}to partial G}$

does exist.

By combining and iterating these constructions, Mitra produced^[16] examples of hyperbolic subgroups of hyperbolic groups H ≤ G where the subgroup distortion of H in G is an arbitrarily high tower of exponentials, and the Cannon–Thurston map

${displaystyle partial i:partial Hto partial G}$

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 ≤ G such that the Cannon–Thurston map

${displaystyle partial i:partial Hto partial G}$

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

${displaystyle partial i:partial Hto partial G}$

does not exist.^[19]

Multiplicity of the Cannon–Thurston 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–Thurston map

${displaystyle partial i:partial Hto partial G}$

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–Thurston map

${displaystyle partial i:partial Hto partial G}$

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–Thurston map

${displaystyle partial i:partial Hto partial G}$

exists then for every concical limit point for H in

${displaystyle partial G}$

has exactly one pre-image under

${displaystyle partial i}$

.^[21] However, the converse fails: If

${displaystyle partial i:partial Hto partial G}$

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

${displaystyle rho :pi _{1}(S)to mathbb {P} SL(2,mathbb {C} ),}$

the Cannon–Thurston map

${displaystyle j:mathbb {S} ^{1}to mathbb {S} ^{2}}$

is known to be uniformly finite-to-one.^[13] That means that for every point

${displaystyle pin mathbb {S} ^{2}}$

, the full pre-image

${displaystyle j^{-1}(p)}$

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

${displaystyle 1to Hto Gto Qto 1}$

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

${displaystyle H=pi _{1}(S)}$

is the fundamental group of a closed hyperbolic surface S, such hyperbolic extensions of H are described by the theory of “convex cocompact” subgroups of the mapping class group Mod(S). Every subgroup Γ ≤ Mod(S) determines, via the Birman short exact sequence, an extension

${displaystyle 1to Hto E_{Gamma }to Gamma to 1}$

Moreover, the group

${displaystyle E_{Gamma }}$

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_Γ 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

${displaystyle Gamma }$

is a convex-cocompact purely atoroidal subgroup of

${displaystyle operatorname {Out} (F_{n})}$

(where

${displaystyle ngeq 3}$

) then for the corresponding extension

${displaystyle 1to F_{n}to E_{Gamma }to Gamma to 1}$

the group

${displaystyle E_{Gamma }}$

is word-hyperbolic. In this setting Dowdall, Kapovich and Taylor proved^[23] that the Cannon–Thurston map

${displaystyle partial i:partial F_{n}to partial E_{Gamma }}$

is uniformly finite-to-one, with point preimages having cardinality

${displaystyle leq 2n}$

. This result was first proved by Kapovich and Lustig^[24] under the extra assumption that

${displaystyle Gamma }$

is infinite cyclic, that is, that

${displaystyle Gamma }$

is generated by an autoroidal fully irreducible element of

${displaystyle operatorname {Out} (F_{n})}$

.

Ghosh proved that for an arbitrary atoroidal

${displaystyle phi in operatorname {Out} (F_{n})}$

(without requiring

${displaystyle Gamma =langle phi rangle }$

to be convex cocompact) the Cannon–Thurston map

${displaystyle partial i:partial F_{n}to partial E_{Gamma }}$

is uniformly finite-to-one, with a bound on the cardinality of point preimages depending only on n.^[25] (However, Ghosh’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

${displaystyle partial i:partial Hto partial G}$

exists, if the map

${displaystyle partial i}$

is finite-to-one.
However, it is known that in this setting for every

${displaystyle pin Lambda _{partial G}H}$

such that p is a conical limit point, the set

${displaystyle (partial i)^{-1}(p)}$

has cardinality 1.

Generalizations, applications and related results[edit]

• As an application of the result about the existence of Cannon–Thurston maps for Kleinian surface group representations, Mj proved^[3] that if
  ${displaystyle Gamma leq mathbb {P} SL(2,mathbb {C} )}$

  is a finitely generated Kleinian group such that the limit set

  ${displaystyle Lambda subseteq partial mathbb {H} ^{3}}$

  is connected, then

  ${displaystyle Lambda }$

  is locally connected.

• Leininger, Mj and Schleimer,^[26] given a closed hyperbolic surface S, constructed a ‘universal’ Cannon–Thurston map from a subset of
  ${displaystyle partial pi _{1}(S)=mathbb {S} ^{1}}$

  to the boundary

  ${displaystyle partial {mathcal {C}}(S,z)}$

  of the curve complex of S with one puncture, such that this map, in a precise sense, encodes all the Cannon–Thurston maps corresponding to arbitrary ending laminations on S. As an application, they prove that

  ${displaystyle partial {mathcal {C}}(S,z)}$

  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
  ${displaystyle mathbb {S} ^{2}}$

  , which is not a lattice and contains no parabolic elements, has discrete commensurator in

  ${displaystyle mathbb {P} SL(2,mathbb {C} )}$

  .

• 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
  ${displaystyle operatorname {Out} t(F_{n})}$

  and of mapping class groups.

References[edit]

Further reading[edit]