On large groups of symmetries of finite graphs embedded in spheres

TL;DR

This paper investigates the symmetry groups of finite graphs embedded in spheres across various dimensions, establishing bounds on the group order relative to the graph's genus and exploring minimal embedding dimensions.

Contribution

It generalizes previous 3D results to arbitrary dimensions, providing polynomial bounds on symmetry group orders and analyzing minimal embedding dimensions.

Findings

Order of symmetry group bounded by polynomial in genus

Degree of polynomial depends on sphere dimension

Optimal bounds established for even dimensions

Abstract

Let G be a finite group acting orthogonally on a pair (S^d,\Gamma) where \Gamma is a finite, connected graph of genus g>1 embedded in the sphere S^d. The 3-dimensional case d=3 has recently been considered in a paper by C. Wang, S. Wang, Y. Zhang and the present author where for each genus g>1 the maximum order of a G-action on a pair (S^3,\Gamma) is determined and the corresponding graphs \Gamma are classified. In the present paper we consider arbitrary dimensions d and prove that the order of G is bounded above by a polynomial of degree d/2 in g if d is even, and of degree (d+1)/2 if d is odd; moreover the degree d/2 is best possible in even dimensions d. We discuss also the problem, given a finite graph \Gamma and its finite symmetry group, to find the minimal dimension of a sphere into which \Gamma embeds equivariantly as above.

Equations

$$1\to F_g\hookrightarrow E={\pi }_1(\bar{\Gamma },\mathcal{G})\to G\to 1.$$

$$|G|=m^k=(g+1{)}^k/k^k$$

Full text

On large groups of symmetries of finite graphs embedded in spheres

Bruno P. Zimmermann

Università degli Studi di Trieste

Dipartimento di Matematica e Geoscienze

34127 Trieste, Italy

Abstract. Let $G$ be a finite group acting orthogonally on a pair $(S^{d},\Gamma)$ where $\Gamma$ is a finite, connected graph of genus $g>1$ embedded in the sphere $S^{d}$. The 3-dimensional case $d=3$ has recently been considered in a paper by C. Wang, S. Wang, Y. Zhang and the present author where for each genus $g>1$ the maximum order of a $G$-action on a pair $(S^{3},\Gamma)$ is determined and the corresponding graphs $\Gamma$ are classified. In the present paper we consider arbitrary dimensions $d$ and prove that the order of $G$ is bounded above by a polynomial of degree $d/2$ in $g$ if $d$ is even and of degree $(d+1)/2$ if $d$ is odd; moreover the degree $d/2$ is best possible in even dimensions $d$. We discuss also the problem, given a finite graph $\Gamma$ and its finite symmetry group, to find the minimal dimension of a sphere into which $\Gamma$ embeds equivariantly as above.

Keywords: Groups acting on finite graphs; equivariant embeddings into spheres.

Mathematics Subject Classification 2010: 57S17, 57S27, 05C10

1. Introduction

We study large finite groups $G$ of automorphisms of a finite, connected graph $\Gamma$ which embeds smoothly into a sphere $S^{d}$ of some dimension $d$ such that the $G$-action on $\Gamma$ extends to an orthogonal action of $G$ on $S^{d}$. In other words, we study large finite groups $G$ of orthogonal transformations of pairs $(S^{d},\Gamma)$ where $\Gamma$ denotes a finite, connected graph smoothly embedded in a sphere $S^{d}$. All actions considered in the present paper will be faithful on both $S^{d}$ and $\Gamma$, and all finite graphs $\Gamma$ will be hyperbolic, i.e. connected, of genus $g>1$ (the rank of its free fundamental group) and without free edges (edges with one vertex of valence 1; note that free edges can be deleted in a $G$-equivariant way without changing the genus of a graph); we allow closed and multiple edges.

The case of dimension $d=3$ is considered in [6]. For a finite subgroup of the orthogonal group SO(4) acting on a pair $(S^{3},\Gamma)$, a regular neighbourhood of $\Gamma$ in $S^{3}$ is a 3-dimensional handlebody $V^{3}_{g}$ of genus $g>1$ on which $G$ acts orientation-preservingly, and by [8], [1] there is the linear bound $|G|\leq 12(g-1)$ for orientation-preserving actions on 3-dimensional handlebodies. In [6], for each genus $g>1$ the maximal possible order of a $G$-action on $(S^{3},\Gamma)$ is determined and the corresponding graphs $\Gamma$ are classified (on the basis of analogous results in [5] for the case of closed surfaces embedded in $S^{3}$); the maximal possible order $12(g-1)$ is obtained only for finitely many values of $g$.

Concerning dimension $d=4$, supppose that $G$ acts orthogonally on a pair $(S^{4},\Gamma)$; now a regular neighbourhood of $\Gamma$ is a 4-dimensional handlebody $V^{4}_{g}$ whose boundary $\partial V^{3}_{g}$ is a connected sum $\sharp_{g}(S^{2}\times S^{1})$ of $g$ copies of $S^{2}\times S^{1}$. Finite group actions on such connected sums are considered in [11] whose results imply the quadratic upper bound $|G|\leq 24g(g-1)$, for $g\geq 15$; moreover there does not exist a linear bound in $g$ for the order of $G$.

In the following main result of the present paper we consider the case of arbitrary dimensions $d\geq 4$.

Theorem 1. Let $G$ be a finite subgroup of the orthogonal group ${\rm O}(d+1)$ acting on a pair $(S^{d},\Gamma)$, for a finite hyperbolic graph of genus $g>1$ embedded in $S^{d}$. Then the order of $G$ is bounded above by a polynomial of degree $d/2$ in $g$ if $d$ is even and of degree $(d+1)/2$ if $d$ is odd. The degree $d/2$ is best possible in even dimensions whereas in odd dimensions the optimal degree is either $(d-1)/2$ or $(d+1)/2$.

So the optimal degree in odd dimensions remains open at present (except for $d=3$ where it is $(d-1)/2=1$). The proof of Theorem 1 will be reduced to an analogous result in [2] about finite group actions on $d$-dimensional handlebodies (see Theorem 2 in the next section).

The maximum order of a finite group of automorphisms of a finite hyperbolic graph of genus $g>2$ is $2^{g}g!$ ([7]), obtained for a graph with one vertex and $g$ closed edges (a ”bouquet of $g$ circles”) whose automorphism group is isomorphic to the semidirect product $({Z}_{2})^{g}\mathchar 9582\relax S_{g}$ (with normal subgroup $({Z}_{2})^{g}$ on which the symmetric group $S_{g}$ acts by permutation of coordinates). At present we don’t know the minimal dimension of a sphere which admits an equivariant embedding of the bouquet of $g$ circles (i.e., invariant under an orthogonal action of $({Z}_{2})^{g}\mathchar 9582\relax S_{g}$), see the question at the end of section 2. For some other graphs with large symmetry groups instead, we determine this minimal dimension of an equivariant embedding in the examples in section 2.

2. Proof of Theorem 1

See [2] for the following. A $d$-dimensional handlebody $V^{d}_{g}$ of genus $g$ can be defined as a regular neighbourhood of a finite connected graph $\Gamma$ of genus g embedded in $S^{d}$. Such a handlebody $V^{d}_{g}$ can be uniformized by a Schottky group ${\cal S}_{g}$, a free group of rank $g$ of Möbius transformations of $S^{d-1}$ which extends naturally to the disk ${{B}}^{d}$ (”Poincaré extension”); the interior of $B^{d}$ is the Poincaré-model of hyperbolic space ${H}^{d}$ on which Möbius transformations act as hyperbolic isometries. The handlebody $V^{d}_{g}$ is obtained as the quotient $(B^{d}-\Lambda({\cal S}_{g}))/{\cal S}_{g}$ where $\Lambda({\cal S}_{g})\subset S^{d-1}$ denotes the set of limit point of the action of ${\cal S}_{g}$ on $B^{d}$ (a Cantor set), in particular $B^{d}-\Lambda({\cal S}_{g})$ is the universal covering of $V^{d}_{g}$. This gives the interior ${H}^{d}/{\cal S}_{g}$ of $V^{d}_{g}$ the structure of a complete hyperbolic manifold, and we say that the Schottky group ${\cal S}_{g}$ uniformizes the hyperbolic handlebody $V^{d}_{g}$. In particular there is the notion of an isometry of such a hyperbolic handlebody meaning that it acts as an isometry on the interior of $V^{d}_{g}$; equivalently, each lift to the universal covering $B^{d}-\Lambda({\cal S}_{g})$ of $V^{d}_{g}$ extends to a Möbius transformation of $B^{d}$.

The following is proved in [2].

Theorem 2. Let $G$ be a finite group of isometries of a hyperbolic handlebody $V^{d}_{g}$ of dimension $d\geq 3$ and of genus $g>1$ which acts faithfully on the fundamental group. Then the order of $G$ is bounded by a polynomial of degree $d/2$ in $g$ if $d$ is even and of degree $(d+1)/2$ if $d$ is odd.

Since a handlebody of dimension $d\geq 4$ admits $S^{1}$-actions there is no upper bound for the order of finite group actions which are not faithful on the fundamental group (however there is a Jordan-type bound for such actions, see [2, Corollary]). On the other hand, finite faithful actions on finite hyperbolic graphs are faithful on the fundamental group, that is inject into the outer automorphism group of the fundamental group ([12, Lemma 1]); conversely, it is observed in [9, p. 478] (as a version of the Nielsen realization problem for free groups) that every finite subgroup $G$ of the outer automorphism group of a free group can be realized by an action of $G$ on a finite graph.

Starting with the Proof of Theorem 1 now, let $G$ be a finite group acting orthogonally on $(S^{d},\Gamma)$, for a finite hyperbolic graph $\Gamma$ of genus $g>1$ embedded in $S^{d}$. A $G$-invariant regular neighbourhood of $\Gamma$ in $S^{d}$ is homeomorpic to a handlebody $V^{d}_{g}$ of dimension $d$ and genus $g$. Since we are assuming that the action of $G$ on the graph $\Gamma$ is faithful, by [12, Lemma 1] also the induced action of $G$ on the fundamental group of $\Gamma$ and hence of $V^{d}_{g}$ is faithful and defines an injection of $G$ into the outer automorphism groups of the fundamental groups of $\Gamma$ and $V^{d}_{g}$. The first part of Theorem 1 is now a consequence of Theorem 2 and the following:

Proposition. Let $G$ be a finite group acting orthogonally on a pair $(S^{d},\Gamma)$, for a finite hyperbolic graph of genus $g>1$ embedded in $S^{d}$, and hence also on a handlebody $V^{d}_{g}$ in $S^{d}$ obtained as a $G$-invariant regular neighbourhood of $\Gamma$. Then $V^{d}_{g}$ can be uniformized by a Schottky group such that $V^{d}_{g}$ admits an isometric $G$-action (inducing the same injection into the outer automorphism group of the fundamental group as the original $G$-action).

Very likely the original $G$-action on $V^{d}_{g}$ is in fact conjugate to an isometric $G$-action; however in order to apply Theorem 2 we just need some isometric $G$-action on $V^{d}_{g}$ which is faithful on the fundamental group, so we don’t follow this here.

Proof of the Proposition. The group $G$ acts as a group of automorphisms of the finite graph $\Gamma$. Let $\tilde{\Gamma}$ be the universal covering tree of $\Gamma$; the group of all liftings of all elements of $G$ to $\tilde{\Gamma}$ defines a group $E$ of automorphisms of the tree $\tilde{\Gamma}$ and a group extension $1\to F_{g}\hookrightarrow E\to G\to 1$ where $F_{g}$ denotes the universal covering group, a free group of rank $g$ isomophic to the fundamental group of $\Gamma$.

By possibly subdividing edges by a new vertex we can assume that $G$ acts on $\Gamma$ without inversions of edges; then the quotient $\bar{\Gamma}=\Gamma/G$ is again a finite graph. Choose a maximal tree in $\bar{\Gamma}$ and lift it isomorphically first to $\Gamma$ and then to $\tilde{\Gamma}$, then lift also the remaining edges of $\bar{\Gamma}$ to $\Gamma$ and $\tilde{\Gamma}$. Associating to the vertices and edges of $\bar{\Gamma}$ the stabilizers in $G$ or $E$ of the lifted vertices and edges in $\Gamma$ and $\tilde{\Gamma}$, this defines a finite graph of finite groups $(\bar{\Gamma},{\cal G})$, with inclusions of the edge groups into the adjacent vertex groups. The fundamental group $\pi_{1}(\bar{\Gamma},{\cal G})$ of the finite graph of finite groups $(\bar{\Gamma},{\cal G})$ is the iterated free product with amalgamation and HNN-extension of the vertex groups amalgamated over the edge groups, first taking the iterated free product with amalgamation over the chosen maximal tree of $\Gamma$ and then associating an HNN-generator to each of the remaining edges. By the standard theory of groups acting on trees, graphs of groups and their fundamental groups (see [3], [4] or [10]), the extension $E$ is isomorphic to the fundamental group $\pi_{1}(\bar{\Gamma},{\cal G})$ of $(\bar{\Gamma},{\cal G})$ and we have a group extension

[TABLE]

We will assume in the following that the graph of groups $(\bar{\Gamma},{\cal G})$ has no trivial edges, i.e. no edges with two different vertices such that the edge group coincides with one of the two vertex groups (by collapsing trivial edges, i.e. by amalgamating the two vertex groups into a single vertex group). We will realize the extension $E=\pi_{1}(\bar{\Gamma},{\cal G})$ as a group of isometries of hyperbolic space ${H}^{d}$, the interior of the $d$-ball $B^{d}$, or equivalently as a group of Möbius transformations of $B^{d}$ or $S^{d-1}=\partial B^{d}$ such that the subgroup $F_{g}$ is realized by a Schottky group ${\cal S}_{g}$. Then $G$ acts as a group of isometries of the hyperbolic handlebody $V^{d}_{g}=(B^{d}-\Lambda({\cal S}_{g}))/{\cal S}_{g}$ proving the Proposition. The realization of $E=\pi_{1}(\bar{\Gamma},{\cal G})$ is by standard combination methods as described in [2, p.247] and [9, p. 479-482]. As an illustration we discuss the case of a graph of groups $(\bar{\Gamma},{\cal G})$ with a single, non-closed edge. We lift the edge to an edge $B$ of $\Gamma$, with vertices $A_{1}$ and $A_{2}$. The corresponding edge and vertex groups of $(\bar{\Gamma},{\cal G})$ (which we denote by the same letters) are defined as the stabilizers of the edge $B$ and its vertices in the group $G$ acting orthogonally on $\Gamma\subset S^{d}$, and $E=\pi_{1}(\bar{\Gamma},{\cal G})$ is the free product with amalgamation $A_{1}*_{B}A_{2}$.

For $i=1$ and 2, the vertex group $A_{i}\subset G$ acts orthogonally on $S^{d}$ and fixes the corresponding vertex $A_{i}$ of the edge $B$ of $\Gamma$. A regular invariant neighbourhood of the fixed point $A_{i}$ is a ball $B_{i}^{d}$ with an action of $A_{i}$. The edge group $B$ is a subgroup of both $A_{1}$ and $A_{2}$ which fixes both vertices $A_{1}$ and $A_{2}$ and pointwise the edge $B$,. We can assume that the intersection $\partial B_{1}^{d}\cap\partial B_{2}^{d}$ of the two boundary spheres $S^{d-1}$ is non-empty and hence a sphere $S^{d-2}$. We conjugate $A_{2}$ by the reflection in this sphere $S^{d-2}$ and obtain a group $A_{2}^{\prime}$ fixing the vertex $A_{1}$; since the reflection commutes with each element of $B$, both $A_{1}$ and $A^{\prime}_{2}$ act on $B_{1}^{d}$ now with a common subgroup $B$.

Identifying $B_{1}^{d}$ with the standard ball $B^{d}$, with the center $A_{1}$ corresponding to the origin 0, we have orthgonal actions of $A_{1}$ and $A_{2}^{\prime}$ on $B^{d}$ with a common subgroup $B$. The orthogonal action of $B$ on $B^{d}$ fixes two diametral points in $\partial B^{d}$ which are not fixed by any other element of $A_{1}$ and $A^{\prime}_{2}$ (corresponding to the intersection of the edge $B$ with $\partial B_{1}^{d}$), and hence $B$ fixes pointwise the hyperbolic line $L$ in hyperbolic space ${H}^{d}$ (the interior of $B^{d}$) connecting these two diametral points. We conjugate $A^{\prime}_{2}$ by a reflection in a hyperbolic hyperplane orthogonal of the line $L$ far from the origin $0\in B^{d}$. The group generated by $A_{1}$ and the reflected group $A^{\prime\prime}_{2}$ is then isomorphic to the free product with amalgamation $A_{1}*_{B}A^{\prime\prime}_{2}$ and realizes $E=A_{1}*_{B}A_{2}$ as a group of hyperbolic isometries of ${H}^{d}$, or equivalently of Möbius transformations of $S^{d-1}=\partial B^{d}$ (by standard combination methods, see [2, p. 247] and [9, p. 480] for more details and some figures).

In a similar way, inductively edge by edge in finitely many steps, one realizes the group $E=\pi_{1}(\bar{\Gamma},{\cal G})$ also in the general case (see again [9, p. 479-482]).

This concludes the proof of the Proposition and, by Theorem 2, also of the first statement of Theorem 1.

For the second statement of Theorem 1, we construct an infinite series of orthgonal actions of finite groups $G$ on finite graphs $\Gamma$ embedded in $S^{d}$ which realize the lower bounds for the polynomial degrees in Theorem 1.

Example 1. For $k>1$, let $G=C_{1}\times\ldots\times C_{k}\cong({Z}_{m})^{k}$, of order $n=m^{k}$, be the product of $k$ cyclic groups $C_{i}\cong{Z}_{m}$ of order $m$. Choose an orthogonal action of $G$ on ${R}^{2k}$ as follows. Decomposing ${R}^{2k}=P_{1}\times\ldots\times P_{k}$ as the product of $k$ orthogonal planes $P_{i}$, each $C_{i}$ acts on $P_{i}$ faithfully by rotations and trivially on the $k-1$ orthogonal planes. This $G$-action on ${R}^{2k}$ extends to an orthogonal $G$-action on the one-point compactification $S^{2k}$ of ${R}^{2k}$, with two global fixed points 0 and $\infty$.

We consider a graph $\Gamma$ in $S^{2k}$ with two vertices 0 and $\infty$ and $km$ connecting edges divided into $k$ groups of $m$ edges. We embed the first group of $m$ edges into $P_{1}$ such that $C_{1}$ permutes these edges cyclically, then the next $m$ edges into $P_{2}$ etc., defining an orthogonal action of $G$ on the graph $\Gamma$ embedded in $S^{2k}$. The graph $\Gamma$ has genus $g=km-1$, hence

[TABLE]

which is a polynomial of degree $k=d/2$ in $g$.

Suppose that $m>2$; then $d=2k$ is the minimal dimension of a sphere which admits a $G$-equivariant embedding of the graph $\Gamma$. In fact, in such an embedding into a sphere $S^{d}$ the group $G\cong({Z}_{m})^{k}$ has a global fixed point and hence acts orthogonally on the boundary $S^{d-1}$ of an invariant regular neighbourhood of a global fixed point, and $d-1=2k-1$ is the minimal dimension of a sphere with a faithful orthogonal action of $({Z}_{m})^{k}$; equivalently, the minimal dimension of a faithful, real, linear representation of $({Z}_{m})^{k}$ is $2k$.

In odd dimensions $d=2k+1$, we extend the orthogonal action of $G$ on ${R}^{2k}$ to an orthogonal action on ${R}^{2k+1}$, trivial on the last coordinate, and then proceed as before; we get a polynomial of degree $k=(d-1)/2$ in $g$ for the order of $G$. As noted in the Introduction, the optimal degree in dimension $d=3$ is $(d-1)/2=1$; for odd dimensions $d>3$ the optimal degree is either $(d-1)/2$ or $(d+1)/2$ but at present we do not know which of these two values occurs.

This completes the proof of Theorem 1.

We present some other infinite series of finite orthogonal group actions on finite graphs embedded in spheres which realize the minimal dimension of such an embedding.

Examples 2. i) Let $\Gamma$ be the complete graph with $d+2$ vertices, or the the 1-skeleton of a $(d+1)$-simplex. We embed the regular $(d+1)$-simplex into $B^{d+1}$, with vertices in $S^{d}=\partial B^{d+1}$, and project its edges radially to $S^{d}$; this defines an embedding of $\Gamma$ into $S^{d}$. The automorphism group of the graph $\Gamma$ is the symmetric group $S_{d+2}$ which extends to an orthogonal action on the sphere $S^{d}$. Again $d$ is the minimal dimension of such an equivariant embedding since it is the minimal dimension of a sphere with a faithful, orthogonal action of $S_{d+2}$ (equivalently, the minimal dimension of a faithful, real, linear representation of the symmetric group $S_{d+2}$ is $d+1$).

ii) Let $\Gamma^{\prime}$ be a graph with two vertices and $d+2$ egdes. Then $\Gamma^{\prime}$ has an embedding into $S^{d+1}$ which is ”dual” to the embedding of $\Gamma$ into $S^{d}\subset S^{d+1}$ in part i) in the following sence : the sphere $S^{d}$ seperates $S^{d+1}$ into two balls $B^{d+1}$; the two vertices of $\Gamma^{\prime}$ are the centers of these two balls, and each of the $d+2$ connecting edges intersects exactly one of the $d+2$ faces of the projected $(d+1)$-simplex in $S^{d}$ in its center.

The symmetric group $S_{d+2}$ acts orthogonally on $(S^{d+1},\Gamma^{\prime})$ with two global fixed points (the double suspension of the action of $S_{d+2}$ on $S^{d}$), in particular it acts on $S^{d}$ and it follows as in i) that $d+1$ is the minimal dimension of a sphere with an equivariant embedding of $\Gamma^{\prime}$.

iii) Let $\Gamma$ be the 1-skeleton of the $(d+1)$-dimensional hypercube now, with a projection to $S^{d}=\partial B^{d+1}$ as in i). The graph $\Gamma$ has an automorphism group $({Z}_{2})^{d+1}$ which extends to an orthogonal action on $S^{d}$ (inversion of coordinates), and $d$ is the minimal dimension of a sphere with an orthogonal action of $({Z}_{2})^{d+1}$ (the minimal dimension of a faithful, real, linear representation of $({Z}_{2})^{d+1}$ is $d+1$).

Dualizing as in ii) we get a graph $\Gamma^{\prime}$ in $S^{d+1}$ with two vertices connected by $2(d+1)$ edges (each intersecting exactly one of the faces of the hypercube), with an action of $({Z}_{2})^{d+1}$ with two global fixed points, and $d+1$ is the minimal dimension of such an embedding (since $d$ is the minimal dimension of an orthgonal action of $({Z}_{2})^{d+1}$ on a sphere). Note that this realizes the minimal dimension in the case $m=2$ of the group $({{Z}_{2}})^{k}$ in Example 1, replacing each plane $P_{i}$ with a rotation by a line with a reflection.

Question. As noted in the introduction, the maximum order of a finite group of automorphisms of a finite hyperbolic graph of genus $g>2$ is $2^{g}g!$ ([7]), obtained for a graph $\Gamma$ with a single vertex and $g$ closed edges whose automorphism group is the semidirect product $({Z}_{2})^{g}\mathchar 9582\relax S_{g}$ (the symmetric group $S_{g}$ acts by permutation of the edges, the nomal subgroup $({Z}_{2})^{g}$ by inversion of the edges). What is the minimal dimension of a sphere with an equivariant embedding of this graph?

Note that the graph $\Gamma$ embeds equivariantly into ${R}^{2}\times\ldots\times{R}^{2}={R}^{2g}$, by embedding each of the $g$ closed edges into a different plane ${R}^{2}$ (the unique vertex corresponds to the origin of ${R}^{2g}$ and also of each plane ${R}^{2}$); the group $({Z}_{2})^{g}$ acts by reflections in lines through the origin on the $g$ planes, inverting the $g$ embedded closed edges, the symmetric group $S_{g}$ acts by permutation of the planes. Then $\Gamma$ admits an equivariant embedding also into the one-point compactification ${{S}}^{2g}$ of ${R}^{2g}$. On the other hand, the minimal dimension of a sphere with an orthgonal action of $({Z}_{2})^{g}\mathchar 9582\relax S_{g}$ is $g-1$ (since $g$ is the minimal dimension of a faithful, real, linear representation of $({Z}_{2})^{g}$). It seems reasonable, however, that the minimal dimension of an equivariant embedding of $\Gamma$ into a sphere is $2g$ (as described above), but this minimal dimension, between $g-1$ and $2g$, remains open at present.

References

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[7] S. Wang, B. Zimmermann, The maximum order finite groups of outer automorphisms of free groups, Math. Z. 216 (1994) 83-87

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[9] B. Zimmermann, Über Homöomorphismen n-dimensionaler Henkelkörper und endliche Erweiterungen von Schottky-Gruppen, Comm. Math. Helv. 56 (1981) 474-486

[10] B. Zimmermann, Generators and relations for discontinuous groups, Generators and Relations in Groups and Geometries (eds. Barlotti, Ellers, Plaumann, Strambach), NATO Advanced Study Institute Series 333, Kluwer Academic Publishers (1991) 407-436

[11] B. Zimmermann, On finite groups acting on a connected sum of 3-manifolds $S^{2}\times S^{1}$, Fund. Math. 226 (2014) 131-142

[12] B. Zimmermann, Finite groups of outer automorphism groups of free groups Glasgow Math. J. 38 (1996) 275-282