Cannon–Thurston map explained

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

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 The preprint remained unpublished until 2007,^[1] but in the meantime had generated numerous follow-up works by other researchers.^[2]

\tildeS

can be identified with the hyperbolic plane

H²

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

H³

. The inclusion

S\subseteqM

lifts to a

\pi_1(S)

-invariant inclusion

\tildeS=H^2\subseteqH^3=\tildeM

. This inclusion is highly distorted because the action of

\pi_1(S)

H³

is not geometrically finite.

Nevertheless, Cannon and Thurston proved that this distorted inclusion

H^2\subseteqH³

extends to a continuous

\pi_1(S)

-equivariant map

j:S^1\toS²

, where

S^1=\partialH²

and

S^2=\partialH³

. 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

j:S^1\toS²

, 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

S²

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

Kleinian representations of surface groups

H=\pi_1(S)

. As a subgroup of

G=\pi_1(M)

, the group H acts on

H^3=\tildeM

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

\rho:H\to

	3)
PSL(2,C)=\operatorname{Isom}
	+(H

The group

H=\pi_1(S)

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

H^2=\tildeS

, with the limit set

ΛH\subseteq\partialH^2=S¹

being equal to

S¹

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

H²

and

H³

induce a continuous H-equivariant map

j:S^1\toS²

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

\rho:\pi_1(S)\toPSL(2,C)

, if there exists an induced continuous map

j:ΛH\toS²

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

H^2=\tildeS

and

	1
\pi
	1(S)=S

(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

H=\pi_1(S)

. Let

\rho:H\toPSL(2,C)

be a discrete faithful representation without accidental parabolics. Then

\rho

induces a continuous H-equivariant map

j:S^1\toS²

Here the "without accidental parabolics" assumption means that for

1\neh\inH

, the element

\rho(h)

is a parabolic isometry of

H³

if and only if

is a parabolic isometry of

H²

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

Λ\rho(\pi_{1(S))\subseteq}S²

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

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

Let

\rho:G\toPSL(2,C)

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

\rho(G)

contains no parabolic isometries of

H³

. Then

\rho

induces a continuous G-equivariant map

j:\partialG\toS²

, where

\partialG

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

S²

Here "induces" means that the map

J:G\cup\partialG\toH³\cupS²

is continuous, where

J|_\partial=j

and

J(g)=gx_0,g\inG

(for some basepoint

x_0\inH³

). 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

H^3/G

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

\hati:H\cup\partialH\toG\cup\partialG

, given by

\hati|_H=i,\hati|_\partial=\partiali

, 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

Λ_\partial(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

A_v

, for the inclusion map

i:A_v\toG

the Cannon–Thurston map

\partiali:\partialA_v\to\partialG

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