Hyperbolic volume and Heegaard distance

All the manifolds considered in this paper are 3-dimensional, compact, connected, and orientable. By hyperbolic manifold we mean a manifold whose interior admits a complete finite volume Riemannian metric locally isometric to hyperbolic 3-space $\mathbb{H}^{3}$.

It is generally agreed that the volume of a hyperbolic manifold $M$, $\mbox{\rm Vol}(M)$, is a good measure of the complexity of $M$. As evidence of that, arguments of M. Gromov, T. Jørgensen and W. Thurston show that the hyperbolic volume is linearly equivalent to the number of tetrahedra needed to triangulate a link exterior in $M$. The argument is based on Thurston’s notes [thurston], for a detailed presentation see [gjt]; throughout this paper we follow the notation and definitions given in that paper. For a precise statement, let $t_{c}(M)$ be the smallest number of tetrahedra needed to triangulate $M\setminus N(L)$, where the minimum is taken over all links $L\subset M$ (possibly, $L=\emptyset$) and all possible triangulations.

Theorem 1.1 implies that manifolds of low volume admit Heegaard splittings of low genus: let $M$ be a hyperbolic manifold, $L\subset M$ a link, and $\mathcal{T}$ a triangulation of $M\setminus N(L)$ that realizes $t_{c}(M)$. Let $\Gamma$ be $\mathcal{T}^{(1)}\cup\partial(M\setminus N(L))$, where $\mathcal{T}^{(1)}$ denotes the 1-skeleton of $\mathcal{T}$. It is easy to see that $\partial N(\Gamma)$ is a Heegaard surface for $M\setminus N(L)$ and its genus is at most the number of tetrahedra plus one, that is, $t_{c}(M)+1$. Since the Heegaard genus does not increase after Dehn filling we get:

Here and throughout this paper, $g(M)$ denotes the Heegaard genus of $M$. The converse is false: it is easy to construct hyperbolic manifolds of arbitrarily high volume and Heegaard genus two (for example, consider Dehn fillings of 2-bridge knots; see [schultens2bridge]).

Our goal is to show that any Heegaard surface for a generic hyperbolic manifold $M$ is “simple”. This is described precisely in Theorem 1.5, and we informally explain it here. In [hempel] J. Hempel defined a complexity of Heegaard surfaces which we will call the distance, denoted by $d(\Sigma)$ (the distance, which is based on Kobayashi’s idea of height of loops [KobayashiHeights], is defined in Section 4). We say that a Heegaard surface $\Sigma$ is simple if either $g(\Sigma)$ is low (in terms of the volume) or $d(\Sigma)\leq 2$. A. Casson and C. Gordon constructed a hyperbolic manifold admitting infinitely many Heegaard surfaces, and showed that these surfaces all have distance at least two (in their language, are strongly irreducible). They further showed that there is no upper bound on the genera of these surfaces; hence this result is best possible.

We now explain what a generic hyperbolic manifold is. Let $X$ be a compact 3-manifold (not necessarily hyperbolic) so that $\partial X$ consists of tori, say $T_{1},\dots,T_{n}$. Let $W$ be a manifold obtained from $X$ by Dehn filling some of its boundary components, say $T_{1},\dots,T_{m}$, $m\leq n$. Note that $X\subset W$ and any Heegaard surface for $X$ is a Heegaard surface for $W$. Rieck and E. Sedgwick [rieck][rs1][rs2] prove that on each $T_{i}$ there is a finite set of slopes, denoted by $B_{i}$, so that if the slope filled on each $T_{i}$ intersects every slope of $B_{i}$ more than once, then any Heegaard surface for $W$ is a Heegaard surface for $X$ (after isotopy if necessary). In that case, we say that $W$ is a generic Dehn filling of $X$. With this in mind, we define:

In this paper we prove that $(\mu,d)$-generic manifolds enjoy the following property:

Our proof of Theorem 1.5 uses Dehn filling and hence forces us to assume that $M$ is $(\mu,d)$-generic. However, this does not seem to be an integral part of the theory. In light of this and Remark 1.6 (2) we ask:

We first show how Theorem 1.5 follows from Theorem 1.8. Fix the notation of Theorem 1.5. Let $\lambda=\frac{1}{A}$, where $A>0$ is the constant given in Theorem 1.1. By Remark 1.2, for any complete finite volume hyperbolic 3-manifold $M$, $N_{d}(M_{\geq\mu})$ can be triangulated using at most $\lambda\mbox{\rm Vol}(M)$ tetrahedra.

Set $\Lambda=76\lambda+29$. Let $\Sigma\subset M$ be a Heegaard surface of genus $g(\Sigma)\geq\Lambda\mbox{\rm Vol}(M)$. Using the definition of $\Lambda$ and the fact that $\mbox{\rm Vol}(M)>.9$ (Gabai, Meyerhoff, and Milley [gmm]) we get:

By assumption, $M$ is a $(\mu,d)$-generic, that is, $M$ is obtained from $N_{d}(M_{\geq\mu})$ by a generic Dehn filling (recall Definition 1.3). Hence, after isotopy if necessary, $\Sigma$ is a Heegaard surface for $N_{d}(M_{\geq\mu})$. By Remark 1.2, $N_{d}(M_{\geq\mu})$ can be triangulated using $t\leq\lambda\mbox{\rm Vol}(M)$ tetrahedra. We see that $g(\Sigma)>76\lambda\mbox{\rm Vol}(M)+26\geq 76t+26$, and by Theorem 1.8 (applied to $\Sigma$ as a Heegaard surface of $N_{d}(M_{\geq\mu})$), $d(\Sigma)\leq 2$. It is elementary to see that distance never increases under Dehn filling, and we conclude that $\Sigma\subset M$ is a Heegaard surface of distance at most 2, completing the proof of Theorem 1.5.

Theorem 1.8 is a constraint on the distance of surfaces of genus $76{t}+26$ or more. There are other constraints on the distance known, and by far the most important is Casson and Gordon’s theorem [cg] that says that no Heegaard surface of an irreducible, non-Haken 3-manifold has distance exactly one. Other examples include W. Haken’s theorem that says that any Heegaard surface of a reducible 3-manifold has distance zero, and T. Li’s theorem [li2] that says that a non-Haken 3-manifold admits only finitely many Heegaard surfaces of positive distance. Another constraint is [schtom, Corollary 3.5], where M. Scharlemann and M. Tomova prove that if $\Sigma_{1}$ and $\Sigma_{2}$ are non isotopic Heegaard surfaces of a closed manifold so that $d(\Sigma_{2})>2g(\Sigma_{1})$, then $d(\Sigma_{1})=0$ (in fact, they show that $\Sigma_{1}$ is obtained from $\Sigma_{2}$ by stabilization).

On the positive side, all but finitely many of the surfaces constructed by Casson and Gordon have distance exactly two (Casson and Gordon’s work show that the distance is at least 2 and Theorem 1.5 provides a new proof that the distance is at most 2). Hempel [hempel], using a construction of Kobayashi [KobayashiHeights], shows that for any $g\geq 2$ there exists a sequence of 3-manifolds $M_{n}$ and Heegaard splittings $\Sigma_{n}$ for $M_{n}$, so that $g(\Sigma_{n})=g$ and $\lim_{n\to\infty}d(\Sigma_{n})=\infty$. T. Evans [evans] improved this by constructing, given $g\geq 2$ and $d\geq 0$, a Heegaard splitting of genus $g$ with distance at least $d$. Recently, Qiu, Zou, and Guo [QiuZouGuo] and, independently, Ido, Jang and Kobayashi [IJK], constructed, given $g\geq 2$ and $d\geq 1$, a compact manifold with Heegaard splitting of genus $g$ and distance exactly $d$. In [yoshi] Yoshizawa shows that when $d$ is even, a Heegaard splitting of distance exactly $d$ can be obtained by applying high powers of a single Dehn twist.

However, the answers to the following questions are not known in general:

Questions (1) and (2) above can naturally be generalized to more that two surfaces by setting $i=1,\dots,n$, for some chosen $n$. The word “distinct” in the questions above can be interpreted as “distinct up to isotopy” or “distinct up to homeomorphism”; both yield interesting questions.

The answer for Question 3.1 (2) is known only in the following cases:

• •

  $d_{1}=d_{2}=2$: As mentioned above, there are examples of Casson and Gordon of 3-manifolds admitting infinitely many Heegaard surfaces of unbounded genera and of distance invariant two. Other examples follow from S. Beiler and Y. Moriah [BleilerMoriah] (see also K. Morimoto and M. Sakuma [MorimotoSakuma]). They show that there exist 2-bridge knots $K$ admitting more than one minimal genus Heegaard surface (up to homeomorphism). Let $\Sigma$ be one of these surfaces. It is easy to see that $d(\Sigma)=2$: first, since $g(\Sigma)=2$, it is easy to see that $d(\Sigma)\geq 2$. Next, $\Sigma$ is constructed by viewing $K$ as a torus 1-bridge knot (that is, there exists a genus 1 Heegaard splitting $T_{1}\cup T_{2}$ so that $K$ intersects each $T_{i}$ in a single unknotted arc) and tubing once. Meridian disks for $T_{i}$ which are disjoint from $K$ and the tube, are also disjoint from the core of the tube, showing that $d(\Sigma)\leq 2$.

• •

  $d_{1}=d_{2}=1$: Let $S$ be a 4-punctured sphere and $M=S\times S^{1}$. J. Schultens [schultens] showed that $g(M)=3$. We note that $M$ admits two minimal genus Heegaard splittings, say $\Sigma_{1}$ and $\Sigma_{2}$, such that $\Sigma_{1}$ is obtained by tubing three boundary parallel tori, and $\Sigma_{2}$ is obtained by tubing two boundary parallel tori, with an extra tube that wraps around a third boundary component. Since $\Sigma_{1}$ and $\Sigma_{2}$ induce boundary partitions with distinct numbers of components, they are distinct up to homeomorphism. By construction, $d(\Sigma_{1})=d(\Sigma_{2})=1$.

• •

  $d_{1}=1,\ d_{2}=2$: In [KRSIWR] the authors constructed a 3-manifold $M$ admitting minimal genus Heegaard splittings $\Sigma_{1}$, $\Sigma_{2}$, with $d(\Sigma_{2})=2$ and $d(\Sigma_{1})=1$. In this example, $g(M)=g(\Sigma_{1})=g(\Sigma_{2})=3$.

• •

  $d_{1}=d_{2}=3$: Scharlemann [MR2823137], based on a preprint by Berge [berge], shows that there exists a closed manifold $M$ admitting two Heegaard splittings $\Sigma_{1}$ and $\Sigma_{2}$, distinct up-to homeomorphism, so that $g(M)=g(\Sigma_{1})=g(\Sigma_{2})=2$ and $d(\Sigma_{1})=d(\Sigma_{2})=3$.

We see that much is known when $d_{1}$, $d_{2}\leq 3$. By contrast, the answers to the following basic questions are unknown:

By manifold we mean compact, connected, orientable 3-manifold. We assume familiarity with the basic notions of 3-manifold topology (see, for example, [hempel-book] or [jaco]) and the basic facts about Heegaard splittings (see, for example, [scharlemann] or [sss]). We use the notation $N(\ )$ for open normal neighborhood, $\partial$ for boundary, and $|\ \ |$ for the number of components. We define:

We refer the reader to, for example, [LackenbyAlgorithm, Section 2] for a detailed discussion of generalized tetrahedra. It is well known that a very large class of 3-manifolds admits generalized triangulations, including all compact 3-manifolds. We outline the proof here. Let $W$ be a compact manifold and $K_{i}\subset\partial W$ ($i=1,\dots,n$) a disjoint, closed, connected subsurfaces. By crushing each $K_{i}$ to a point $p_{i}$, we obtain a 3-complex $X$. We can triangulate $X$ so that each $p_{i}$ is a vertex of the triangulation. Removing $p_{i}$ we obtain a semi-ideal triangulation of $N\setminus(\cup_{i}K_{i})$. We conclude that (with $K$ corresponding to $\cup_{i}K_{i}$):

The following lemma is easy and well known (see, for example [schleimer, Remark 2.6]):

Fix $M$ as in the statement of Theorem 1.8 and let $V_{1}\cup_{\Sigma}V_{2}$ be a Heegaard splitting for $M$ with $g(\Sigma)\geq 76{t}+26$. Let $\mathcal{T}$ be a generalized triangulation of $M\setminus K$ (where $K\subset\partial M$ is a closed subsurface) with ${t}$ generalized tetrahedra.

If $\Sigma$ weakly reduces, then $d(\Sigma)\leq 1$; we assume as we may that $\Sigma$ is strongly irreducible. Rubinstein [rubinstein] (see also Stocking [stocking] and Lackenby [lackenby][LackenbyAlgorithm] when $M$ is not closed) show that $\Sigma$ is isotopic to an almost normal surface, that is, after isotopy the intersection of $\Sigma$ with the generalized tetrahedra of $\mathcal{T}$ consists of normal faces, of which there are two types:

1. (1)

   normal disks (normal triangles and normal quadrilaterals)

2. (2)

   an exceptional component, which is either an octagonal disk or an annulus obtained by tubing together two normal disks; at most one normal face of $\Sigma$ is an exceptional component.

Let $N$ be a regular neighborhood of $\mathcal{T}^{(1)}$, the 1-skeleton of $\mathcal{T}$. For each $v\in\mathcal{T}^{(1)}\cap\Sigma$, let $D_{v}$ be the component of $\Sigma\cap N$ containing $v$. Then $D_{v}$ is a disk properly embedded in $N$, called the vertex disk corresponding to $v$. Let $\widehat{F}$ be a normal face contained in a generalized tetrahedron $T$. Then $F=\widehat{F}\setminus\mbox{int}N$ is obtained from $\widehat{F}$ by removing a neighborhood of the vertices of $\widehat{F}$. $F$ is called a truncated normal face. For the remainder of this paper, by a face we mean a truncated normal face or a vertex disk.

Let $v,\ v^{\prime}\in\mathcal{T}^{(1)}\cap\Sigma$ be two vertices and $D_{v}$, $D_{v^{\prime}}$ the corresponding vertex disks. Then $D_{v}$ and $D_{v^{\prime}}$ are called $I$-adjacent if $v$ and $v^{\prime}$ are contained in the same edge $e\in\mathcal{T}^{(1)}$ and $v$ is adjacent to $v^{\prime}$ along $e$. Note that $D_{v}$ is $I$-adjacent to $D_{v^{\prime}}$ if and only if $v$ and $v^{\prime}$ are contained in the same edge $e\in\mathcal{T}^{(1)}$ and there exists an $I$-bundle over $D^{2}$ with total space $Q\subset N$, so that $\partial Q\setminus(D_{v}\cup D_{v^{\prime}})\subset\partial N$, $Q\cap\Sigma=D_{v}\cup D_{v^{\prime}}$, and $D_{v}\cup D_{v^{\prime}}$ is the associated $\partial I$-bundle.

Let $F$ and $F^{\prime}$ be truncated normal faces. Then $F$ and $F^{\prime}$ are called $I$-adjacent if the corresponding normal faces are parallel and there is no normal face between the two. Note that $F$ and $F^{\prime}$ are $I$-adjacent if and only if they are contained in the same generalized tetrahedron $T$, and there exists an $I$-bundle with total space $Q\subset T\setminus\mbox{int}N$, so that $\partial Q\setminus(F\cup F^{\prime})\subset\partial(T\setminus\mbox{int}N)$ and is disjoint from the vertices, truncated vertices, and missing vertices, $Q\cap\Sigma=F\cup F^{\prime}$, and $F\cup F^{\prime}$ is the associated $\partial I$-bundle.

Clearly $I$-adjacency is symmetric but not, in general, transitive. The equivalence relation generated by $I$-adjacency is called $I$-equivalence, and its equivalence classes are called $I$-equivalent families. For example, suppose that a tetrahedron contains four quadrilaterals and denote the corresponding truncated normal faces by $q_{1},q_{2},q_{3},q_{4}$ (listed in order). If there is a truncated exceptional piece between $q_{2}$ and $q_{3}$, then the truncated quadrilaterals form exactly two $I$-equivalent families: $\{q_{1},q_{2}\}$ and $\{q_{3},q_{4}\}$.

Let $\mathcal{F}$ be an $I$-equivalent family. Then $I$-adjacency induces a linear ordering on the faces in $\mathcal{F}$, ordered as $F_{1},\dots,F_{n}$, so that $F_{i}$ is $I$-adjacent to $F_{i+1}$ ($i=1,\dots,n-1$). This order is unique up-to reversing. We color the faces of $\mathcal{F}$ as follows:

1. (1)

   $F_{1}$, $F_{2}$, $F_{n-1}$, and $F_{n}$ are colored red.

2. (2)

   If $n\geq 5$, then $F_{3},\dots,F_{n-2}$ are colored blue and yellow alternately. Note that this leaves us the freedom to exchange the blue and yellow colors of the faces of $\mathcal{F}$.

By construction, any yellow or blue face is $I$-adjacent to two distinct faces.

Let $B$ (resp. $Y$, $R$) denote the union of the blue (resp. yellow, red) faces; note that faces are closed, so $R$, $Y$, and $B$ are compact and may intersect along their boundaries. By Remark 5.1, $B$, $Y$, $R$, and $B\cup Y$ are subsurfaces of $\Sigma$.

By Remark 5.1, $B\cap Y$ is a compact 1-manifold properly embedded in $B\cup Y$. Let $\Gamma\subset B\cup Y$ be the union of the arc components of $B\cap Y$. Endpoints of $\Gamma$ are the vertices of $\Sigma$ where red, blue, and yellow faces meet. By Remark 5.2 around any vertex of $\Sigma$ that is on the boundary of a red vertex disk all the colors are red; therefore the vertex disk at an endpoints of $\Gamma$ is yellow or blue.

Let $\mathcal{V}$ be the set of vertices of red truncated normal faces. We subdivide $\mathcal{V}$ into 3 disjoint sets as follows: $\mathcal{V}_{0}$ are vertices that are on the boundary of at least two red faces; $\mathcal{V}_{+}$ are vertices that are on the boundary of three faces so that one is red, one is yellow, and one is blue; $\mathcal{V}_{-}$ are vertices that are on the boundary of three faces so that one is red and two are yellow, or one is red and two are blue. By construction, $\mathcal{V}_{+}$ is exactly the set of endpoints of $\Gamma$.

By construction, at every vertex exactly one face is a vertex disk. We exchange the colors of the blue vertex disks with the colors of the yellow vertex disks; let $R^{\prime}$, $B^{\prime}$, $Y^{\prime}$ and $\Gamma^{\prime}$ be defined as above, using the new coloring. By Remark 5.2, $\mathcal{V}_{-}$ is exactly the set of endpoints of $\Gamma^{\prime}$ (we emphasize that $\mathcal{V}_{-}$ is the set of vertices defined above using the original coloring). Hence, by exchanging colors if necessary, we may assume that the number of endpoints of $\Gamma$ is at most $\frac{1}{2}|\mathcal{V}|$. Since every arc of $\Gamma$ has two distinct endpoints and $\Gamma$ has at most $\frac{1}{2}|\mathcal{V}|$ endpoints, we get that $|\Gamma|\leq\frac{1}{4}|\mathcal{V}|$.

There are at most 16 vertices in $\mathcal{V}$ from the truncated exceptional component, at most 6 from each truncated red triangle, and at most 8 from each truncated red quadrilateral. By Lemma 5.3 we get:

Hence:

Let $F_{1},\dots,F_{k}$ be the components of $B\cup Y$ cut open along $\Gamma$ (note that $F_{1},\dots,F_{k}$ are not, in general, faces). Cutting along $\Gamma$ increases the Euler characteristic by $|\Gamma|$ and increases the number of boundary components by at most $|\Gamma|$. Using Lemma 5.4 we get:

And using Lemma 5.5 we get:

Combining these inequalities we get:

Since $X\subset\mbox{int}(B)$ it is on the boundary of the total space of an $I$-bundle in $V_{i}$ ($i=1,2$). The other component of the associated $\partial I$-bundle is a pair of pants denoted by $X_{i}$. The components of $\partial X_{i}$ are denoted by $\alpha_{i}$, $\beta_{i}$, and $\gamma_{i}$ so that $\alpha_{i}$ is parallel to $\alpha$, $\beta_{i}$ is parallel to $\beta$, and $\gamma_{i}$ is parallel to $\gamma$. Since $X\subset\mbox{int}(B)$, every point of $X_{i}$ is yellow or red; we conclude that $X\cap X_{i}=\emptyset$. Hence the $I$-bundle in $V_{i}$ is trivial. The annulus extended from $\alpha$ to $\alpha_{i}$ (resp. $\beta$ to $\beta_{i}$, $\gamma$ to $\gamma_{i}$) in $V_{i}$ is denoted by $A_{i}$ (resp. $B_{i}$, $C_{i}$). By construction, these annuli are embedded. Note that $X_{1}\cap X_{2}\neq\emptyset$ is possible.