The remaining case of McMullen's Conjecture 1.3: Cannon-Thurston maps from the Floyd boundary for Kleinian groups with degenerate compressible ends and parabolics

McMullen (Invent. math. 2001) proved that the limit set of a hyperbolic 3-manifold with fundamental group a free group $\langle A,B\rangle$ whose commutator is parabolic (a cusped-torus group) is locally connected, by constructing a continuous $\pi_1$-equivariant map from the circle at infinity onto the limit set, and posed Conjecture 1.3: for any hyperbolic 3-manifold $N$ with finitely generated fundamental group there exists a continuous, $\pi_1(N)$-equivariant map from the Floyd boundary $\partial\pi_1(N)$ onto the limit set $\Lambda\subset S^2_\infty$, a conjecture Floyd had proved for geometrically finite groups in 1980. The subsequent literature develops as follows: the model-manifold program of Minsky (2010) and Brock-Canary-Minsky (2012) supplied the higher-genus quasi-isometric models McMullen identified as the missing ingredient; Mj (2014, Ann. of Math.) constructed Cannon-Thurston maps for all simply and doubly degenerate surface Kleinian groups and deduced that connected limit sets of finitely generated Kleinian groups are locally connected; Mj (2017, Forum Math. Pi) proved the conjecture for arbitrary finitely generated Kleinian groups without parabolics and reduced the general statement to the existence of a bi-Lipschitz Minsky model for each degenerate end. The single surviving case, finitely generated Kleinian groups with parabolics and degenerate ends of compressible boundary type, is open in the published literature: the required model rests on unpublished preprints of Bowditch and the in-preparation Brock-Canary-Minsky classification. The problem stated here is that residual core: establish the Minsky model for degenerate ends with compressible boundary in the presence of parabolics (the general Ending Lamination Theorem) and the equivariant Floyd-boundary map to the limit set in the remaining generality.

Background

A Kleinian group is a discrete subgroup $G$ of $\mathrm{PSL}_2(\mathbb{C})$, acting by isometries on hyperbolic 3-space $\mathbb{H}^3$ and by Moebius transformations on its sphere at infinity $S^2_\infty=\partial\mathbb{H}^3$. Its limit set $\Lambda\subset S^2_\infty$ is the set of accumulation points of any orbit $Gx$; it is the minimal closed $G$-invariant set and the locus of the chaotic part of the action. Write $N=\mathbb{H}^3/G$. An element of $G$ is parabolic if it fixes exactly one point of $S^2_\infty$; parabolics correspond to cusps (maximal rank-two cusps, or accidental rank-one cusps) of $N$. The group, or $N$, is geometrically finite if $N$ has a finite-sided fundamental domain, and degenerate (geometrically infinite) otherwise. By the tameness theorem (Agol 2004; Calegari-Gabai 2006), such an $N$ with finitely generated $\pi_1$ is homeomorphic to the interior of a compact 3-manifold $M$, its compact core; the ends of $N$ are either cusp ends or degenerate ends modeled on $S\times[0,\infty)$ for a finite-area hyperbolic surface $S$. A boundary surface of $M$ is incompressible if it injects on $\pi_1$ (surface groups are the prototype) and compressible otherwise (handlebody groups, for example). Each degenerate end is encoded, following Thurston, Bonahon and Canary, by an ending lamination: a geodesic lamination (a closed union of disjoint simple geodesics) on $S$ obtained as the limit of simple closed curves whose geodesic representatives exit the end. The ending lamination theorem of Minsky and Brock-Canary-Minsky states that for ends incompressible away from cusps (in particular all Kleinian surface groups), $N$ is determined up to isometry by its core and end invariants, and admits a model manifold, built combinatorially from the end invariants via the Masur-Minsky hierarchy machine (hierarchies of tight geodesics in the complexes of curves — simplicial complexes whose vertices are isotopy classes of essential simple closed curves — of the subsurfaces of $S$), that is bi-Lipschitz to $N$; this model is the ingredient Mj's general theorems call a bi-Lipschitz Minsky model for a degenerate end. The Floyd boundary $\partial G_F$ of a finitely generated group $G$ is the boundary of the metric completion of the Cayley graph after scaling the metric by the conformal factor $d(e,x)^{-2}$ (Floyd, 1980); it compactifies $G$ and carries the natural boundary action of $G$. The finitely generated Kleinian groups considered here are hyperbolic relative to their parabolic subgroups — the extension of Gromov hyperbolicity in which the parabolic subgroups are treated as peripheral — and for such relatively hyperbolic groups the Floyd boundary maps canonically onto the Bowditch (relative hyperbolic) boundary by theorems of Gerasimov and Gerasimov-Potyagailo; without parabolics it is identified with the Gromov boundary. A Cannon-Thurston map for $N$ is a continuous $G$-equivariant extension of the orbit map $G\to\mathbb{H}^3$ to a map $\partial G_F\to\Lambda$; the sphere-filling Peano curves of Cannon and Thurston for fibered groups were the first examples. Since the continuous image of a compact locally connected space is locally connected, such a map implies local connectivity of $\Lambda$, the topological property motivating McMullen's paper and the older question (going back to Abikoff) of whether connected limit sets of finitely generated Kleinian groups are locally connected. Floyd (1980) constructed the equivariant boundary map for geometrically finite groups; Cannon and Thurston for closed surface groups fibered over the circle; Minsky (1994) for groups with injectivity radius bounded below; Bowditch (2007) for punctured surface groups of bounded geometry (injectivity radius bounded below away from the cusps); McMullen (2001) for the cusped torus; and Mj (2010) for ends of i-bounded geometry, meaning that the boundary tori of the tubes around the short geodesics of the end have uniformly bounded diameter, a class generalizing the cusped-torus case. McMullen then posed Conjecture 1.3: for any hyperbolic 3-manifold $N$ with finitely generated fundamental group there exists a continuous, $\pi_1(N)$-equivariant map $F:\partial\pi_1(N)\to\Lambda\subset S^2_\infty$, identifying the quasi-isometry type (model) of $N$ as the missing ingredient for higher genus. Minsky (2010) and Brock-Canary-Minsky (2012) supplied the model for all surface groups; Mj (2014) constructed Cannon-Thurston maps for all simply and doubly degenerate surface Kleinian groups and deduced local connectivity of all connected limit sets; Mj (2017) proved the conjecture for all finitely generated Kleinian groups without parabolics and showed (Theorem 4.22) that in general it follows from the existence of a bi-Lipschitz Minsky model for each degenerate end. What remains open in the published literature is the class of groups with parabolics and degenerate ends of compressible boundary type, for which the model, that is, the general ending lamination theorem beyond incompressible boundary, exists so far only in unpublished preprint form.

Problem Statement

Resolve Conjecture 1.3 of McMullen (2001) in the generality that remains open: the class of finitely generated Kleinian groups $G$ whose quotient manifold $N=\mathbb{H}^3/G$ contains parabolics and has at least one degenerate (geometrically infinite) end whose boundary surface is compressible in the compact core of $N$; degenerate handlebody groups with cusps are typical examples. Prove that each degenerate end of such an $N$, in particular ends with compressible boundary in the presence of parabolics, admits a bi-Lipschitz Minsky model, i.e. that the combinatorial model manifold built from the end invariants is bi-Lipschitz to a neighborhood of the end; this is the Ending Lamination Theorem (classification of hyperbolic 3-manifolds with finitely generated fundamental group) beyond the incompressible-boundary and parabolic-free cases. Prove that there is a continuous, $G$-equivariant map $F:\partial G_F\to\Lambda\subset S^2_\infty$ from the Floyd boundary of $G$ onto the limit set of $N$, for every group $G$ in the class.

References

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