Bordered surfaces in the 3-sphere with maximum symmetry

# Bordered surfaces in the 3-sphere with maximum symmetry

## Abstract

We consider orientation-preserving actions of finite groups on pairs , where denotes a compact connected surface embedded in . In a previous paper, we considered the case of closed, necessarily orientable surfaces, determined for each genus the maximum order of such a for all embeddings of a surface of genus , and classified the corresponding embeddings.

In the present paper we obtain analogous results for the case of bordered surfaces (i.e. with non-empty boundary, orientable or not). Now the genus gets replaced by the algebraic genus of (the rank of its free fundamental group); for each we determine the maximum order of an action of , classify the topological types of the corresponding surfaces (topological genus, number of boundary components, orientability) and their embeddings into . For example, the maximal possibility is obtained for the finitely many values and .

## 1 Introduction

We study smooth, faithful actions of finite groups on pairs where denotes a compact, connected, bordered surface with an embedding (so is a finite group of diffeomorphisms of a pair ). We also say that such a -action on is extendable (w.r.t. ).

Let denote the orientable compact surface of (topological) genus with boundary components, writing also instead of ; for , let denote the non-orientable compact surface of genus with boundary components. is obtained from the connected sum of real projective planes by creating boundary components (by deleting the interiors of disjoint embedded disks), and it is well-known that each compact surface is either or .

For , and are bordered surfaces, and we use to denote their algebraic genus equal to the rank of the free fundamental group ; this is also the genus of a regular neighbourhood of in which is a 3-dimensional handlebody. We have and . We will always assume that in the present paper.

We will consider only orientation-preserving finite group actions on ; then, referring to the recent geometrization of finite group actions on (see [BMP] for the case of non-free actions and [Pe] for the general case), we can restrict to orthogonal actions of finite groups on , i.e. to finite subgroups of the orthogonal group .

Let denote the maximum order of such a group acting on a pair , for all embeddings of bordered surfaces of a fixed algebraic genus into . In the present paper we will determine and classify all surfaces which realize the maximum order .

A similar question for the pair , where is the closed orientable surface of genus , was studied in [WWZZ1]. The corresponding maximum order of finite groups acting on for all possible embeddings was obtained in that paper.

Let denote the handlebody of genus . Each bordered surface of algebraic genus has a regular neighborhood which is homeomorphic to . We note that, similar as for handlebodies, the maximal possibilities for the orders of groups of homeomorphisms of compact bordered surfaces of algebraic genus are , , , , … (see section 3 of [MZ]), and these are exactly the values occurring in the next theorem. A classification of all finite group actions on compact bordered surfaces up to algebraic genus is given in [BCC], and the lists in that paper may be compared with the list in the next theorem. Concerning other papers considering symmetries of surfaces immersed in 3-space, see [CC], [CH] and [LT].

Our main result is:

###### Theorem 1.1.

For each , and the surfaces realizing are listed below.

-3.2em0em , , , , , , , , , , , , , , , , , , , the remaining numbers , ( even), ( odd),

Acknowledgement. This work was supported by National Natural Science Foundation of China (Grant Nos. 11371034 and 11501239).

## 2 The case of closed surfaces: 3-orbifolds and main results in [Wwzz1]

For orbifold theory, see [Th], [Du1] or [BMP]. We give a brief introduction here for later use.

All of the -orbifolds that we consider have the form . Here is an orientable -manifold and is a finite group acting faithfully on , preserving orientation. For each point , denote its stable subgroup by , its image in by . If , is called a singular point with index , otherwise it is called a regular point. If we forget the singular set we get the topological underlying space of the orbifold.

We can also define covering spaces and the fundamental group of an orbifold. There is an one-to-one correspondence between orbifold covering spaces and conjugacy classes of subgroups of the fundamental group, and regular covering spaces correspond to normal subgroups. A Van-Kampen theorem is also valid, see [BMP, Corollary 2.3]. In the following, automorphisms, covering spaces and fundamental groups always refer to the orbifold setting.

We call (resp. a discal (resp. spherical, handlebody) orbifold. Here (resp. ) denotes the -dimensional ball (resp. sphere). By classical results, is a disk, possibly with one singular point; belongs to one of the five models in Figure 1, corresponding to the five classes of finite subgroups of . Here the labeled numbers denote indices of interior points of the corresponding edges. can be obtained by pasting finitely many along some in their boundaries. It is easy to see that the singular set of a 3-orbifold is always a trivalent graph .

Suppose acts on , where is a closed surface. Call a 2-orbifold allowable if . A sequence of observations about allowable 2-orbifolds were made in [WWZZ1] (Lemma 2.4, 2.7, 2.8, 2.9, 2.10), in particular: Suppose is allowable, then

(i) ;

(ii) is -surjective;

(iii) with four singular points having one of the following types: , , , ;

(iv) bounds a handlebody orbifold which is a regular neighborhood of either an edge of the singular set or a dashed arc, presented in (a) or (b) of Figure 2 (so a dashed arc does not belong to the singular set). Here labels are omitted in (a), and more description of (b) will be given later.

(i) allows us to consider only Dunbar’s famous list in [Du1] of all spherical 3-orbifolds with underlying space . Searching for all possible 2-suborbifolds that satisfy the conditions (ii), (iii) and (iv) by further analysis from topological, combinatoric, numerical, and group theoretical aspects leads to a list in Theorem 6.1 of [WWZZ1], presented here as Theorem 2.1. We will first need to explain the terminology in the statement of Theorem 2.1 and the notation in the accompanying tables.

Since all the spherical 3-orbifolds we consider have underlying space , they are determined by their embedded labeled singular trivalent graphs. From now on, a singular edge always means an edge of , the singular set of the orbifold; singular edges with index are not labeled; and a dashed arc is always a regular arc with two ends at two edges of with indices 2 and 3 as in Figure 2(b). An edge/dashed arc is allowable if the boundary of its regular neighborhood is an allowable 2-orbifold.

For each 3-orbifold in the list of Theorem 2.1, the order of is given first. Then singular edges/dashed arcs are listed, which are marked by letters to denote the boundaries of their regular neighborhoods. Then singular types of the boundaries and genera of their pre-images in are given. When the singular type is , there are two subtypes denoted by I and II, corresponding to Figure 2(a) and Figure 2(b) (exactly the dashed arc case).

We say that an orientable separating 2-suborbifold (2-subsurface) in an orientable 3-orbifold (3-manifold) is unknotted or knotted, depending on whether it bounds handlebody orbifolds (handlebodies) on both sides. A singular edge/dashed arc is unknotted or knotted, depending on whether the boundary of its regular neighborhood is unknotted or knotted.

If a marked singular edge/dashed arc is knotted, then it has a subscript ’’. If a marked dashed arc is unknotted, then there also exists a knotted one (indeed infinitely many) and it has a subscript ’’. Call two singular edges/dashed arcs equivalent, if there is an orbifold automorphism sending one to the other, or the boundaries of their regular neighborhoods as 2-orbifolds are orbifold-isotopic.

The way to list orbifolds in Theorem 2.1 is influenced by the lists of [Du1] and [Du2]. The labels below the orbifold pictures come from [WWZZ1, Tables I, II and III]. In picture 15E and picture 19, the letter refers to particular choices of parameters in infinite families.

###### Theorem 2.1.

Up to equivalence, the following tables list all allowable singular edges/dashed arcs except those of type II. In the type II case, if there exists an allowable dashed arc in some , then and one such arc in it are listed, and this arc in the list will be unknotted if there exists an unknotted one in .

: II, : II,

: II, : II,

: II, : II,

: I, : II, : I,

: I, : I, : I, : II,

: I, : I, : II,

:(2,2,3,4), :(2,2,3,5),

:(2,2,2,n) :(2,2,2,n)

:(2,2,2,3), :(2,2,2,3), :(2,2,2,3),

:(2,2,2,4), :(2,2,3,4),

:(2,2,2,3), :(2,2,2,3), :(2,2,2,5), :(2,2,2,5),

:(2,2,3,5),

: I, :(2,2,2,3), : I, : II,

:(2,2,2,4), : I, :(2,2,2,3), : I, : II,

:(2,2,2,3), :(2,2,2,4), :(2,2,3,4), :(2,2,2,3), :(2,2,2,5), :(2,2,3,5), : II,

:(2,2,2,3), :(2,2,3,4), :(2,2,2,3), :(2,2,3,5),

No allowable 2-suborbifold : I, : II,

:(2,2,2,3), :(2,2,2,3), :(2,2,2,3), : II,

: I, : II, : I, : II,

: I, : II, :(2,2,2,3) : II,

: I, : II, : I, : I, : II,

## 3 The case of bordered surfaces: Edges in 3-orbifolds providing bordered surface embeddings with maximum symmetry

Suppose now that is a finite group acting on some bordered surface ; then each singular point in the orbifold is of one of three types (see Figure 3): (a) an isolated singular point lying in the inner of , corresponding to a cyclic stable subgroup; (b) a singular point lying on a reflection boundary, corresponding to a stable subgroup; (c) a corner point, corresponding to a dihedral stable subgroup.

Figure 3 Possible singular points of 2-orbifolds

We first recall an example of [WWZZ1], which gives a lower bound for .

###### Example 3.1.

For every , we will construct a group of order which acts on . Let be the equator sphere of with punctured holes, see Figure 4 for . We choose the holes all on the equator of , centered at the vertices of a regular -polygon. There is a dihedral group acting on which keeps invariant. And there is also a action on changing the inner and outer of , whose fixed point set is the equator of the 2-sphere in Figure 4. So there is a action on . This group has order and corresponds to the orbifold 15E in Table 2 of Theorem 2.1.

Figure 4 The 2-sphere with five boundary components

Suppose acts on . Consider a -equivariant regular neighborhood of which is a handlebody ; then acts on . By the previous example, if this action realizes , then we have . So as discussed in Section 2, can be of two types as in Figure 2.

If is of type (a) then since is a regular neighborhood of , the two degree- singular points (i.e., valence three vertices of the singular trivalent graphs) together with the middle singular edge joining them must lie in . So the singular edge corresponds to a reflection boundary of , and the two degree- singular points correspond to two corner points of . So in this case, is a disk with three reflection boundaries and two corner points. Consider ; this boundary arc is isotopy to an arc on the boundary of , and it is an arc joining two singular points on . is a sphere with four singular points, and we are now considering an arc joining two of them and the arc is disjoint from the other two. Since a sphere minus two points has fundamental group , the isotopy class of is in the sequence indicated in Figure 5(a).

If is of type (b) then the index- singular arc may be either a reflection boundary of or intersect in an isolated point, and the index- singular point must be isolated in . So there are two cases as shown in Figure 5(b).

Figure 5 Possible embeddings of 2-orbifolds into 3-orbifolds

From the discussion above, not all the allowable orbifolds given in Theorem 2.1 corresponds to some bordered surface orbifold . For type (I), we have seen that if the singular edge corresponds to some , then it must have index , and the stable subgroups corresponding to its two ends must be dihedral groups.

Call two singular edges/dashed arcs equivalent if there is an orbifold automorphism sending one to the other. However, if two (nonequivalent) singular edges/dashed arcs have regular neighborhoods with isotopic boundaries (so that an unknotted -orbifold splits the spherical -orbifold into two handlebody orbifolds), then we say that the edges/arcs are dual to each other.

In the study of maximum symmetry of closed surfaces, it is reasonable to call two dual edges equivalent since they produce the same allowable -orbifold. But it is not good to call them equivalent in the study of maximum symmetry of bordered surfaces since they may correspond to different -orbifolds or to different bordered surfaces.

Then by a routine checking of Theorem 2.1, we have the following Theorem 3.2 for our further study of the realizations of the maximum symmetry of in the following sense: (1) we only pick information from Theorem 2.1 related to (so among 40 orbifolds listed in Theorem 2.1, only 16 orbifolds appear in Theorem 3.2); and list the orbifolds according to the sizes of (so one figure in Theorem 2.1 can become several figures in Theorem 3.2, for example, 15E or 27). (2) In Table 4 of Theorem 3.2, , , … denote edges which are dual to the edges , , … in the tables of Theorem 2.1; for example, the edge of orbifold 26 in Table 3 of Theorem 2.1 is not allowable since it has singular order three but its dual edge has singular order two and is allowable (see Table 4 of Theorem 3.2).

The topological type of the bordered surfaces are also listed in the table. The method to determine their topological genus and the number of boundary components will be explained in the next section.

###### Theorem 3.2.

Up to equivalent embeddings, all the edges/arcs corresponding to bordered surfaces with maximum symmetry are listed below (for dashed arcs only one position is showed).

and

,

Others,

, belongs to remaining numbers

Others, , ,

## 4 The bordered surfaces realizing the maximum symmetry

We use the following lemma from [WWZZ2] to compute the number of boundary components of the preimage of .

###### Lemma 4.1 (Lemma 2.7 [Wwzz2]).

Suppose that is simply connected with an action of , and that is a suborbifold of such that is connected. Let be the inclusion map; then the preimage of in has connected components.

Then we use the following lemmas to determine whether the preimage of is oriented.

###### Lemma 4.2 (Lemma 2.5 [Wwzz2]).

Suppose that acts on a compact surface such that each singular point in the orbifold is isolated and the underlying space is orientable; then is orientable.

###### Lemma 4.3.

Suppose that acts on a compact surface such that has at least one reflection boundary and the underlying space is orientable. Then is oriented if and only if there is a group homomorphism sending each element corresponding to a reflection boundary of to the non-trivial element in .

###### Proof.

If is oriented, let be the subgroup of which contains all elements preserving the orientation of ; then is the desired homomorphism. Conversely, if is such a homomorphism, then consider a double cover of corresponding to the subgroup of the kernel of . has only isolated singular points, so by the previous lemma is oriented. ∎

So using these lemmas, we check all the allowable coming from allowable singular edges/dashed arcs in Table 4 of Theorem 2.2 and determine the number of boundary components, orientability and the topological genus of their preimages in .

###### Proof of Theorem 1.1.

Note that in Table 4 only two orbifolds E and have a free parameter ; for these two orbifolds, we check the preimages of possible embeddings of as follows.

In the orbifold E, we lift the embedding of to a double cover , where is an abelian group of order :

 ˜G=⟨x,y∣x2,yn,xyx−1y−1⟩.

Here and are the generators corresponding to the index and index singular circles in .

One possible embedding of is as in Figure 6; by lifting to a double cover, the fundamental group of the boundary of in corresponds to the subgroup generated by and (the image of the fundamental group of the orbifold corresponding to the boundary of ). So this subgroup has index in , which means has boundary components by Lemma 4.1. Furthermore, in this case,

 π1(˜X)=⟨x,y⟩

where corresponds to the reflection boundary and corresponds to the isolated singular point in . If we take

 Z2≅⟨t∣t2⟩,

then is a homomorphism. So by Lemma 4.3, is oriented. Then by the relation , we have .

Figure 6 Embeddings of 2-orbifolds into 3-orbifolds

Another possible embedding of is as in Figure 7; by lifting to a double cover, the image of the fundamental group of the boundary of in corresponds to the subgroup generated by . This subgroup has index if is odd, and index is is even. By Lemma 4.2, is oriented, so for odd and for even.

Figure 7 Embeddings of 2-orbifolds into 3-orbifolds

For other possible embeddings of in the orbifold E (we mean in the sequence given in Figure 5(a)), it is easy to see is diffeomorphic to one case discussed above.

In the orbifold , one possible embedding of is as in Figure 8. By lifting to a -sheet cover, the boundary of in corresponds to the subgroup generated by , where

 ˜G=⟨x,y∣xn,yn,xyx−1y−1⟩.

This subgroup has index . By Lemma 4.2, is oriented. So . For other possible embeddings of in the orbifold , it is easy to see that we get the same .

Figure 8 Embeddings of 2-orbifolds into 3-orbifolds

We also remark that in these orbifolds, one can also directly lift to to see which surface is. But we think it is a good example here to show how the Lemmas given above work. In all other orbifolds given in Table 4, there are no parameters and only finitely many groups , so we can check the situation by some computer software case by case. We present two such examples, one for the case of a singular edge and one for a dashed arc; in all other cases, the computations are similar. ∎

###### Example 4.4 (Singular edge case).

Let us show how we use [GAP] to determine the bordered surface type for the singular edge in orbifold . First we use the Wirtinger presentation with generators as in Figure 9 to get

Figure 9 Generators of the fundamental group of a tetrahedral 3-orbifold

 G=π1(O28)=⟨x,y,z∣x5,y2,z2,(xz)3,(xy)2,(yz−1)2⟩.

Then consider the possible embeddings of with as one of its reflection edge. The other reflection edge on the left corner must correspond to the edge labeled , and the other reflection edge on the right corner may correspond to the edge labeled or . Note also that the embedding of may have some twist around , so the subgroup corresponding to the boundary of is one of the following:

 G1=⟨xy,xyx−1⟩⊂G,
 G2=⟨xy,xzx−1⟩⊂G,
 G3=⟨xy,(xyz−1)y(xyz−1)−1⟩⊂G,
 G4=⟨xy,(xyz−1)z(xyz−1)−1⟩⊂G.

Then we use [GAP] to compute the index to get the number of boundary components: and . Also we use Lemma 4.3 to determine if is oriented: the surfaces corresponding to and are oriented, the surfaces corresponding to and are non-oriented. So we get and for this case. Codes for [GAP] are listed here.

f := FreeGroup( "x","y","z");
G := f / [f.1^5, f.2^2, f.3^2, (f.1*f.3)^3, (f.1*f.2)^2,
(f.2*f.3^-1)^2];
Print(Size(G),"\n"); #120
f2 := FreeGroup( "t"); Z2 := f2 / [f2.1*f2.1]; x:=G.1;
y:=G.2; z:=G.3; midarc:=x*y*z^-1*x^-1; rightArc1:=x*y*x^-1;
rightArc2:=x*z*x^-1; leftArc:=x*y; algebGenus := 11;
iso1:=IsomorphismPermGroup(G); iso2:=IsomorphismPermGroup(Z2);
g1:=Image(iso1); z2:=Image(iso2); img1:=Image(iso1,midarc);
img2:=Image(iso1,rightArc1); img3:=Image(iso1,leftArc);
img22:=Image(iso1, rightArc2); img4:=Image(iso2,Z2.1); l:=[];
if GroupHomomorphismByImages(g1,z2,[img1,img2,img3],[img4,img4,img4])<>fail then
Print("oriented!\n"); oriented:=1;
else
Print("non-oriented!\n"); oriented:=0;
fi;
borderSubG := GroupWithGenerators([rightArc1,leftArc]);
b:=Index(G, borderSubG);
if oriented=1 then
g:=(algebGenus+1-b)/2;
else
g:=algebGenus+1-b;
fi;
borderSubG := GroupWithGenerators([rightArc1,midarc*leftArc*midarc^-1]);
b:=Index(G, borderSubG);
if oriented=1 then
g:=(algebGenus+1-b)/2;
else
g:=algebGenus+1-b;
fi;
if GroupHomomorphismByImages(g1,z2,[img1,img22, img3],[img4,img4,img4])<>fail then
Print("oriented!\n"); oriented:=1;
else
Print("non-oriented!\n"); oriented:=0;
fi;
borderSubG := GroupWithGenerators([rightArc2,leftArc]);
b:=Index(G, borderSubG);
if oriented=1 then
g:=(algebGenus+1-b)/2;
else
g:=algebGenus+1-b;
fi;
borderSubG := GroupWithGenerators([rightArc2,midarc*leftArc*midarc^-1]);
b:=Index(G, borderSubG);
if oriented=1 then
g:=(algebGenus+1-b)/2;
else
g:=algebGenus+1-b;
fi;
Print(l, "\n");

Remark. In this case the group is of order , and all computations can be easily done by hand, by working with permutations and cycles in (and similarly for various other small orders). However, for larger orders it is convenient to use some computer algebra.

###### Example 4.5 (Dashed arc case).

We use again the orbifold , considering now the embeddings of corresponding to the dashed arc as shown in Figure 10.

Figure 10 Dashed arc in a tetrahedral 3-orbifold

We use the same Wirtinger presentation as in the previous example. The dashed arc may be any arc in the orbifold with the same endpoints as that in Figure 10. So the subgroup corresponding to some embedding of should be generated by and , here may be any element in . [GAP] can systematically create all elements of the group by some standard procedure. For each element we first need to check if and generates the whole group of , which means the preimage of is connected. Then if is as Figure 5(b) left, then the subgroup corresponding to the boundary of is generated by or . If is as Figure 5(b) right, then the subgroup corresponding to the boundary of is generated by or . Then we use Lemma 4.1 to determine the number of boundary components by the index corresponding to their subgroups: in fact, all these subgroups have index .. We use Lemma 4.2 or 4.3 to determine if it is oriented: they are all oriented. So we get for this case.

f := FreeGroup( "x","y","z");
G := f / [f.1^5, f.2^2, f.3^2, (f.1*f.3)^3, (f.1*f.2)^2,
(f.2*f.3^-1)^2];
Print(Size(G),"\n"); #120
x:=G.1; y:=G.2; z:=G.3; f2 := FreeGroup( "t");
Z2 := f2 / [f2.1*f2.1]; iso1:=IsomorphismPermGroup(G);
iso2:=IsomorphismPermGroup(Z2); g1:=Image(iso1);
z2:=Image(iso2); img1:=Image(iso1,y); img2:=Image(iso1,x*z);
img3:=Image(iso2,Z2.1); img4:=Image(iso2,Z2.1*Z2.1^-1);
l:=[]; oriented:=0; algebGenus := 1 + Size(G)/6;
if GroupHomomorphismByImages(g1,z2,[img1,img2],[img3,img4])<>fail then
Print("oriented!\n"); oriented:=1;
else
Print("non-oriented!\n");
fi;
for e in G do
conjEle:=e^-1*x*z*e;
subG:=GroupWithGenerators([y,e^-1*x*z*e]);
if Index(G, subG)=1 then
borderSubG := GroupWithGenerators([y*e^-1*x*z*e]);
b:=Index(G, borderSubG);
g:=(algebGenus+1-Index(G, borderSubG))/2;
if ([g,b,1] in l)<>true then
fi;
borderSubG := GroupWithGenerators([y^-1*e^-1*x*z*e]);
b:=Index(G, borderSubG);
g:=(algebGenus+1-Index(G, borderSubG))/2;
if ([g,b,1] in l)<>true then
fi;
borderSubG := GroupWithGenerators([y, conjEle^-1*y*conjEle]);
b:=Index(G, borderSubG);
if oriented=1 then
g:=(algebGenus+1-Index(G, borderSubG))/2;
else
g:=(algebGenus+1-Index(G, borderSubG));
fi;
if ([g,b,oriented] in l)<>true then
fi;
conjEle:=e^-1*(x*z)^-1*e;
borderSubG := GroupWithGenerators([y, conjEle^-1*y*conjEle]);
b:=Index(G, borderSubG);
if oriented=1 then
g:=(algebGenus+1-Index(G, borderSubG))/2;
else
g:=(algebGenus+1-Index(G, borderSubG));
fi;
if ([g,b,oriented] in l)<>true then
fi;
fi;
od;
Print(l,"\n");

Chao Wang, School of Mathematical Sciences, University of Science and Technology of China, 230026 Hefei, CHINA

Shicheng Wang, School of Mathematical Sciences, 100871 Beijing, CHINA

Yimu Zhang, Mathematics School, Jilin University, 130012 Changchun, CHINA

Bruno Zimmermann, Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, 34127 Trieste, ITALY

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