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 ${\displaystyle H=\pi _{1}(S)}$. 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 ${\displaystyle \rho :H\to \mathbb {P} SL(2,\mathbb {C} )=\operatorname {Isom} _{+}(\mathbb {H} ^{3})}$.

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 ${\displaystyle \Lambda H\subseteq \partial H^{2}=\mathbb {S} ^{1}}$ 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 ${\displaystyle j:\Lambda H\to \mathbb {S} ^{2}}$.

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 ${\displaystyle \rho :H\to \mathbb {P} SL(2,\mathbb {C} )}$ 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 ${\displaystyle 1\neq h\in H}$, 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

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

Let ${\displaystyle \rho :G\to \mathbb {P} SL(2,\mathbb {C} )}$ 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 ${\displaystyle j:\partial G\to \mathbb {S} ^{2}}$, 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 ${\displaystyle J:G\cup \partial G\to \mathbb {H} ^{3}\cup \mathbb {S} ^{2}}$ is continuous, where ${\displaystyle J|_{\partial G}=j}$ and ${\displaystyle J(g)=gx_{0},g\in G}$ (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

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 ${\displaystyle {\hat {i}}:H\cup \partial H\to G\cup \partial G}$, 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-equivariant, 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 ${\displaystyle \partial i:\partial H\to \partial G}$ 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 ${\displaystyle \partial i:\partial H\to \partial G}$ 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 ${\displaystyle \partial i:\partial H\to \partial G}$ does not exist.^[19]

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 ${\displaystyle \partial i:\partial H\to \partial G}$ 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 ${\displaystyle \partial i:\partial H\to \partial G}$ 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 ${\displaystyle \partial i:\partial H\to \partial G}$ exists then for every conical limit point for H in ${\displaystyle \partial G}$ has exactly one pre-image under ${\displaystyle \partial i}$.^[21] However, the converse fails: If ${\displaystyle \partial i:\partial H\to \partial G}$ 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 ${\displaystyle p\in \mathbb {S} ^{2}}$, 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 ${\displaystyle 1\to H\to G\to Q\to 1}$ 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

${\displaystyle 1\to H\to E_{\Gamma }\to \Gamma \to 1}$

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 ${\displaystyle n\geq 3}$) then for the corresponding extension ${\displaystyle 1\to F_{n}\to E_{\Gamma }\to \Gamma \to 1}$ 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 atoroidal 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 ${\displaystyle \partial i:\partial H\to \partial G}$ exists, if the map ${\displaystyle \partial i}$ is finite-to-one. However, it is known that in this setting for every ${\displaystyle p\in \Lambda _{\partial G}H}$ such that p is a conical limit point, the set ${\displaystyle (\partial i)^{-1}(p)}$ has cardinality 1.