Totally Geodesic Surfaces in Hyperbolic 3-Manifolds: Algorithms and Examples

First, we give a brief introduction to the general theory of hyperbolic 3-manifolds along with details on some of the more specific computational background. A complete background is outside of the scope of this paper but readers can find much of the theory in [35] and [5].

We start by reviewing some of the fundamentals of hyperbolic space and hyperbolic manifolds. Recall that $n$-dimensional hyperbolic space $\mathbb{H}^{n}$ is the unique simply connected space of constant curvature $-1$. It can be modeled as the $n$-dimensional upper half-space, $\{\vec{x}\in\mathbb{R}^{n}:x_{n}>0\}$ with the Riemannian metric determined by $ds=\frac{dx_{1}^{2}+\dots+dx_{n}^{2}}{x_{n}^{2}}$. With this model of $\mathbb{H}^{n}$, we define the boundary of $\mathbb{H}^{n}$ to be $\{\vec{x}\in\mathbb{R}^{n}:x_{n}=0\}\cup\{\infty\}$. Note that topologically the boundary is homeomorphic to $S^{n-1}$. The orientation-preserving isometries of $\mathbb{H}^{n}$, $\text{Isom}^{+}(\mathbb{H}^{n})$ correspond bijectively to conformal (angle-preserving) maps on the boundary. In particular, for $n=3$, if we identify the embedded plane $\{\vec{x}\in\mathbb{R}^{3}:x_{n}=0\}$ with $\mathbb{C}$, the boundary then becomes the Riemann sphere and the orientation-preserving isometries of $\mathbb{H}^{3}$ can be identified with the group of Möbius transformations which is isomorphic to $PSL(2,\mathbb{C})$. An orientable hyperbolic 3-manifold $M$ is then a quotient of 3-dimensional hyperbolic space $\mathbb{H}^{3}$ by a torsion-free discrete subgroup $\Gamma$ of $PSL(2,\mathbb{C})$. This manifold has $\mathbb{H}^{3}$ as its universal cover with the deck transformations corresponding to the elements of $\Gamma$. We say that $M$ is finite-volume if the volume of $M$ with respect to the metric inherited on $M$ from $\mathbb{H}^{3}$ is finite. Every finite-volume hyperbolic 3-manifold can be decomposed uniquely into a compact core and some collection of cusps [22]. The compact core consists of a compact manifold whose boundary is a possibly empty disjoint union of tori. The cusps consist of disjoint unions of $T^{2}\times[1,\infty)$. The two parts are glued together along their boundary tori to form $M$.

In order to work with manifolds concretely, it is often convenient to decompose them into simple pieces. For cusped hyperbolic 3-manifolds, it is generally more convenient to use an ideal triangulation. Before that, we must introduce the concept of an ideal tetrahedron. Topologically, an ideal tetrahedron is simply a tetrahedron with its vertices removed. We can endow these with a hyperbolic geometric structure by taking an ideal tetrahedron to be the “convex hull” of any 4 distinct points in the boundary of $\mathbb{H}^{3}$. The use of quotes here denotes the fact that we want to take the portion of that tetrahedron only in $\mathbb{H}^{3}$ and not on the boundary. For every edge $e$ connecting the “ideal” vertices $a$ and $b$ of an ideal tetrahedron, we can find an orientation-preserving isometry which takes $a$ to $0$, and $b$ to $\infty$, and maps the remaining two ideal vertices to $1$ and some complex number $z$ lying in $\{x+iy\in\mathbb{C}:y\geq 0\}$. This number $z$ is called the shape parameter of the edge of the tetrahedron.

We mention here that for cusped hyperbolic manifolds $M$ we deal with throughout this paper, its ideal triangulation admits a partially flat angle structure as defined in [26].

More information about how the shape parameters are used to build up the hyperbolic structure and the holonomy representation can be found in the Appendix A.1.

A compressing disk for a closed, connected, embedded surface $F$ in a 3-manifold $M$ is a disk $D\subset M$ such that $D\cap F=\partial D$ and $\partial D$ does not bound a disk in $F$. A closed orientable surface is incompressible if it does not have a compressing disk and is not a 2-sphere. A closed orientable surface $F$ in $M$ is $\partial$-parallel if $F$ is isotopic into $\partial M$. A closed orientable surface is essential if it is incompressible and is not $\partial$-parallel.

For a closed non-orientable surface $F$, we say that $F$ is incompressible or essential if the boundary of a regular neighborhood of $F$ has that property. For a hyperbolic 3-manifold, an equivalent definition for incompressible is that the fundamental group of $F$, where $F$ is not a 2-sphere, injects into the fundamental group of $M$, i.e. $\pi_{1}(F)\hookrightarrow\pi_{1}(M)$ is injective. We note that even for non-orientable $F$, $\pi_{1}(F)\hookrightarrow\pi_{1}(M)$ being injective implies that $F$ is incompressible. Let $F$ be an embedded surface in $M$. $F$ is said to be totally geodesic if and only if every geodesic arc in $F$ with its induced Riemannian metric is also a geodesic in $M$.

Checking for totally geodesic surfaces is deeply related to the holonomy representation and we review the work of [31] for notions we use in our algorithms. For a complete hyperbolic manifold $M$, there is a unique (up to conjugation) discrete, faithful representation $\rho:\pi_{1}(M)\to PSL(2,\mathbb{C})$ called the holonomy representation. This is the group of isometries used to construct the 3-manifold. It is often more convenient to work with a lift of the holonomy representation to $SL(2,\mathbb{C})$ which always exists by [13]. For a surface $F$ embedded in a 3-manifold $M$, the inclusion map $i:F\to M$ induces a map $i_{*}:\pi_{1}(F)\to\pi_{1}(M)$ which composed with the holonomy representation, gives a representation from the fundamental group of $F$ to $PSL(2,\mathbb{C})$ (or $SL(2,\mathbb{C})$ when considering the lift of the holonomy representation).

The group of orientation-preserving isometries of $\mathbb{H}^{3}$ is $PSL(2,\mathbb{C})$. Let $G\leq PSL(2,\mathbb{C})$ be a discrete group and let $x\in\mathbb{H}^{3}$ be any point. The limit set of $G$ is the set of accumulation points on $\partial\mathbb{H}^{3}$ of the orbit $G(x)$. A discrete subgroup of $PSL(2,\mathbb{C})$ is called a Kleinian group. A Kleinian group $\Gamma$ acting on the Riemann sphere $\mathbb{C}\cup\infty$ is quasi-Fuchsian if its limit set is a Jordan curve. $\Gamma$ is called Fuchsian if the Jordan curve is a geometric circle.

A surface $F$ is called Fuchsian if the induced representation $\rho(\pi_{1}(F))$ is Fuchsian. For an embedded non-orientable essential surface $F$ in $M$, if the induced representation $\rho(\pi_{1}(F))$ is contained in the normalizer of $PSL(2,\mathbb{R})$ in $PSL(2,\mathbb{C})$ then $F$ is totally geodesic. Note that for non-orientable surfaces, the property of being isotopic to a totally geodesic surface is equivalent to being Fuchsian. This is not true for orientable surfaces since the double cover $\tilde{F}$ of a non-orientable Fuchsian surface $F$ will also be Fuchsian, but $\tilde{F}$ cannot be totally geodesic in $M$. However, we have that if $F$ is Fuchsian, then either $F$ is isotopic to a totally geodesic surface or is a double cover of a non-orientable totally geodesic surface. Moreover, there is a topological obstruction to the existence of essential non-orientable surfaces.

Being totally geodesic implies Fuchsian, giving a necessary condition for a surface to be totally geodesic. For totally geodesic surfaces, we also have the following well-known lemma.

The above also holds for $SL(2,\mathbb{C})$. Since it is more convenient to work with matrices compared to projective matrices, we will deal with a lift of the holonomy representation into $SL(2,\mathbb{C})$ throughout this paper. Checking whether or not an $SL(2,\mathbb{C})$ representation conjugates into $SL(2,\mathbb{R})$ is difficult, so we reduce it to something simpler.

If $F$ is a closed essential surface, then $\rho$ restricted to $\pi_{1}(F)$ is an irreducible representation. Elements of $SU(2)$ as isometries of $\mathbb{H}^{3}$ all have fixed points so cannot represent elements of the fundamental group of $M$ and therefore cannot represent elements of the fundamental group of $F$. Hence, if $\rho$ restricted to $\pi_{1}(F)$ has all real traces, then it must be conjugate into $SL(2,\mathbb{R})$ and thus $F$ is totally geodesic or is a double cover of a totally geodesic surface.

The following lemma helps to simplify things further.

From the above, we get the following corollary.

In this section we review some basic normal surface theory following the work in [42]. Let $M$ be a $3$-manifold with a triangulation $\mathcal{T}$. An elementary disk $E$ in a tetrahedron $\Delta$ of $\mathcal{T}$ is a properly embedded disk that meets each edge of $\Delta$ in at most one point and each face of $\Delta$ in at most one line. A surface is said to be normal if it is in general position with the 1-skeleton $\mathcal{T}^{(1)}$ of $\mathcal{T}$ and meets each tetrahedron only in elementary disks. We follow the convention that if $E\cap\mathcal{T}^{(1)}$ is a planar set then $E$ is planar and if not then $E$ is the cone $b\ast\partial E$ where $b$ is the centroid of the 3-simplex $\Delta$ spanned by $E\cap\mathcal{T}^{(1)}$. Elementary disks and hence normal surfaces are uniquely determined by their intersection points with $\mathcal{T}^{(1)}$. A normal isotopy of $M$ is an isotopy that leaves every simplex of $\mathcal{T}$ invariant. There are exactly 7 normal isotopy classes of elementary disks: four triangles that each cut off a vertex and three quadrilaterals that separate the three pairs of disjoint edges. We call such normal isotopy classes the disk types of a tetrahedron. Similarly, the normal isotopy classes of arcs t from the intersection of elementary disks and each $2$-face of $\Delta$ are called the arc types.

The following demonstrates the generality of normal surfaces.

The significance of understanding normal surfaces is that they can be expressed uniquely as a tuple of non-negative numbers. Fix an ordering $d_{1},d_{2},\dots,d_{7t}$ on all disk types in $\mathcal{T}$ ($t$ is the number of tetrahedra in $\mathcal{T}$). To a normal surface $F$ we can assign a $7t$-tuple $\overrightarrow{F}=(x_{1},x_{2},\dots,x_{7t})$ where $x_{i}$ is the number of elementary disk types of type $d_{i}$ in $F$. $\overrightarrow{F}$ is called the normal coordinate of $F$. Any normal surface is uniquely determined by its normal coordinate up to normal isotopy.

A normal coordinate $\overrightarrow{x}$ is said to be admissible if the corresponding normal surface has at most one nonzero quadrilateral disk type in every tetrahedron. There is a one-to-one correspondence between all admissible solutions in the solution space of $\mathbb{R}^{7t}$ and normal surfaces in $M$. More details about solution spaces are available in Appendix A.3.

We first introduce the specific computational descriptions of the mathematical objects we use in our algorithms in this section.

A cusped hyperbolic 3-manifold can be represented by an ideal triangulation. This is a list of ideal tetrahedra, their shape parameters, and how each face of each tetrahedron is glued. Fundamental groups of 3-manifolds and surfaces can be written using a specific group presentation: a set of generators $\{a_{1},\dots,a_{n}\}$ for the group and a specific set of relations $\{r_{1},\dots,r_{m}\}$ expressed as trivial words in the group. A detailed treatment of generating sets for fundamental groups will be given in Section 3.2. Every fundamental group element can be represented as a word in the alphabet consisting of generators and their inverses.

The holonomy representation of an element of the fundamental group described in Section 2.1 can be found from its group presentation and a set of matrices representing the generators as follows: any element $a_{i_{1}}a_{i_{2}}\cdots a_{i_{k}}$ in the group presentation is represented by the product of matrices that correspond to each of the generators $a_{i_{1}},\dots,a_{i_{k}}$. Due to the following result, holonomy representations of manifolds are of a certain form:

This allows us to avoid some of the theoretical downsides of working with approximate numbers by using a representation of a number field in the following way, as found in [12]. Given an arbitrary algebraic field extension $\mathbb{Q}(\alpha)$ where $\alpha$ is a root of the irreducible degree $n$ polynomial $p(x)\in\mathbb{Q}[x]$, every element $\beta\in\mathbb{Q}[x]$ can be uniquely represented as $c_{0}+c_{1}\alpha+c_{2}\alpha^{2}+\ldots+c_{n-1}\alpha^{n-1}$. More information about this way of representing algebraic field elements can be found in Appendix A.4.

In addition to fields, we also need information about specific embeddings of a field into $\mathbb{C}$. Specifically, we need to determine whether a given embedding of a field into $\mathbb{C}$ is contained in $\mathbb{R}$. This can be done numerically as follows. Every embedding of $\mathbb{Q}(\alpha)$ into $\mathbb{C}$ is completely determined by which root $\alpha_{0}$ of $p(x)$ $\alpha$ is sent to. Therefore, it suffices to determine whether or not $\alpha_{0}$ is a real root of $p(x)$ or not. Using Sturm’s theorem or one of its generalizations (as in [39]), the number of real roots $r$ of $p(x)$ can be determined exactly. The complex roots can be approximated by one’s favorite convergent numerical method which will ensure that the root will not fall on the real line. Once $n-r$ complex roots have been determined, the remaining roots must be real.

We state two lemmas on finding the fundamental group of a space from a cellulation.

We present algorithms that determine whether or not a given surface is totally geodesic.

Now given a 3-manifold, we describe an algorithm that checks whether it contains a totally geodesic surface. The idea is to enumerate all normal surfaces in the 3-manifold and implement the algorithm in the previous section. The 3-manifold and normal surfaces should first satisfy certain volume constraints. By a result of Miyamoto (Theorem 4.2 [32]), we obtain a lower bound on the volume of all 3-manifolds containing a closed totally geodesic surface. Moreover, if a given 3-manifold $M$ contains such a surface, the result also gives an upper bound on the Euler characteristic of a totally geodesic surface.

It remains to find all surfaces in a given 3-manifold $M$ that satisfy this volume bound to check if it is totally geodesic.

Upon finding all normal surfaces, by using Theorem 4.2 of [25] we can determine whether each normal surface is essential in order to obtain a finite list of all essential normal surfaces up to a certain Euler characteristic.

Our computations were all performed in SageMath [38] on KEELING, the School of Earth, Society & Environment computer cluster at the University of Illinois Urbana-Champaign. It made use of the programs Regina [9] and SnapPy [14]. In particular, all surface enumeration was done by Regina based on the work in [7]. The data sets which we ran our algorithm on come from a variety of censuses in SnapPy. More details for these censuses can be found in Section 4.2. Geometric information about the manifolds included in these censuses were provided by SnapPy.

While the algorithms specify an exact solution to the gluing equations, finding and working with an exact form of the holonomy representation is computationally expensive. To speed up and simplify our computations, we instead chose to work with double precision and then check that the traces are real by ensuring that the imaginary parts are less than a specified threshold. A middle ground between these two approaches would be to use interval arithmetic along with verified computations of the holonomy representation as found in [16]. This would have the advantage of being able to provably rule out any non-totally geodesic surfaces with only very minor performance losses. However, we opted for double precision because the observed values used to rule out the surfaces were high enough to be unambiguous (nearly always at least 1). With the large proportion of negative results we expected our code to find, we chose the threshold to be quite large (.01) so that if the algorithm ruled out the surface we could be confident that it did so correctly. However, even with this large threshold, it is still unlikely for the algorithm to return a false positive, given that the trace of every checked matrix (and recall that the number of these matrices is cubic in the number of generators of the fundamental group of the surface) would have to have its imaginary part below this threshold. Because of this high threshold, we still decided to do further checks to ensure that every surface the algorithm marked as positive was, in fact, totally geodesic. Some additional sanity checks on our code are also detailed in Appendix D

To confirm that a surface identified as possibly totally geodesic is indeed totally geodesic, we plotted the limit set of the Kleinian group $\Gamma$ corresponding to the surface’s fundamental group. Applying an isometry corresponding to a long geodesic in the surface to any point in $\partial\mathbb{H}^{3}$ will result in a point very close to one of the ends of the geodesic on the boundary. In order to get these points, we need only find the isometries corresponding to long words in the fundamental group of $M$ and then apply them on any point in $\partial\mathbb{H}^{3}$. As mentioned in Section 2, if the surface is totally geodesic, then the limit set of $\Gamma$ must be a geometric circle, not just a topological one. We present some examples of approximations of limit sets of surfaces that are not totally geodesic. Figure 7 is the limit set of an essential quasi-Fuchsian surface that is not totally geodesic found in [3]. In particular, it is a surface of genus 2 in the exterior of the knot in Figure 7(a) of [3]. Figure 8 are limit sets of essential surfaces in the exterior of alternating knots. They are surfaces of genus 2 found in the exteriors of knots $8_{16}$ and $9_{29}$, names for these knots follow the online database, KnotInfo:Table of Knots [29]. These surfaces are all known to have an accidental parabolic. Details on the parameters used to plot these approximations is given in the description of Figure 13 of Section 4.3.

It should be noted that some manifolds took significantly longer amounts of time to run the program on than other manifolds. Thus, in order to save time and increase the quantity of manifolds computed, the program terminated for manifolds after exceeding a certain chosen runtime. This runtime was chosen to be 5,000 seconds, based on the average runtime of census manifolds. To ensure the algorithm ran on all knot exteriors with at most 12 crossings, a few manifolds whose runtime exceeded 5,000 were allowed but their data is not included in the average calculation or plots in Section 4.2.

We observed a variety of 3-manifolds provided by databases from SnapPy. We mainly looked at link exteriors and covers of orientable cusped hyperbolic manifolds. The knot and link exteriors that were considered came from the census of 15 crossing knots and links of Hoste-Thistlethwaite-Weeks [23]. We only take link exteriors in $S^{3}$ with volume larger than 7.2 (Lemma 3.7) as these link exteriors do not contain non-orientable essential surfaces since $S^{3}$ does not have such surfaces. Moreover, we exclude alternating links [31] where the result is already known, to get that the list contained 279,649 manifolds. The triangulation information we used came from the census HTLinkExteriors in SnapPy.

The second census that was considered are covers of 3-manifolds provided by the OrientableCuspedCensus in SnapPy [8]. This census consists of orientable cusped hyperbolic 3-manifolds that can be triangulated with at most 9 ideal tetrahedra. It contains 61,911 manifolds, some of which contain multiple cusps. To give ourselves plenty of examples to work with we took $n$-fold covers of these manifolds where $n$ allows the volume to be large enough to admit a totally geodesic surface but small enough so that the volume of the resulting cover is at most 20 (at this volume, our algorithm’s runtime starts to get prohibitively long).

We have currently completed computation for 142,409 manifolds in the HTLinkExteriors census in SnapPy and 15,992 covers of manifolds in the OrientableCuspedCensus in SnapPy. The average total runtime was 1483.37 seconds.

Figure 9 are histograms of the runtimes of manifolds for which we completed our computation. We have made the distinction between the time it takes to enumerate surfaces in a manifold and applying Algorithm 2 to these found surfaces. On average, enumerating surfaces constituted 87% of the the entire runtime whereas applying Algorithm 2 constituted 13%. Since for small manifolds the runtimes for both procedures are very small and have almost no difference this average can be a somewhat coarse indicator. A more detailed comparison is given in Figure 10. Notice in Figure 10 that in general, runtimes for enumerating surfaces are relatively larger than the runtime for applying Algorithm 2. It is also interesting to examine results regarding runtimes of enumeration of normal surfaces in [6]. In the worst case, finding all normal surfaces below the given genus bound of Algorithm 3 is known to scale exponentially with the volume of the manifold. Even in the average case, there is evidence that the number of normal surfaces scales exponentially with the number of tetrahedra, limiting the size of manifolds we can run the algorithm on.

We speculate that the number of tetrahedra and volume of the manifold have the biggest effect on runtimes. Relations between runtimes of our algorithm and number of tetrahedra and volume are presented in Figures 11 and 12 respectively. We chose to plot the log of runtimes to make the relation more explicit. Moreover, in Figures 10, 11 and 12, manifolds where no surfaces were detected (hence Algorithm 2 did not run at all) were not plotted.

Determining whether or not a given normal surface is essential can be an immensely time consuming computation. Algorithm 2 usually takes much less time than checking if a surface is essential, hence we chose to run our algorithm on all normal surfaces only excluding the simple compressible surfaces identified by the method provided by Letscher explained in Appendix C. For surfaces found to be totally geodesic, we check their limit sets to make sure that the surface found was indeed totally geodesic or homotopic to a totally geodesic surface. This surface could be homotopic to a totally geodesic surface because for any totally geodesic surface $F$ (which would be essential by Lemma 2.2) we can add a trivial handle to that surface to create a non-essential and non-totally geodesic surface $F^{\prime}$. However, our algorithm would still detect $F^{\prime}$ as a totally geodesic surface because the image of the fundamental group through the holonomy representation would be the same as $F$. If our algorithm does find such an $F^{\prime}$, we do know that there exists some $F$ in $M$ which is totally geodesic.

In Table 1 we present a list of manifolds that we found to contain a totally geodesic surface. All manifolds were covers of a manifold in the OrientableCuspedCensus provided by SnapPy. Through private correspondence with Nathan Dunfield, he has informed us that all of these manifolds are commensurable to the figure-8 knot complement m004 which, as mentioned in Appendix A.2, is known to have infinitely many immersed closed totally geodesic surfaces.

It is worth mentioning that of the 142,409 link exteriors we ran the algorithm on, none contained totally geodesic surfaces. Our algorithm finished running for all non-alternating knot exteriors of up to 12 crossings and we anticipate more to be completed as our computations are ongoing. These results give strong support for Menasco and Reid’s conjecture.

Here we specifically look in detail at the last manifold in Table 1, the 3-fold cover of the census manifold m412(0,0)(0,0) which we will call $Y$. The triangulation information we used for our computation is recorded as a tight encoding of a Triangulation3 class in Regina:

This triangulation had 15 tetrahedra and its volume was $15.2241240961448$. Algorithm 3 found one potential totally geodesic surface which we will call $S$. $S$ was a surface of genus $2$ whose corresponding normal coordinate $\overrightarrow{S}$ is given as the following vector:

Let $\Gamma$ denote the fundamental group representation of $S$ into $PSL(2,\mathbb{C})$ using the holonomy representation of $Y$ as in Algorithm 3. An approximation of the limit set of $S$ is presented in Figure 13. All limit set approximations in this paper consist of $10,000$ points obtained by applying up to $2000$ randomly chosen matrices in $\Gamma$ to the complex point $(1,0)$. It is evident from Figure 13 that the approximation resembles a geometric circle.

We also computed the trace field of $\Gamma$. To compute the trace field of $\Gamma$, we computed the traces of all generators and products of pairs and triples of all of the generators. Using SageMath, we calculated the smallest field containing these traces. The result was that the trace field of $\Gamma$ is $\mathbb{Q}(\sqrt{3})$. The traces of all elements of $\Gamma$ are real and the surface $S$ is Fuchsian. Note that all coordinates of the vector of $S$ are even. Hence $S=2F$ for some non-orientable surface $F$. By above, we have that $F$ is totally geodesic while $S$ is just Fuchsian.