On large orientation-reversing finite group-actions on 3-manifolds and equivariant Heegaard decompositions

TL;DR

This paper characterizes 3-manifolds and finite groups that achieve the maximal order of orientation-reversing actions on Heegaard splittings, linking geometric structures to group actions.

Contribution

It provides a classification of 3-manifolds and groups realizing the maximal finite group-action order on Heegaard splittings, including reducible and irreducible cases.

Findings

Maximal order of group actions is 24(g-1) for genus g > 1.

Reducible manifolds are doubles of handlebodies with maximal actions.

Irreducible manifolds are quotients of geometries by Coxeter groups.

Abstract

We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g. The maximal possible order of a finite group-action of an orientable or nonorientable handlebody of genus g > 1 is 24(g-1), and in the present paper we characterize the 3-manifolds M and groups G for which the maximal possible order |G| = 24(g-1) is obtained, for some G-invariant Heegaard splitting of genus g > 1. If M is reducible then it is obtained by doubling an action of maximal possible order 24(g-1) on a handlebody of genus g. If M is irreducible then it is a spherical, Euclidean or hyperbolic manifold obtained as a quotient of one of the three geometries by a normal subgroup of finite index of a Coxeter group associated to a Coxeter tetrahedron, or of a twisted version of such a Coxeter group.

Equations

$${\bar{D}}_2{\ast }_{{\bar{Z}}_2}{\bar{D}}_3,{\bar{D}}_3{\ast }_{{\bar{Z}}_3}{\bar{A}}_4,{\bar{D}}_4{\ast }_{{\bar{Z}}_4}{\bar{S}}_4,{\bar{D}}_5{\ast }_{{\bar{Z}}_5}{\bar{A}}_5,$$

$$D_{2\ast 2}{\ast }_{{\bar{Z}}_2}{\bar{D}}_3,D_{2\ast 3}{\ast }_{{\bar{Z}}_3}{A}_4,D_{2\ast 4}{\ast }_{{\bar{Z}}_4}{\bar{S}}_4,D_{2\ast 5}{\ast }_{{\bar{Z}}_5}{\bar{A}}_5;$$

$${\mathcal{H}}_2=\mathcal{H}({\bar{D}}_2{\ast }_{{\bar{Z}}_2}{\bar{D}}_3),{\mathcal{H}}_3=\mathcal{H}({\bar{D}}_3{\ast }_{{\bar{Z}}_3}{\bar{A}}_4),{\mathcal{H}}_4=\mathcal{H}({\bar{D}}_4{\ast }_{{\bar{Z}}_4}{\bar{S}}_4),$$

$${\mathcal{H}}_5=\mathcal{H}({\bar{D}}_5{\ast }_{{\bar{Z}}_5}{\bar{A}}_5),$$

$${\tilde{\mathcal{H}}}_2=\mathcal{H}(D_{2\ast 2}{\ast }_{{\bar{Z}}_2}{\bar{D}}_3),{\tilde{\mathcal{H}}}_3=\mathcal{H}(D_{2\ast 3}{\ast }_{{\bar{Z}}_3}{\bar{A}}_4),{\tilde{\mathcal{H}}}_4=\mathcal{H}(D_{2\ast 4}{\ast }_{{\bar{Z}}_4}{\bar{S}}_4),$$

$${\tilde{\mathcal{H}}}_5=\mathcal{H}(D_{2\ast 5}{\ast }_{{\bar{Z}}_5}{\bar{A}}_5).$$

$$D_2{\ast }_{Z_2}{D}_3,D_3{\ast }_{Z_3}{A}_4,D_4{\ast }_{Z_4}{S}_4,D_5{\ast }_{Z_5}{A}_5$$

Full text

On large orientation-reversing finite group-actions on 3-manifolds

and equivariant Heegaard decompositions

Bruno P. Zimmermann

Università degli Studi di Trieste

Dipartimento di Matematica e Geoscienze

34127 Trieste, Italy

Abstract. We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of $M$ of some genus $g$. The maximal possible order of a finite group-action of an orientable or nonorientable handlebody of genus $g>1$ is $24(g-1)$, and in the present paper we characterize the 3-manifolds $M$ and groups $G$ for which the maximal possible order $|G|=24(g-1)$ is obtained, for some $G$-invariant Heegaard splitting of genus $g>1$. If $M$ is reducible then it is obtained by doubling an action of maximal possible order $24(g-1)$ on a handlebody of genus $g$. If $M$ is irreducible then it is a spherical, Euclidean or hyperbolic manifold obtained as a quotient of one of the three geometries by a normal subgroup of finite index of a Coxeter group associated to a Coxeter tetrahedron, or of a twisted version of such a Coxeter group.

1. Introduction

The maximal possible order of a finite group $G$ of orientation-preserving diffeomorphisms of an orientable handlebody of genus $g>1$ is $12(g-1)$ ([Z1], [MMZ]); for orientation-reversing finite group-actions on an orientable handlebody and for actions on a nonorientable handlebody, the maximal possible order is $24(g-1)$; we will always assume $g>1$ in the present paper.

Let $G$ be a finite group of diffeomorphisms of a closed, orientable or nonorientable 3-manifold $M$. We define the (equivariant) Heegaard genus of such a $G$-action as the minimal genus $g>1$ of a Heegaard decomposition of $M$ into two handlebodies of genus $g$ (nonorientable if $M$ is nonorientable) such that both handlebodies are invariant under the $G$-action. Then $|G|\leq 24(g-1)$, and in the maximal case $|G|=24(g-1)$ we call both the $G$-action and the 3-manifold strongly maximally symmetric (the term maximally symmetric is used in various papers for the case of orientation-preserving actions of maximal possible order $12(g-1)$ on orientable 3-manifolds, see [Z2], [Z7] or the survey [Z4]). In the present paper we characterize the strongly maximally symmetric finite group-actions, using the approach to finite orientation-reversing group-actions on handlebodies in [Z3].

In order to state our results, we introduce some notation (see [T1], [T2] for the following). A Coxeter tetrahedron is a tetrahedron all of whose dihedral angles are of the form $\pi/n$ (denoted by a label $n$ of the edge, for some integer $n\geq 2$) and, moreover, such that at each of the four vertices the three angles of the adjacent edges define a spherical triangle (i.e., $1/n_{1}+1/n_{2}+1/n_{3}>1$). Such a Coxeter tetrahedron can be realized as a spherical, Euclidean or hyperbolic tetrahedron in the 3-sphere $S^{3}$, Euclidean 3-space ${{R}}^{3}$ or hyperbolic 3-space ${H}^{3}$, and will be denoted by ${\cal C}(n,m;a,b;c,d)$ where $(n,m),(a,b)$ and $(c,d)$ are the labels of pairs of opposite edges. We denote by $C(n,m;a,b;c,d)$ the Coxeter group generated by the reflections in the faces of ${\cal C}(n,m;a,b;c,d)$, a properly discontinuous group of isometries of one of these three geometries.

We are interested in particular in the Coxeter tetrahedra ${\cal C}(n,m)={\cal C}(n,m;2,2;2,3)$ and the corresponding Coxeter groups $C(n,m)={\cal C}(n,m;2,2;2,3)$ which are exactly the following:

spherical:      $C(2,2),\;\;C(2,3),\;\;C(2,4),\;\;C(2,5),\;\;C(3,3),\;\;C(3,4),\;\;C(3,5)$;

Euclidean:    $C(4,4)$;

hyperbolic:   $C(4,5),\;\;C(5,5).$

We consider also the Coxeter tetrahedra of type ${\cal C}(n,m;3,3;2,2)$; such a Coxeter tetrahedron has a rotational symmetry $\tau$ of order two which exchanges the opposite edges with labels 3 and 2 and acts as an inversion on the two edges with lables $n$ and $m$. The involution $\tau$ can be realized by an isometry and hence defines a group of isometries $C_{\tau}(n,m)$ containing the Coxeter group $C(n,m;3,3;2,2)$ as a subgroup of index two; we call $C_{\tau}(n,m)$ a twisted Coxeter group. The twisted Coxeter groups of type $C_{\tau}(n,m)$ are the following:

spherical:      $C_{\tau}(2,2),\;\;C_{\tau}(2,3),\;\;C_{\tau}(2,4)$;

Euclidean:    $C_{\tau}(3,3)$;

hyperbolic:   $C_{\tau}(2,5),\;\;C_{\tau}(3,4),\;\;C_{\tau}(3,5),\;\;C_{\tau}(4,4),\;\;C_{\tau}(4,5),\;\;C_{\tau}(5,5).$

Our main result is:

Theorem. i) A reducible, strongly maximally symmetric $G$-manifold $M$ is obtained by doubling a $G$-action of maximal possible order $24(g-1)$ on an orientable or nonorientable handlebody of genus $g>1$ (i.e., by taking the double along the boundary of both the handlebody and its $G$-action).

ii) An irreducible, strongly maximally symmetric $G$-manifold $M$ is spherical, Euclidean or hyperbolic and obtained as a quotient of the 3-sphere, Euclidean or hyperbolic 3-space by a normal subgroup of finite index, acting freely, of a Coxeter group $C(n,m)$ or a twisted Coxeter group $C_{\tau}(n,m)$; the $G$-action is obtained as the projection of the Coxeter or twisted Coxeter group to $M$.

There are only finitely many possibilities in the spherical case, and most of the finite group-actions are on the 3-sphere. The genera of the strongly maximally symmetric group-actions on $S^{3}$ can be computed from the orders of the spherical Coxeter groups (see [CM], [Z2, Table 1]), and one obtains:

Corollary. The genera of the strongly maximally symmetric finite group actions on the 3-sphere are $g$ = 2, 3, 5, 11, 6, 17 and 601 in the untwisted cases, and $g$ = 4, 11, and 97 in the twisted cases.

For orientation-preserving actions of maximal possible order $12(g-1)$ on the 3-sphere, the genera are determined in [WZ1, Theorem 3.1] and [WZ2], and there are in addition the values $g$ = 9, 25, 121 and 241.

In section 3 we consider also the case of $G$-actions of maximal possible order $48(g-1)$, allowing $G$-actions which interchange the two handlebodies of a Heegaard splitting of a 3-manifold $M$. We use methods from [Z3], in particular we correct and complete results in [Z3] where the twisted cases are missing. Our results for the spherical case are obtained also in [WWZ] where more general group actions of large orders on the 3-sphere are considered.

2. Proof of the Theorem

Let $G$ be a finite group of maximal possible order $24(g-1)$ which acts on a closed 3-manifold $M$ and leaves invariant the handlebodies $V_{g}$ and $V_{g}^{\prime}$ of a Heegaard decomposition $M=V_{g}\cup_{\partial}V_{g}^{\prime}$ of genus $g>1$. By [Z3], each of $V_{g}/G$ and $V_{g}^{\prime}/G$ is a handlebody orbifold obtained by glueing two 3-disk orbifolds $D^{3}/G_{1}$ and $D^{3}/D_{2}$ (quotients of the closed 3-disk $D^{3}$ by spherical groups $G_{1}$ and $G_{2}$) along a common 2-disk suborbifold $D^{2}/{{D}}_{n}$ of their boundaries (a quotient of the 2-disk $D^{2}$ by a dihedral group ${{D}}_{n}$ of order $2n$), with orbifold Euler characteristic $\;\;1/|{{D}}_{n}|-1/|G_{1}|-1/|G_{2}|$ (see [T1], [T2] for basic facts about orbifolds, and [Z7] for the orientation-preserving case). For the case of maximal possible order $24(g-1)$, there are exactly eight such handlebody orbifolds which are the handlebody orbifolds of orbifold Euler characteristic -1/24, the largest possible value smaller than 0; since the orbifold Euler characteristic is multiplicative under finite orbifold coverings, $(-1/24)|G|=1-g$, $|G|=24(g-1)$. These minimal handlebody orbifolds can best be codified by their orbifold fundamental groups $G_{1}*_{{{D}}_{n}}G_{2}$ which, by [Z3, Theorem 1], are exactly the following eight free products with amalgamation:

[TABLE]

[TABLE]

here $\bar{{D}}_{n}=[2,2,n]$, $\bar{{A}}_{4}=[2,3,3]$, $\bar{{S}}_{4}=[2,3,4]$ and $\bar{{A}}_{5}=[2,3,5]$ denote the extended dihedral, tetrahedral, octahedral and dodecahedral groups (generated by the reflections in the corresponding spherical triangles with angles $\pi/m$). The group ${{D}}_{2*n}$ is a spherical group of order $4n$ (isomorphic to the dihedral group ${{D}}_{2n}$), a subgroup of index two in the extended dihedral group $\bar{{D}}_{2n}$, with the standard dihedral action by rotations and reflections on the equatorial section $S^{1}$ of the 2-sphere $S^{2}$, the reflections corresponding alternatingly to rotations and reflections in great circles of $S^{2}$. The extended cyclic group $\bar{Z}_{n}$ of order $2n$ is isomorphic to the dihedral group $\bar{D}_{n}$ and acts in the standard way by rotations and reflections on $D^{2}$, and also on $D^{3}$.

For example, in the case of $\bar{{D}}_{5}*_{\bar{{Z}}_{5}}{\bar{{A}}_{5}}$ the associated handlebody orbifold ${\cal H}(\bar{{D}}_{5}*_{\bar{{Z}}_{5}}{\bar{{A}}_{5}})$ is obtained as follows. The 3-disk orbifold $D^{3}/\bar{{D}}_{5}$ is a cone over its orbifold boundary, the spherical 2-orbifold $S^{2}/\bar{{D}}_{5}$ which is just a triangle with angles $\pi/2,\pi/2$ and $\pi/5$; the singular points of this triangle consists of its three sides which are reflection axes with local groups ${{Z}}_{2}$, seperated by the three vertices with local groups ${{D}}_{2}$, ${{D}}_{2}$ and ${{D}}_{5}$. In a similar way, the 3-disk orbifold $D^{3}/\bar{{A}}_{5}$ is constructed. In their boundaries, both 3-disk orbifolds $D^{3}/\bar{{D}}_{5}$ and $D^{3}/\bar{{A}}_{5}$ have a 2-disk suborbifold $D^{2}/\bar{{Z}}_{5}$ which is a triangle: two of its sides are reflection axes meeting in a dihedral point ${{D}}_{5}$, the third side is a nonsingular arc (except for its endpoints) which constitutes the orbifold boundary of $D^{2}/\bar{{Z}}_{5}$. The handlebody orbifold ${\cal H}_{5}={\cal H}(\bar{{D}}_{5}*_{\bar{{Z}}_{5}}{\bar{{A}}_{5}})$ is then obtained by glueing $D^{3}/\bar{{D}}_{5}$ and $D^{3}/\bar{{A}}_{5}$ along the 2-suborbifolds $D^{2}/\bar{{Z}}_{5}$ in their boundaries (or better, by connecting the two 3-disk orbifolds by a 1-handle orbifold $(D^{2}/\bar{{Z}}_{5})\times[-1,1]$); note that the orbifold boundary of this handlebody orbifold is a square ${\cal D}^{2}([2,2,2,3])$ whose sides are reflection axes meeting in three dihedral points ${{D}}_{2}$ and one ${{D}}_{3}$.

Applying the construction to the first four amalgamated free products, one obtains the four handlebody orbifolds

[TABLE]

[TABLE]

and each of these four handlebody orbifolds has the square ${\cal{{D}}}^{2}([2,2,2,3])$ as its orbifold boundary.

For the remaining four free products with amalgamation one constructs in a similar way the handlebody orbifolds

[TABLE]

[TABLE]

For example, the quotient orbifold $D^{3}/{{D}}_{2*5}$ is a cone over its orbifold boundary $S^{2}/{{D}}_{2*5}$ which is a disk with a singular point ${{Z}}_{2}$ in its interior whose boundary consists of a unique reflection axis starting and finishing in a singular point ${{D}}_{5}$. Since $S^{2}/{{D}}_{2*5}$ has again a 2-disk suborbifold $D^{2}/\bar{{Z}}_{5}$, one can construct the handlebody orbifold ${\cal H}({{D}}_{2*5}*_{\bar{{Z}}_{5}}{\bar{{A}}_{5}})$ as before.

Note that the orbifold boundary of each of the four handlebody orbifolds $\tilde{\cal H}_{n}$ is a 2-disk ${\cal D}^{2}(2,[2,3])$ whose boundary consists of two reflection axes intersecting in two dihedral points ${{D}}_{2}$ and ${{D}}_{3}$, and with a singular point ${{Z}}_{2}$ in its interior; note that ${\cal D}^{2}(2,[2,3])$ is a quotient of the square orbifold ${\cal D}^{2}([2,2,2,3])$ by a rotational involution $\tau$.

Remark. In the case of orientation-preserving actions on orientable handlebodies and 3-manifolds the maximal order is $12(g-1)$, and the orbifold fundamental groups of the minimal orientable handlebody orbifolds, of Euler characteristic $-1/12$, are the following four amalgamated free products:

[TABLE]

where ${{D}}_{n}=(2,2,n)$, ${{A}}_{4}=(2,3,3)$, ${{S}}_{4}=(2,3,4)$ and ${{A}}_{5}=(2,3,5)$ denote the dihedral, tetrahedral, octahedral and dodecahedral groups, spherical triangle groups which are the orientation-preserving subgroups of index two in the corresponding extended groups (see [Z7]).

Returning to the $G$-action on the 3-manifold $M=V_{g}\cup_{\partial}V_{g}^{\prime}$, the quotient orbifold $M/G=(V_{g}/G)\cup_{\partial}(V_{g}^{\prime}/G)$ is obtained by identifying the minimal handlebody orbifolds $V_{g}/G$ and $V_{g}^{\prime}/G$ along their boundaries, and both $V_{g}/G$ and $V_{g}^{\prime}/G$ are of one of the eight minimal types described above. Since the boundary of an orbifold of type ${\cal H}_{n}$ is not homeomorphic to that of an orbifold of type $\tilde{\cal H}_{m}$, both orbifolds $V_{g}/G$ and $V_{g}^{\prime}/G$ are of the same type.

Suppose first that $V_{g}/G={\cal H}_{n}$ and $V_{g}^{\prime}/G={\cal H}_{m}$. The boundary of both ${\cal H}_{n}$ and ${\cal H}_{m}$ is the square orbifold ${\cal D}^{2}([2,2,2,3])$; up to isotopy, this square has exactly two orbifold homeomorphisms which are the identity map and a reflection in a diagonal of the square (connecting the two opposite vertices of type ${{D}}_{2}$ and ${{D}}_{3}$ and exchanging the other two vertices of type ${{D}}_{2}$).

Suppose that the boundaries of the handlebody orbifolds ${\cal H}_{n}$ and ${\cal H}_{m}$ are identified by the identity map of ${\cal D}^{2}([2,2,2,3])$. As explained before, both handlebody orbifolds ${\cal H}_{n}$ and ${\cal H}_{m}$ are constructed by identifying two 3-disk orbifolds along a 2-disk suborbifold $D^{2}/\bar{{Z}}_{n}$ and $D^{2}/\bar{{Z}}_{m}$ in their boundaries. Identifying the boundaries of ${\cal H}_{n}$ and ${\cal H}_{m}$ by the identity map, these 2-disk suborbifolds $D^{2}/\bar{{Z}}_{n}$ and $D^{2}/\bar{{Z}}_{m}$ fit together along their boundaries and create a 2-disk suborbifold of $M/G$ whose boundary consists of two reflection axes meeting in dihedral points $\bar{{Z}}_{n}\cong{{D}}_{n}$ and $\bar{{Z}}_{m}\cong{{D}}_{m}$. If $n\neq m$, this 2-disk is a bad 2-orbifold, i.e. not covered by a manifold, which is a contradiction since we have the manifold covering $M$ of $M/G$. Hence $n=m$ and $M/G$ is obtained by doubling ${\cal H}_{n}$ along the boundary. Then also $M$ is obtained by doubling the handlebody $V_{g}$ and its $G$-action along the boundary, so we are in part i) of the Theorem.

Suppose then that the boundaries of ${\cal H}_{n}$ and ${\cal H}_{m}$ are identified by a reflection in a diagonal of the square ${\cal D}^{2}([2,2,2,3])$. The Coxeter group $C(n,m)$ acts on the 3-sphere, Euclidean or hyperbolic 3-space, and the quotient orbifold of this action is the Coxeter tetrahedron ${\cal C}(n,m)$: the underlying topological space is the 3-disk, and the singular set consists of the boundary of the tetrahedron (the local group associated to a point is its stabilizer in the Coxeter group). The Coxeter orbifold ${\cal C}(n,m)$ has a 2-suborbifold ${\cal D}^{2}([2,2,2,3])$ (a square whose vertices are on the four edges of the tetrahedron with labels 2,2,2 and 3, seperating the two edges with labels $n$ and $m$), and ${\cal D}^{2}([2,2,2,3])$ seperates ${\cal C}(n,m)$ into the two handlebody orbifolds ${\cal H}_{n}$ and ${\cal H}_{m}$. So in this case the quotient $M/G$ is the Coxeter tetrahedral orbifold ${\cal C}(n,m)$ and we are in part ii) of the Theorem.

We are left with the cases $V_{g}/G=\tilde{\cal H}_{n}$ and $V_{g}^{\prime}/G=\tilde{\cal H}_{m}$. The boundary of each of these minimal handlebody orbifolds is the 2-disk ${\cal D}^{2}(2,[2,3])$, and every orbifold homeomorphism of this 2-orbifold is isotopic to the identity map or to a reflection in a segment which connects the two singular points of types ${{D}}_{2}$ and ${{D}}_{3}$ and has the singular point ${{Z}}_{2}$ in its interior.

As in the first case, if the boundaries of $\tilde{\cal H}_{n}$ and $\tilde{\cal H}_{m}$ are identified by the identity map then either $n\neq m$ and there is a bad 2-suborbifold, or $n=m$ and $M/G$ is obtained by doubling $\tilde{\cal H}_{n}$ along its boundary and we are in case i) of the Theorem.

Finally suppose that the boundaries of $\tilde{\cal H}_{n}$ and $\tilde{\cal H}_{m}$ are identified by a reflection. Similar as before, the Coxeter orbifold ${\cal C}(n,m;3,3;2,2)$ has a square 2-suborbifold ${\cal D}^{2}([3,3,2,2])$ (seperating the two edges with lables $n$ and $m$) which is invariant under the involution $\tau$ of ${\cal C}(n,m;3,3;2,2)$. The projection of ${\cal D}^{2}([3,3,2,2])$ to the twisted Coxeter orbifold ${\cal C}_{\tau}(n,m)={\cal C}(n,m;3,3;2,2)/\tau$ is the quotient ${\cal D}^{2}([3,3,2,2])/\tau$ which is homeomorphic to the orbifold ${\cal D}^{2}(2,[2,3])$ and seperates ${\cal C}_{\tau}(n,m)$ into two handlebody orbifolds $\tilde{\cal H}_{n}$ and $\tilde{\cal H}_{m}$ (e.g., the quotient of the 1-handle orbifold $D^{2}/\bar{{Z}}_{5}\times[-1/2,1/2]$ by the involution $\tau$ is the 3-disk orbifold $D^{3}/{{D}}_{2*5}$, and hence the quotient of ${\cal H}(\bar{{A}}_{5}*_{\bar{{Z}}_{5}}{\bar{{A}}_{5}})$ by $\tau$ gives the handleboldy orbifold $\tilde{\cal H}_{5}={\cal H}({{D}}_{2*5}*_{\bar{{Z}}_{5}}{\bar{{A}}_{5}})$). So by identifying $\tilde{\cal H}_{n}$ and $\tilde{\cal H}_{m}$ along their boundaries we obtain the twisted Coxeter orbifold ${\cal C}_{\tau}(n,m)$, and we are in case ii) of the Theorem.

This completes the proof of the Theorem.

3. Examples and comments

The maximal possible order of a $G$-action of a closed 3-manifold $M$ which leaves invariant a Heegaard surface of genus $g>1$ is $48(g-1)$; in this maximal case, some element of $G$ has to interchange the two handlebodies of the Heegaard splitting, and the subgroup of index two preserving both handlebodies gives a strongly maximally symmetric $G$-action on $M$.

Suppose that $M$ is irreducible. Then the subgroup of index two preserving both handlebodies of the Heegaard splitting is obtained from a Coxeter group $C(n,m)$ or twisted Coxeter group $C_{\tau}(n,m)$ as in the Theorem. By the geometrization of finite group actions in dimension 3, we can assume that the whole group $G$ acts by isometries; lifting $G$ to the universal covering, we obtain a group of isometries of $S^{3}$, ${{R}}^{3}$ of ${H}^{3}$ containing the Coxeter group $C(n,m)$ or the twisted Coxeter group $C_{\tau}(n,m)$ as subgroup of index two, and in the second case it contains the Coxeter group $C(n,m;3,3;2,2)\subset C_{\tau}(n,m)$ as a subgroup of index 4. Now any 2-fold or 4-fold extension of such a Coxeter group is obtained by adjoining the symmetry group ${{Z}}_{2}$ or ${{Z}}_{2}\times{{Z}}_{2}$ of rotations of the Coxeter tetrahedron to the Coxeter group (see [M] for the hyperbolic case, the other cases are similar), and clearly the presence of such symmetries requires $n=m$.

We denote by $C_{\mu}(n,n)$ the twisted group generated by $C(n,n)=C(n,n;2,2;2,3)$ and the involution $\mu$ of the Coxeter tetrahedron ${\cal C}(n,n;2,2;2,3)$ which exchanges the two edges with label $n$ and inverts the other four edges, and by $C_{\tau\mu}(n,n)$ the doubly twisted group generated by $C(n,n;3,3;2,2)$ and the involutions $\tau$ and $\mu$ of the Coxeter tetrahedron ${\cal C}(n,n;3,3;2,2)$. The possibilities are then the following:

spherical:      $C_{\mu}(2,2),\;\;C_{\mu}(3,3);\;\;\;\;C_{\tau\mu}(2,2);$

Euclidean:    $C_{\mu}(4,4);\hskip 28.45274ptC_{\tau\mu}(3,3)$;

hyperbolic:   $C_{\mu}(5,5);\;\;\;\;\;C_{\tau\mu}(4,4),\;\;C_{\tau\mu}(5,5).$

Summarizing, we have:

Proposition. Let $M$ be closed, irreducible 3-manifold with a $G$-action of maximal possible order $48(g-1)$ which leaves invariant the Heegaard surface of a Heegaard splitting of genus $g>1$ of $M$. Then $M$ is obtained as the quotient of the 3-sphere, Euclidean or hyperbolic 3-space by a normal subgroup of finite index, acting freely, of a twisted Coxeter group $C_{\mu}(n,n)$ or a doubly twisted Coxeter group $C_{\tau\mu}(n,n)$, and the $G$-action is obtained as the projection of $C_{\mu}(n,n)$ or $C_{\tau\mu}(n,n)$ to $M$. If $M$ is the 3-sphere, the possible genera are $g$ = 2, 4 and 6.

Finally, we discuss an explicit example of a hyperbolic, strongly maximally symmetric 3-manifold $M$ for which, moreover, also the bound $48(g-1)$ is obtained: this is the hyperbolic 3-manifold $M$ considered in [Z5], [Z7], see also [Z6] for some further properties. The universal covering group of $M$ is a normal torsionfree subgroup of smallest possible index 120 in the Coxeter group $C(5,5;3,3;2,2)$, and of index $240=24(g-1)$ in its twisted version $C_{\tau}(5,5)$. In particular, $M$ is a strongly maximally symmetric $G$-manifold of genus $g=11$ (and, by [Z5], also the ordinary Heegaard genus of $M$ is equal to 11). Moreover, it follows from [Z6, Propositions 3.2 and 3.3] that the universal covering group of $M$ is a normal subgroup also of the doubly twisted Coxeter group $C_{\tau\mu}(n,n)$, so $C_{\tau\mu}(n,n)$ projects to an isometry group of order $480=48(g-1)$ of $M$ (which is, in fact, the full isometry group of $M$), and we are in the situation of the Proposition.

We believe that the genus $11$ of $M$ is the smallest equivariant genus of a hyperbolic $G$-manifold for which the bound $48(g-1)$ in the Proposition is obtained and, more generally, also the smallest genus of any strongly maximally symmetric hyperbolic 3-manifold. For this, one has to check the minimal indices of the torsionfree normal subgroups of the other hyperbolic Coxeter and twisted Coxeter groups.

References

[CM] H.S.M. Coxeter, W.O.J. Moser, Generators and Relations for Discrete Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete 14, Springer-Verlag 1984

[M] A.D. Mednykh, Automorphism groups of three-dimensional hyperbolic manifolds. Soviet Math. Dokl. 32 (1985), 633-636

[MMZ] D. McCullough, A. Miller, B. Zimmermann, Group actions on handlebodies. Proc. London Math. Soc. 59 (1989), 373-415

[T1] W. Thurston, The Geometry and Topology of 3-Manifolds. Lecture Notes Princeton Univ. 1977

[T2] W. Thurston, Three-dimensional Geometry and Topology. Revised and extended version of a part of [T1], Princeton 1990

[WWZ] C. Wang, S. Wang, Y. Zhang, Maximum orders of extendable actions on surfaces. Acta Math. Sin. (Engl. Ser.) 32 (2016), 54-68

[WZ1] C. Wang, S. Wang, Y. Zhang, B. Zimmermann, Extending finite group actions on surfaces over $S^{3}$. Top. Appl. 160 (2013), 2088-2103

[WZ2] C. Wang, S. Wang, Y. Zhang, B. Zimmermann, Embedding surfaces into $S^{3}$ with maximum symmetry. Groups, Geometry, and Dynamics 9 (2015), 1001-1045

[Z1] B. Zimmermann, Über Abbildungsklassen von Henkelkörpern. Arch. Math. 33 (1979), 379-382

[Z2] B. Zimmermann, Finite group actions on handlebodies and equivariant Heegaard genus for 3-manifold. Top. Appl. 43 (1992), 263-274

[Z3] B. Zimmermann, Finite maximal orientation reversing group actions on handlebodies and 3-manifolds. Rend. Circ. Mat. Palermo 48 (1999), 549-562

[Z4] B. Zimmermann, A survey on large finite group actions on graphs, surfaces and 3-manifolds. Rend. Circ. Mat. Palermo 52 (2003), 47-56

[Z5] B. Zimmermann, On a hyperbolic 3-manifold with some special properties. Math. Proc. Camb. Phil. Soc. 113 (1993), 87-90

[Z6] B. Zimmermann, On maximally symmetric hyperbolic 3-manifolds. Progress in Knot Theory and Related Topics, Travaux En Cours, Hermann, Paris 1997, 143-153

[Z7] B. Zimmermann, Genus actions of finite groups on 3-manifolds. Mich. Math. J. 43 (1996), 593-610