A classification of maximally symmetric surfaces in the 3dimensional torus
Abstract.
If a finite group of orientationpreserving diffeomorphisms of the 3dimensional torus leaves invariant an oriented, closed, embedded surface of genus and preserves the orientation of the surface, then its order is bounded from above by . In the present paper we classify (up to conjugation) all such group actions and surfaces for which the maximal possible order is achieved, and note that the unknotted surfaces can be realized by equivariant minimal surfaces in a 3torus.
Key words and phrases:
surfaces embedded in the 3torus, finite group action, Euclidean 3orbifold2010 Mathematics Subject Classification:
Primary 57M60; Secondary 57S251. Introduction
All manifolds and maps considered in the present paper are smooth, and group actions are faithful and orientationpreserving.
Definition 1.1.
Let be a closed, connected, orientable surface of genus and be a finite group. A action on is extendable over a 3manifold with respect to an embedding if there is a action on such that , for all . Identifying with , we will also shortly say that acts on the pair . We will always assume in the present paper.
A classical result of Hurwitz says that the order of a finite group action on a surface is bounded by [Hu], and an action realizing this bound is usually called a Hurwitz action; in general, the Hurwitz actions are not classified. For actions on surfaces which extend to a 3dimensional handlebody, the upper bound is [Zi1], and again the actions realizing this upper bound are not classified. More generally, if we require that the actions on surfaces extend to a certain 3manifold , then there will be an upper bound and hopefully one can classify the actions realizing this bound.
The most natural choices of include the 3dimensional Euclidean space and the 3dimensional sphere . In each case, the classification does exist, and it is stronger in the sense that for each given the maximum of the group order can be obtained and the actions realizing the maximum can be classified (see [WWZZ3] for and [WWZZ1, WWZZ2] for ).
The 3dimensional torus is another natural choice. In this case, the upper bound is again as shown in [BRWW], as a consequence of the equivariant loop theorem [MY] (since an embedded surface of genus in the 3torus has to be compressible) and the formula of RiemannHurwitz. In [BRWW], various series of actions of maximal possible order are constructed and a conjectural picture of the situation is given. In the present paper, we obtain a complete classification for the maximal case ; in particular, this confirms the conjecture in [BRWW].
Definition 1.2.
A closed subsurface in a closed 3manifold is unknotted if the surface separates the 3manifold into two handlebodies (so it is a Heegaard surface of a Heegaard splitting of the 3manifold), otherwise the surface is knotted.
Theorem 1.3.
If a action on with order is extendable over , then has one of the forms: , , , , where is a positive integer. Up to conjugation, such actions on the pair are listed below:
Here each number represents an action on . If appears times, then there are different actions. The actions in the first three columns correspond to unknotted surfaces, all others to knotted ones.
For example, since , there are five actions on realizing the maximal order . For three of them the surface is unknotted and for two of them the surface is knotted. We will derive Theorem 1.3 from a stronger classification result Theorem 3.6 in Section 3, and the nine columns in Theorem 1.3 correspond to the nine cases of Theorem 3.6. Theorem 3.6 shows that all actions realizing the bound are actually listed in the examples of [BRWW], in particular it follows that all the unknotted surfaces can be realized by equivariant minimal surfaces.
Corollary 1.4.
If a Heegaard surface of is invariant under a finite group action of order , then it can be realized by an equivariant minimal surface for some Euclidean structure on .
This confirms the following natural question for Euclidean 3manifolds. Actually, the 3torus and the HantzscheWendt manifold (see [Zi2]) are the only orientable closed Euclidean 3manifolds containing such surfaces. The question is also partly confirmed for spherical 3manifolds. By [La], [KPS] and [BWW], it is true for the case of with three possible exceptions.
Question 1.5.
If a Heegaard surface of an orientable closed geometric 3manifold is invariant under a finite group action of order , can it be realized by an equivariant minimal surface for some geometric structure on ?
An example of a hyperbolic 3manifold with such a Heegaard surface is the SeifertWeber dodecahedral space, obtained by identifying opposite faces of a regular hyperbolic dodecahedron with dihedral angles , after a twist by of each face ([SW], [Th2,p.36]). After the identifications, the boundary of a regular neighborhood of the 12 edges connecting the center of the dodecahedron with the centers of its 12 faces gives a Heegaard surface of genus 6, and by [Po,Figure 2(d)], this Heegaard surface can be realized by an equivariant minimal surface, invariant under the action of isometry group of the dodecahedron. Applying the same construction to the regular spherical dodecahedron with dihedral angles and twisting by , one obtains the spherical Poincaré sphere, with a Heegaard surface of genus 6 invariant under the dodecahedral group , and by [KPS] this can again be realized by an equivariant minimal surface. Finally, applying the construction to the Euclidean cube instead, one obtains the 3torus with a Heegaard surface of genus 3 which can be realized by minimal surface invariant under the isometry group of the cube (corresponding to the case of the first item in the first row of Theorem 1.3).
To classify the actions in Theorem 1.3, we need the orbifold theory (see [BMP, Du, Th1]). After identifying with , an extendable action gives an orbifold pair . Conversely, given a 2orbifold in a 3orbifold and a regular orbifold covering , if is connected, then the group acts on the pair . If is or , then finding all the pairs is enough (as in [WWZZ2, WWZZ3]), because and are simply connected and is determined by . For further information about the covering is needed.
Let be a finite group which acts on a pair and has order ; by the RiemannHurwitz formula, the quotient 2orbifold is a sphere with four singular points of indices ; by the geometrization of finite group actions on 3manifolds, we can assume that acts by Euclidean isometries on , for some Euclidean structure on . Then the quotient 3orbifold is a Euclidean orbifold; by a Bieberbach theorem, there is a minimal covering such that for any covering there is a covering satisfying . Hence the classification factors into two steps:
Step 1.6.
List all pairs such that is a Euclidean 3orbifold and is a sphere with four singular points of indices .
Step 1.7.
For a given pair in Step 1.6 find all coverings such that is a regular covering and is connected.
2. List the pairs
The way to list the orbifold pairs in Step 1.6 is similar to [WWZZ2]. First, we need some general results and conventions from [Du].
In [Du], the Euclidean 3orbifolds are classified. There are two classes: the fibred ones and the nonfibred ones. The fibred ones are the Seifert fibred orbifolds having Euler number and base 2orbifold with Euler characteristic . The nonfibred ones are listed in [Du]. Moreover, the singular sets, which are trivalent graphs, of the Euclidean orbifolds with underlying space are pictured, and the names of the fundamental groups of the orbifolds are given.
Definition 2.1.
Let denote the 2orbifold which is a sphere with singular points of indices ; let denote the 2orbifold which is a disk with singular points of indices in the interior.
Let be the 2orbifold which is a disk with singular points of indices in the interior and corner points of groups in the boundary. Here denotes the dihedral group of order , and the boundary points other than the corner points are reflection points.
Lemma 2.2.
If is a pair as in Step 1.6 and is fibred, then the base 2orbifold of is . As a consequence, has underlying space .
Proof.
Let be the base 2orbifold of . Then the Euler characteristic of is . Since has singular points with indices and , is one of , , and . If is , since the Euler number of is , then can only be which has no suborbifolds of type . If is or , then by the discussion in section 4 and 5 of [Du] the underlying space of is a lens space or . Then separates . Since the singular set of index in consists of circles (with degree singular points removed), it cannot intersect three times. Hence is and by the discussion in section 4 and 5 of [Du] the underlying space of is . ∎
Lemma 2.3.
If is a pair as in Step 1.6 and is nonfibred, then has underlying space .
Proof.
By the classification result in [Du], the only nonfibred Euclidean 3orbifold with underlying space not homeomorphic to has underlying space , and its singular set of index consists of a circle. Then separates and cannot intersect the circle only once. ∎
Lemma 2.4.
If is a pair as in Step 1.6, then bounds a handlebody orbifold which is a regular neighborhood of an edge of the singular set, with boundary .
Proof.
As in [BRWW], the equivariant loop theorem [MY] gives a compression disk of . Since is isomorphic to , the compression disk splits into two orbifolds and . Then .
If , then has negative Euler characteristic and is incompressible, which contradicts the equivariant loop theorem. If , then both and are spherical and bound discal 3orbifolds (as in [WWZZ2]), because is irreducible. Since each one of and cannot lie in the discal 3orbifold bounded by the other one, the union of the two discal 3orbifolds is the handlebody orbifold bounded by .
If , then by the classification result in [Du] the orbifold is fibred. Then by Lemma 2.2 the base 2orbifold of is . There is only one such having singular points of index . Its singular set is pictured as in Figure 1, where can only be the boundary of a regular neighborhood of the edge , up to isomorphism between the pairs . ∎
Proposition 2.5.
Up to isomorphism between orbifold pairs, all orbifold pairs in Step 1.6 are obtained as follows. The underlying topological space of the orbifold is , and its singular set is given by one of the six pictures in Figure 1. The 2suborbifold , of type , is obtained as the boundary of a regular neighborhood of one of the nine marked singular edges , or .
Proof.
By Lemma 2.2 and 2.3, has underlying space . Hence its singular set belongs to the list of pictures in [Du]. By Lemma 2.4, is the boundary of a regular neighborhood of a singular edge in the singular set of . Hence all the possible pairs can be found by enumerating the possible singular edges, which are exactly the marked edges in Figure 1. ∎
Remark 2.6.
Consider the complement of a regular neighborhood of a marked singular edge in Figure 1. It is a handlebody orbifold if and only if the singular edge has mark . Hence only the singular edges correspond to unknotted surfaces; the edges and correspond to knotted ones.
3. Find the coverings
For a given pair in Step 1.6, we will first list the finite index normal translation subgroups of , then we will use a lemma in [WWZZ2] to verify the connectedness.
For each in Figure 1, the representation of as a space group can be found in [Ha]. In the present paper we will use a slightly different representation of . First we need to introduce some notation (following [BRWW]).
Definition 3.1.
Any element can act on as the translation:
Let be the following elements in respectively:
For , let be the following subgroups of :
Let be the following isometries of :
Note that when is one of , , , then is homeomorphic to with volume , and when is one of , , then is homeomorphic to with volume . Moreover, we have
The isometries , and are rotations about the directions , and respectively. The isometries and are righthand rotations about the directions and respectively.
Lemma 3.2.
The universal covering groups of the 3orbifolds in Proposition 2.5 are generated by the following elements, starting with the translation groups (whose indices in the whole groups are always except in the last case where the index is ):

:

:

:

:

:

:
To list the finite index normal translation subgroups of the above groups, we need the following two lemmas.
Lemma 3.3.
For a translation of , its conjugates under , , , , are the following translations:
Lemma 3.4.
Let be a discrete group consisting of translations of .
(1) If is invariant under the conjugation of , then there is such that is one of the three groups:
(2) If is invariant under the conjugation of , then there are such that is one of the two groups:
Proof.
We can assume that is nontrivial.
(1) Since is discrete, there is an element of having nonzero minimum distance to . Since is also an element of , by Lemma 3.3 we can assume that . Since and are elements of ,
is a nonzero element of . By the choice of , we have
Hence . We can also have and other similar inequalities about and . Hence the nonzero ones in are equal. Let it be .
If there are two zeros in , then contains as a subgroup. For any , there is such that . Then
By the choice of , we have and equals .
If there is exactly one zero in , then contains as a subgroup. For any , there is such that . Then
By the choice of , we have and equals .
Otherwise, contains as a subgroup. For any , there is such that . Then
By the choice of , we have and equals .
(2) By Lemma 3.3, for any element in the two elements
belong to . Hence and belong to . Consider the subgroups
Then is the direct sum of and . Clearly there is such that .
We can assume that is nontrivial. Then there is in having nonzero minimum distance to . By Lemma 3.3 we can assume that . Since
is an element of , if , then by the choice of we have
Hence . Similarly we can have . Hence .
If , let . Then contains as a subgroup. By the choice of , it is easy to see that equals . Hence is .
If , let . Then is . ∎
Proposition 3.5.
All finite index normal translation subgroups of the fundamental groups of the 3orbifolds in Proposition 2.5 are as below, where .

: , , (with indices , , ).

: , , (with indices , , ).

: , , (with indices , , ).

: , , (with indices , , ).

: , , (with indices , , ).

: , (with indices , ).
Proof.
By Lemma 3.3, all the listed groups are normal translation subgroups with finite index. Note that in the representations in Lemma 3.2 are the maximal translation subgroups respectively, because for such a group in the corresponding space group the action on contains no translations. Then the proof can be finished by using Lemma 3.4, because the generators of the required group can be uniquely presented by the generators of the maximal translation subgroup and the parameters must have certain forms.
As an example, let be a finite index normal translation subgroup of the space group . By Lemma 3.4, there is such that is one of
If it is , then since , we have . Hence and .
If it is , then since
we have and . Hence and .
If it is , then since
we have and . Hence and . ∎
Theorem 3.6.
Up to conjugation, all actions of maximal possible order on a pair , with , are obtained as the regular coverings of the orbifolds corresponding to the following normal translation subgroups of (where is the boundary of a regular neighborhood of one the nine singular edges denoted by , or in Figure 1).

: , , .

: , , .

: , , .

: .

: , .

: , .

: .

: , , .

: , .
Proof.
Note that for each in Proposition 2.5 the minimal covering corresponds to the maximal translation subgroup of , and a regular covering as in Step 1.7 corresponds to a finite index normal translation subgroup of .
Let denote a marked singular edge in . To finish the proof, we need to check for each of the normal translation subgroups in Proposition 3.5 whether is connected. In the unknotted cases, i.e. for each of the three edges in Figure 1, this follows immediately from Lemma 3.7 (since the fundamental group of a regular neighborhood of an edge clearly surjects onto the fundamental group of , or onto the fundamental groups of the two handlebody orbifolds into which splits).
The general case is a consequence of the following two claims.
Claim 1.
is a connected graph in .
Claim 2.
Let denote the embedding of in , then in each case and the image of in are given in the list below.

: , .

: , .

: , .

: , .

: , .

: , .

: , .

: , .

: , .
Now let be a subgroup in Proposition 3.5 and be its corresponding covering. Then there is a covering such that . Since is connected, by Lemma 3.7 the graph is connected if and only if
Hence assuming the two claims one can check this condition case by case to obtain Theorem 3.6.
The two claims can be shown as following.
Since the representation of as a space group is given in Lemma 3.2, one can get the prefundamental domain of the action on , which consists of points in satisfying
Modular the action of the stable subgroup of one can get the fundamental domain of the action, and folding up the fundamental domain the 3orbifold can be obtained. Then the position of the singular edge can be determined and the part of in a fundamental domain of the action on can be obtained. Finally, the two claims can be checked.
In section 4, we will give an explicit example to illustrate this procedure. ∎
Lemma 3.7 ([Wwzz2]).
Suppose that a finite group acts on , where is a 3manifold with an embedding of a surface. We have diagrams:
Suppose that is connected. Then is connected if and only if
Remark 3.8.
The nine classes in Theorem 3.6 correspond to the nine examples in [BRWW]. Note that each of the examples must correspond to some or or . In and the graphs and can be distinguished by the local stable subgroups. In the graphs and can be distinguished by the property of whether the corresponding surface is knotted or not. Then we have the following correspondence where