on almost geodesic surfaces in hyperbolic three-manifolds

Overview

Kahn and Markovic proved that every closed hyperbolic three-manifold contains a closed immersed surface whose induced map on fundamental groups is injective (Kahn & Markovic, 2012). Equivalently, if

$$M={\mathbb{H}}^3/\Gamma$$

is closed hyperbolic, then $\Gamma$ contains a subgroup isomorphic to the fundamental group of a closed surface of genus at least two.

The proof constructs the surface from many nearly totally geodesic pairs of pants. Each cuff has large length, each pair of pants has controlled complex half-lengths, and the pants are glued so that the complex twist-bend parameters are extremely close to the Fuchsian value. The existence of enough pants with well-distributed gluing data is proved by exponential mixing of the frame flow. A geometric matching argument then turns the distribution statement into an actual closed surface.

Surface Subgroups

Let $M$ be a closed hyperbolic $3$-manifold. Write

$$M={\mathbb{H}}^3/\Gamma ,\Gamma <\mathrm{PSL}(2,\mathbb{C})$$

for the corresponding torsion-free cocompact Kleinian group. A surface subgroup of $\Gamma$ is a subgroup isomorphic to

$${\pi }_1(S_g),g\ge 2,$$

where $S_g$ is a closed orientable surface.

Geometrically, a surface subgroup is equivalent to a map

$$f:S_g\to M$$

that is ${\pi }_1$-injective. Such a surface is called essential or incompressible. The surface subgroup conjecture asked whether every closed hyperbolic $3$-manifold has such a subgroup. After geometrization, this was the remaining hyperbolic case of a broad family of problems about finding essential surfaces in $3$-manifolds.

Kahn and Markovic prove a stronger statement. For every small $ϵ>0$, they produce a Fuchsian group

$$F_{ϵ}<\mathrm{PSL}(2,\mathbb{R})$$

and a $(1+ϵ)$-quasiconformal map

$$g:\partial {\mathbb{H}}^3\to \partial {\mathbb{H}}^3$$

such that

$$g{F}_{ϵ}{g}^{-1}<\Gamma .$$

Here ${\mathbb{H}}^2$ is identified with an oriented totally geodesic plane in ${\mathbb{H}}^3$. The subgroup $g{F}_{ϵ}{g}^{-1}$ is quasi-Fuchsian: it is a small quasiconformal deformation of a Fuchsian surface group.

The associated immersed surface is almost geodesic. Its lift to ${\mathbb{H}}^3$ is embedded, and the boundary circle of the lift is a quasicircle close to a round circle. This implies ${\pi }_1$-injectivity.

Kahn-Markovic Surface Subgroup Theorem

Every closed hyperbolic $3$-manifold contains a closed immersed surface $S$ such that

$${\pi }_1(S)\to {\pi }_1(M)$$

is injective. More precisely, for every $ϵ>0$ there is an immersed $(1+ϵ)$-quasigeodesic surface in $M$ whose image subgroup is quasi-Fuchsian (Kahn & Markovic, 2012).

Complex Length and Skew Pants

A loxodromic element

$$A\in \mathrm{PSL}(2,\mathbb{C})$$

has an oriented axis in ${\mathbb{H}}^3$. Its complex translation length is

$$\ell (A)=L+i\theta \in \mathbb{C}/2\pi i\mathbb{Z},$$

where $L>0$ is translation distance along the axis and $\theta$ is the rotation angle around the axis. After conjugating the axis to the geodesic from $0$ to $\infty$, the boundary action is

$$z\mapsto e^{\ell (A)}z.$$

The real part is ordinary hyperbolic length; the imaginary part records holonomy around the geodesic.

A topological pair of pants is a sphere with three open disks removed. A skew pair of pants in $M$ is, in Kahn and Markovic’s language, an injective homomorphism

$$\rho :{\pi }_1({\Pi }^0)\to {\pi }_1(M),$$

up to conjugacy, whose three boundary elements map to loxodromic elements. Equivalently, it is a homotopy class of maps from a pair of pants into $M$ whose cuffs are represented by closed geodesics.

For a pair of pants in a hyperbolic surface, three orthogeodesic seams cut the pants into two right-angled hexagons. The same picture persists in complex hyperbolic notation for skew pants in $M$. If ${\gamma }_i$ is a cuff and ${\eta }_{i-1},{\eta }_{i+1}$ are the two seams meeting it, define the complex half-length by the signed complex distance along ${\gamma }_i$:

$$\mathrm{hl}({\gamma }_i)={\mathbf{d}}_{{\gamma }_i}({\eta }_{i-1},{\eta }_{i+1}).$$

For an admissible skew pants this is half of the complex cuff length, with the correct branch chosen:

$$\mathrm{hl}({\gamma }_i)=\frac{\ell ({\gamma }_i)}{2}.$$

A pants is good at scale $R$ if

$$\left|\mathrm{hl}({\gamma }_i)-\frac{R}{2}\right|<ϵ$$

for each of its three cuffs. Thus every cuff has real length close to $R$, and the rotational part is small.

Feet and Normal Tori

The gluing data of a pants along a cuff is not determined only by the cuff length. One must also know where the seams land on the cuff.

Let $\gamma$ be a closed geodesic in $M$. Its unit normal bundle is a flat torus. After choosing an orientation of $\gamma$, it is naturally written as

$$N^1(\gamma )\simeq \mathbb{C}/(\ell (\gamma )\mathbb{Z}+2\pi i\mathbb{Z}).$$

Translation in the real direction moves along $\gamma$; translation in the imaginary direction rotates the normal vector.

For half-length gluing, Kahn and Markovic quotient by the half translation and use

$$N^1(\sqrt{\gamma })\simeq \mathbb{C}/\left(\frac{\ell (\gamma )}{2}\mathbb{Z}+2\pi i\mathbb{Z}\right).$$

If a pants has cuff $\gamma$, the two seams meeting $\gamma$ determine two normal vectors. These differ by the half-length translation, so they define a single point

$${\mathrm{foot}}_{\gamma }(\Pi )\in N^1(\sqrt{\gamma }).$$

This point is the foot of the pants on the cuff.

When two pants are glued along opposite orientations of the same cuff, their feet determine the reduced complex twist-bend parameter. If the two feet differ by $i\pi$, the normal directions are reversed as required for an oriented gluing. The Fuchsian model used in the paper also includes a real twist of size $1$. Hence the desired matching condition is

$${\mathrm{foot}}_{\gamma }({\Pi }_2)\approx {\mathcal{A}}_{1+i\pi }{\mathrm{foot}}_{\gamma }({\Pi }_1),$$

where ${\mathcal{A}}_{\zeta }$ denotes translation by $\zeta$ on $N^1(\sqrt{\gamma })$.

The number $1$ here is not a full Dehn twist. It is a fixed real displacement in the half-length torus. Since $R$ is large, an error of order $1/R$ in this coordinate is very small relative to the cuff length.

Complex Fenchel-Nielsen Criterion

For a closed surface with a pants decomposition, the reduced complex Fenchel-Nielsen coordinates consist of

$$\mathrm{hl}(C),s(C),$$

for each cuff $C$. Here $\mathrm{hl}(C)$ is the common half-length assigned by the two adjacent pants, and $s(C)$ is the reduced twist-bend parameter obtained from the difference of their feet. These coordinates are the complex analogue of real Fenchel-Nielsen length and twist coordinates (Kourouniotis, 1994; Tan, 1994; Series, 2001).

The geometric input from Section 2 of Kahn-Markovic is the following criterion.

Quasi-Fuchsian Criterion

There are universal constants $\widehat{ϵ},K_0>0$ such that if a viable representation of a pants-decomposed closed surface satisfies

$$\left|\mathrm{hl}(C)-\frac{R}{2}\right|<ϵ,|s(C)-1|<\frac{ϵ}{R}$$

for every cuff $C$, with $R$ sufficiently large and $0<ϵ<\widehat{ϵ}$, then the representation is faithful and quasi-Fuchsian. The boundary conjugacy is $(1+K_0ϵ)$-quasiconformal after adjusting constants (Kahn & Markovic, 2012).

The Fuchsian reference surface has all cuff lengths equal to $R$ and all reduced twists equal to $1$. The theorem says that a small complex perturbation of this highly regular large-pants surface remains quasi-Fuchsian.

The proof is not a local compactness argument in Teichmuller space alone, because the genus of the constructed surface is not fixed. Kahn and Markovic use uniform control of the lifted cuff lamination. Roughly, the lifts of the cuffs form a locally finite lamination in ${\mathbb{H}}^2$. If the complex Fenchel-Nielsen errors are small, then nearby leaves have almost the same complex distances after applying the boundary map. A compactness argument for invariant unflippable laminations shows that a limiting boundary map preserving these complex distances must be Mobius. Here unflippability is the condition that the endpoint pattern of the lamination cannot be globally reversed by a continuous boundary involution; it rules out a false symmetry in the limiting argument. Quantitatively, this forces the quasisymmetric constant to be close to one.

This criterion converts the construction problem into a finite matching problem: build many good pants and pair their cuffs so that the feet match after the shift $1+i\pi$ up to an error much smaller than $1/R$.

The Matching Problem for Pants

Let ${\mathcal{P}}_{ϵ,R}$ denote the set of good oriented skew pants whose half-lengths are within $ϵ$ of $R/2$. For each oriented closed geodesic ${\gamma }^{\ast }$, consider all marked pants

$$(\Pi ,{\gamma }^{\ast })$$

that have ${\gamma }^{\ast }$ as one cuff.

To build a closed surface, one must pair every occurrence of ${\gamma }^{\ast }$ with an occurrence of $-{\gamma }^{\ast }$. The pairing must satisfy

$$\mathrm{dist}\left({\mathcal{A}}_{1+i\pi }\mathrm{foot}(\Pi ,{\gamma }^{\ast }),\mathrm{foot}({\Pi }^{\prime },-{\gamma }^{\ast })\right)\ll \frac{1}{R}.$$

If such pairings exist for every cuff geodesic, the resulting glued surface has

$$\mathrm{hl}(C)\approx \frac{R}{2},s(C)\approx 1,$$

and the quasi-Fuchsian criterion applies.

Thus the main construction is a distribution theorem for feet. For every closed geodesic $\gamma$ that occurs as a cuff in the construction, the feet of good pants ending at $\gamma$ must be nearly uniformly distributed in the torus $N^1(\sqrt{\gamma })$. Uniformity is needed not merely qualitatively, but at the scale required to match after translating by $1+i\pi$.

Measures and Hall Matching

Kahn and Markovic encode the supply of pants by a positive finite measure

$$\mu$$

on the set of oriented skew pants. It is symmetric under reflection of orientation. From such a measure they form the boundary-foot measure

$$\hat{\partial }\mu$$

on the disjoint union of normal tori

$$N^1(\sqrt{\Gamma })=\underset{\gamma }{⨆}{N}^1(\sqrt{\gamma })$$

by placing one atom at each foot of each cuff of each pants, weighted by $\mu (\Pi )$.

The desired matching follows if

$$\hat{\partial }\mu \text{and}({\mathcal{A}}_{1+i\pi }{)}_{\ast }\hat{\partial }\mu$$

are close in a transport sense. For atomic measures, this means that mass can be moved from one measure to the other by distances at most a small error. After rationalizing the weights and multiplying by a common denominator, the measure becomes an integral formal sum of pants. Hall’s marriage theorem then gives an actual bijection between positive and negative cuff occurrences with the required foot control (Calegari, 2009; Kahn & Markovic, 2012).

This is an important structural step. The proof does not try to choose pants one by one. It first constructs a nearly invariant measure on all good pants, then extracts a finite integral matching from the measure.

Frame Flow and Good Connections

The source of the measure is the frame flow. Let

$$\mathcal{F}(M)$$

be the bundle of oriented orthonormal $2$-frames

$$F=(p,u,n),$$

where $u,n\in T_{p}M$ are orthonormal. The frame flow $g_t$ moves $p$ along the geodesic determined by $u$, parallel transports the frame, and keeps track of the normal direction.

For closed hyperbolic $3$-manifolds, the frame flow is exponentially mixing. In the form used by Kahn and Markovic, there is $q>0$ such that for $C^1$ functions $\varphi ,\psi$ on $\mathcal{F}(M)$,

$$\left|\Lambda (\mathcal{F}(M)){\int }_{\mathcal{F}(M)}(g_r^{\ast }\psi )(x)\varphi (x)d\Lambda (x)-\int \psi d\Lambda \int \varphi d\Lambda \right|\le C{e}^{-q|r|}.$$

Here $\Lambda$ is Liouville measure, and $C$ depends on the regularity norms of $\varphi$ and $\psi$ (Moore, 1987; Pollicott, 1992). Brin and Gromov established mixing of frame flows in a broader negative-curvature setting (Brin & Gromov, 1980).

The mixing theorem implies that, for large $r$, any two small neighborhoods in the frame bundle are connected by many orbit segments of length about $r$, with nearly the expected frequency. Kahn and Markovic package this into an affinity function

$$a_{\gamma }(F_1,F_2)$$

that measures whether a geodesic segment $\gamma$ connects two frames well. If $a_{\gamma }(F_1,F_2)>0$, then the segment has length close to $r$, its initial and terminal tangent directions agree with the prescribed frame directions up to exponentially small angular error, and the normal vector is transported close to the prescribed terminal normal.

Tripods and Good Pants

Rotate a frame by $2\pi /3$ around its normal vector. From a frame $F_p$ one obtains a tripod

$$T_p=(F_p,\omega F_p,{\omega }^2{F}_p).$$

From another frame $F_q$ one obtains an anti-tripod using the inverse rotation. A triple of geodesic segments

$$\gamma =({\gamma }_0,{\gamma }_1,{\gamma }_2)$$

from $p$ to $q$ well connects the two tripods if each rotated frame at $p$ is well connected to the corresponding oppositely rotated frame at $q$.

The three segments form a theta graph in $M$. A topological pair of pants deformation retracts onto a theta graph, so this graph determines a homotopy class of maps from a pants into $M$. Kahn and Markovic show that if the two tripods are well connected, then the induced representation of the pants group is admissible and the resulting skew pants has half-lengths close to

$$r-\log \frac{4}{3}.$$

Thus, setting

$$R=2\left(r-\log \frac{4}{3}\right),$$

the pants lies in ${\mathcal{P}}_{Dϵ,R}$ for a universal constant $D$.

The shift by $\log (4/3)$ comes from the hyperbolic geometry of the ideal tripod model: when three long nearly geodesic connections are assembled into a right-angled hexagon, the cuff half-length is not exactly the connection length $r$, but differs by the constant determined by the limiting ideal configuration (Fenchel, 1989; Kahn & Markovic, 2012).

Define a measure on well-connected tripod pairs by

$$d\tilde{\mu }(T_p,T_q,\gamma )=\prod _{i=0}^2{a}_{{\gamma }_i}({\omega }^i{F}_p,{\omega }^{-i}{F}_q)d{\lambda }_T,$$

where $d{\lambda }_T$ is Liouville measure in the two frame variables and counting measure in the connection variables. Pushing this measure forward by the map from a well-connected tripod pair to its skew pants gives a measure

$$\mu$$

on good skew pants.

The measure is reflection-invariant by construction. The remaining issue is to prove that its feet are nearly uniformly distributed on every normal torus.

Predicted Feet and Equidistribution

Fix a cuff geodesic $\delta$. A pants with cuff $\delta$ is built from three connections. Two of the connections determine $\delta$; the third connection closes the tripod pair into a pants. The actual foot of the pants on $\delta$ depends on all three connections, but Kahn and Markovic introduce a predicted foot that depends only on the first two.

More precisely, a well-connected bipod consists of two connections

$${\gamma }_0,{\gamma }_1$$

between the same endpoints, together with the relevant two legs of the tripods. If the loop

$${\gamma }_0\cup {\gamma }_1$$

is freely homotopic to $\delta$, the bipod determines a point

$${\mathbf{f}}_{\delta }(B_p,B_q,{\gamma }_0,{\gamma }_1)\in N^1(\sqrt{\delta }).$$

This is the predicted foot. Hyperbolic geometry shows that, after the third connection is added, the predicted foot differs from the actual foot of the resulting pants by at most

$$O(e^{-r/4}).$$

The advantage is that the predicted foot has an exact symmetry. The solid torus cover with core $\delta$ acts on the space of relevant bipods by the torus

$$\mathbb{C}/(\ell (\delta )\mathbb{Z}+2\pi i\mathbb{Z}),$$

and the predicted-foot map is equivariant for this action. Therefore the pushforward of the natural bipod measure is exactly a constant multiple of Euclidean measure on

$$N^1(\sqrt{\delta }).$$

The third connection is then counted using exponential mixing. Given the first two connections, the total affinity of possible third connections is

$$\frac{1}{\Lambda (\mathcal{F}(M))}(1+O(e^{-qr})),$$

uniformly. Hence forgetting the third connection changes the bipod measure only by an almost constant density. Combining this with the predicted-foot estimate gives a measure ${\beta }_{\delta }$ on each normal torus such that

$$K_{\delta }\le \frac{d{\beta }_{\delta }}{d\lambda }\le K_{\delta }(1+D{e}^{-qR}),$$

and

$$\hat{\partial }\mu {|}_{N^1(\sqrt{\delta })}$$

is close to ${\beta }_{\delta }$ in transport distance.

This proves the equidistribution theorem for feet. Since Euclidean measure on the normal torus is invariant under translation by $1+i\pi$, the boundary-foot measure is close to its shifted copy. The Hall matching step can now be applied.

Assembly of the Surface

The final assembly is a chain of implications.

First, frame-flow mixing produces a finite-support reflection-invariant measure $\mu$ on good pants, with feet nearly equidistributed on every cuff torus.

Second, this measure is rationalized. After multiplying by an integer, it becomes a finite integral multiset of oriented good pants. Reflection symmetry gives equal supply of pants with opposite orientations.

Third, Hall’s theorem pairs each occurrence of an oriented cuff ${\gamma }^{\ast }$ with an occurrence of $-{\gamma }^{\ast }$ so that their feet differ from the target shift $1+i\pi$ by at most

$$DR{e}^{-qR}.$$

For large $R$, this is much smaller than $1/R$.

Fourth, the glued surface has reduced complex Fenchel-Nielsen coordinates satisfying

$$\left|\mathrm{hl}(C)-\frac{R}{2}\right|<ϵ,|s(C)-1|<\frac{ϵ}{R}.$$

The quasi-Fuchsian criterion applies, so the resulting representation of the surface group into $\Gamma$ is faithful.

If the formal gluing produces more than one connected component, one passes to a connected component. The restriction to that component is still a closed quasi-Fuchsian surface subgroup. This completes the construction.

Consequences and Position

Before Kahn and Markovic, essential surfaces were known in several important classes, including cusped finite-volume hyperbolic $3$-manifolds and arithmetic cases (Cooper, Long & Reid, 1997; Lackenby, 2010). Their theorem settled the closed hyperbolic case in full.

The result became a central input to the virtual Haken and virtual fibering breakthroughs. Agol’s proof of the virtual Haken conjecture uses cubulation and special cube complex technology, with Kahn-Markovic surface subgroups entering through the cubulation strategy for closed hyperbolic $3$-manifold groups (Agol, 2013). Kahn-Markovic supplies the geometric surfaces; later work supplies the separability and finite-cover machinery needed to promote immersed surfaces to embedded surfaces in finite covers.

The theorem also introduced a powerful method into low-dimensional topology: use dynamics on the frame bundle to manufacture geometric building blocks with nearly uniform boundary data, then use a discrete matching theorem to assemble a global topological object.

Scope and Limitations

The surfaces constructed by the proof are immersed, not embedded in $M$. They are embedded on the universal cover and quasi-Fuchsian, which is exactly what gives ${\pi }_1$-injectivity. Embedded surfaces generally require passing to finite covers and using separability results not proved in this paper.

The construction is highly nonconstructive at the scale of explicit examples. The parameter $R$ is taken very large, the genus of the resulting surface is enormous, and the proof depends on quantitative mixing constants for the frame flow.

The argument is also tailored to closed hyperbolic $3$-manifolds. Cusped manifolds require additional control near the cusps, and variable negative curvature lacks the same constant-curvature complex Fenchel-Nielsen and frame-flow estimates in this exact form. The method nevertheless strongly influenced later good-pants and good-assembly arguments.

Transferable Mechanisms

The first mechanism is the conversion of a geometric existence problem into a coordinate tolerance theorem. Once Kahn and Markovic prove that small complex Fenchel-Nielsen errors imply a quasi-Fuchsian representation, the rest of the proof only has to build pants satisfying explicit numerical inequalities (Kourouniotis, 1994; Tan, 1994; Kahn & Markovic, 2012).

The second mechanism is measure before selection. The proof constructs a measure with the desired distributional properties before choosing a finite object. Rationalization and Hall matching then turn the measure into an actual glued surface (Calegari, 2009; Kahn & Markovic, 2012).

The third mechanism is mixing as a source of pseudorandom finite geometry. Exponential mixing of the frame flow supplies the near-independence needed to make feet equidistribute on every cuff torus (Moore, 1987; Pollicott, 1992; Brin & Gromov, 1980).

The fourth mechanism is the use of a predicted observable. The actual foot of a pants depends on all three connections, which is awkward for equidistribution. The predicted foot depends on only two connections, has an exact torus symmetry, and is exponentially close to the actual foot. This separates symmetry from error control (Kahn & Markovic, 2012).

The fifth mechanism is global topology from local uniformity. Uniform distribution of feet is a local statement on normal tori; after matching, it produces a closed surface with a faithful global representation. This local-to-global assembly pattern is the core reason the method travels beyond the original surface subgroup problem (Kahn & Markovic, 2012; Agol, 2013).

See Also

• on orbit closures in moduli space — both arguments use dynamical equidistribution to control geometric objects globally; Kahn-Markovic use frame-flow mixing, while Eskin-Mirzakhani-Mohammadi use measure classification and non-divergence in strata.
• on min-max theory and the Willmore conjecture — both posts concern existence of distinguished surfaces in $3$-dimensional geometry, but the mechanisms are complementary: dynamical good-pants assembly here, variational min-max there.
• on degree of mapping and Lipschitz constant — the final object is certified by a global topological conclusion, namely injectivity of the surface group representation, after quantitative geometric estimates have supplied local control.
• on bounded gaps between primes — the analogy is structural: both proofs construct weighted local objects, prove a distribution theorem, and then use a selection or positivity step to extract a discrete global conclusion.

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