H. Zieschang

ON AUTOMORPHISMS OF PLANAR GROUPS

(Presented by Academician P. S. Aleksandrov on 6 XI 1963)

Let a group \(\mathfrak{G}\) of automorphisms preserving orientation\(^*\) act on a cellular decomposition \(N\) of the plane. \(\mathfrak{G}\) is called a planar group. In what follows we consider only groups with bicompact fundamental domain\(^ {**}\), among which are found all fundamental groups of orientable closed surfaces. For them J. Nielsen \((^1)\) proved that every automorphism (up to an inner automorphism) is induced by a homeomorphism of the surface. It turns out that this theorem is quite equivalent to a purely algebraic proposition \((^2)\), and this connection, in my opinion, is of particular interest.

Here we shall establish the corresponding equivalence for all planar groups, and then prove an algebraic proposition from which follows a geometric theorem generalizing Nielsen’s theorem.

§ 1. Let \(\mathfrak{G}\) be a planar group with bicompact fundamental domain. We glue the fundamental domain along equivalent arcs and obtain an orientable closed surface \(N^*\). \(N\) covers \(N^*\), generally speaking, with branching. The group \(\mathfrak{G}\) may have some fixed points. To each of their equivalence classes there belongs a vertex on \(N^*\)—these are the branch points.

The structure of the planar group \(\mathfrak{G}\) is the following\(^ {***}\):

\[ \text{generators}\qquad S_1,\ldots,S_m,T_1,U_1,\ldots,T_g,U_g; \tag{E} \]

\[ \text{defining relations}\quad S_1^{k_1}=\ldots=S_m^{k_m}=S_1\cdot\ldots\cdot S_m\prod_{i=1}^{g}[T_i,U_i]=1. \]

For a group \(\mathfrak{G}\) with the given generators and defining relations one can construct a planar group image, i.e. a two-dimensional complex \(K\), whose vertices and arcs from a single vertex can be denoted by the elements and generators of the group \(\mathfrak{G}\) in such a way that an arc denoted by \(X\) leads from the element \(W\) to \(WX\). Then the one-dimensional skeleton is the group image in the sense of Dehn, or the Cayley diagram. To every path in the one-dimensional skeleton there corresponds a word in the generators. The relations are obtained from closed paths. If, moreover, the boundaries of the disks of \(K\) give the defining relations from (E), then \(K\) is called a planar group image with the given generators and relations. Since every relation is an expression in the defining relations, \(K\) is a planar complex.

It can be proved that every group of structure (E) has a group image that is planar or “spherical.”

\(^*\) The cells of the decomposition are called, according to their dimension, vertices, arcs, and disks. We assume that there are no fixed disks.

\(^ {**}\) A planar closed subcomplex is called a fundamental domain of the group \(\mathfrak{G}\) if it is connected and contains exactly one disk from each equivalence class (under automorphisms from \(\mathfrak{G}\)), together with its entire boundary.

\(^ {***}\) Here \([T_i,U_i]\) denotes the commutator \(T_iU_iT_i^{-1}U_i^{-1}\); \(m\) is the number of inequivalent fixed points, \(k_1,\ldots,k_m\) are their orders; \(g\) is the genus of the surface \(N^*\); \(m\) or \(g\) may be zero.

If a vertex of the group image and a word in the generators are given, then there corresponds to them exactly one path on the one-dimensional skeleton which starts from the chosen point.

§ 2. Let \(\hat{\mathfrak G}\) be the free group with free generators \(\hat S_1,\ldots,\hat S_m,\hat T_1,\hat U_1,\ldots,\hat T_g,\hat U_g\). The standard homomorphism \(\hat{\mathfrak G}\to\mathfrak G\) throws away the hats. An endomorphism of the group \(\hat{\mathfrak G}\) induces an endomorphism of the group \(\mathfrak G\), if it maps the kernel of the standard homomorphism into itself.

Theorem 1. Every automorphism \(\alpha\) of the group \(\mathfrak G\) is induced by an automorphism \(\hat\alpha\) of the group \(\hat{\mathfrak G}\), which satisfies the following equations:

\[ \hat\alpha\hat S_i=\hat L_i\hat S_{n_i}\hat L_i^{-1},\qquad i=1,\ldots,m; \]

\[ \hat\alpha(\hat S_1\cdots \hat S_m\prod[\hat T_i,\hat U_i]) =\hat L\cdot\hat S_1\cdots \hat S_m\prod[\hat T_i,\hat U_i]\cdot\hat L^{-1}. \]

Here
\[ \begin{pmatrix} 1\ldots m\\ n_1\ldots n_m \end{pmatrix} \]
is a permutation with \(k_{n_i}=k_i\); \(\hat L_1,\ldots,\hat L_m\) and \(\hat L\) are elements of \(\hat{\mathfrak G}\).

A homeomorphism \(N^*\to N^*\) is called admissible if it maps each branching vertex to a vertex of the same order.

Theorem 2. Every automorphism of the group \(\mathfrak G\) is induced by an admissible homeomorphism of the surface \(N^*\) onto itself.

If \(\mathfrak G\) is the fundamental group of a surface (i.e. \(m=0\)), then every homeomorphism of this surface is admissible. Consequently, we obtain Nielsen’s theorem.

Other proofs of his theorem were given by H. Seifert \((^5)\) and W. Mangler \((^6)\) with the aid of Hopf’s theory \((^7)\) on deformations of continuous mappings between surfaces. Our theorems were proved by W. Vollmerhaus \((^3)\) in the case \(g=0\) and \(k_i\geqslant5\).

§ 3. Let
\[ \hat K_1=\hat L_1\hat S_{i_1}^{\varepsilon_1}\hat L_1^{-1},\ldots, \hat K_m=\hat L_m\hat S_{i_m}^{\varepsilon_m}\hat L_m^{-1}, \hat K_{m+1},\ldots,\hat K_{m+2g} \]
be arbitrary elements of \(\hat{\mathfrak G}\) \((\varepsilon_i=\pm1)\). By \(\Pi_{\hat K}\) we denote a formal word in \(\hat K_1,\ldots,\hat K_{m+2g}\), in which each symbol \(\hat K_i\) \((1\leqslant i\leqslant m)\) occurs exactly once and each symbol \(\hat K_j\) \((j>m)\) occurs twice, to the powers \(+1\) and \(-1\). Then \(\{\hat K_1,\ldots,\hat K_{m+2g};\Pi_{\hat K}\}\) is called an alternating product.** \(\Pi_{\hat K}(\hat K_1,\ldots,\hat K_{m+2g})\), in contrast to \(\Pi_{\hat K}\), denotes an element of \(\hat{\mathfrak G}\).

If in \(\Pi_{\hat K}\) there is a part \(\hat K_i^\varepsilon\hat K_j^\eta\) with \(i>m\), \(i\ne j\), and \(\varepsilon,\eta=\pm1\), then take the elements \(\hat L_i^\varepsilon=\hat K_i^\varepsilon\hat K_j^\eta\) and \(\hat L_t=\hat K_t\) for \(t\ne i\). We change \(\Pi_{\hat K}\) in a suitable way and obtain a new alternating product. If \(i\leqslant m\), then put \(\hat L_i=\hat K_j^{-\eta}\hat K_i\hat K_j^\eta\) and \(\hat L_t=\hat K_t\). These processes, and the corresponding substitutions of the first \(m\) and the last \(2g\) elements for the other side, and transitions from \(\hat K_i\) to \(\hat K_i^{-1}\), are called elementary.

Two alternating products are regarded as equivalent if one can pass from one to the other by elementary processes. An alternating product has the Nielsen property if no element in it (as a word in the generators) cancels completely with both neighbors or with one more than halfway, and if more than half of the first element remains in the expression \(\Pi_{\hat K}\). If only the second condition is satisfied, then the alternating product is called reduced.

* If \(g=0\), then we require that all \(k_i\geqslant5\) (see Vollmerhaus’s paper \((^3)\), for which these conditions were needed). Already from the fact alone that \(\hat\alpha\) is an automorphism carrying the kernel of the standard homomorphism into itself, the equalities of Theorem 1 \((^4)\) are often obtained.

** This generalizes the concept from \((^2)\), where the case \(m=0\) is considered. The proofs of paper \((^2)\) carry over to the most general case.

To every alternating product there belongs a related product with the Nielsen property.

The assertion of equivalence of Theorems 1 and 2 is also carried over: if the alternating product \(\{\hat K_1,\ldots,\hat K_{m+2g};\Pi_{\hat K}\}\) satisfies the equality

\[ \Pi_{\hat K}(\hat K_1,\ldots,\hat K_{m+2g}) = \hat S_1\cdots \hat S_m \prod [\hat T_i,\hat U_i], \]

then it is related to \(\{\hat S_1,\ldots,\hat T_1,\hat U_1,\ldots;\hat S_1\cdots \hat S_m \prod [\hat T_i,\hat U_i]\}\). Hence it follows, in particular, that \(\hat K_1,\ldots,\hat K_{m+2g}\) are free generators of the group \(\hat{\mathfrak G}\).

Alternating products are obtained from cuts of the surfaces \(N^*\). Elementary processes are induced by replacements of cuts or by homeomorphisms. This is described in \((^2)\) for \(m=0\), and for \(g=0\) it is a well-known fact from the theory of braids (see, for example, \((^8)\)). Thus, from Theorem 1 it follows that every automorphism of the group \(\mathfrak G\) (up to an inner one) is induced by an admissible homeomorphism \(N^*\to N^*\). It is easy to prove that inner automorphisms are also induced by homeomorphisms; hence Theorem 2 follows from Theorem 1. Conversely, Theorem 1 is obtained from Theorem 2—this is the basic fact from the theory of closed surfaces and spheres with holes.

§ 4. Let \(\alpha\) be an automorphism of the group \(\mathfrak G\). Since only the elements \(L S_j^a L^{-1}\) have finite order, \(\alpha\) can be induced by an endomorphism \(\hat\alpha\) with
\[ \hat\alpha \hat S_i=\hat M_i \hat S_{n_i}^{\alpha_i}\hat M_i^{-1}. \]
By \(\Pi_0\) we denote the word
\[ \hat S_1\cdots \hat S_m \prod_{i=1}^{g}[\hat T_i,\hat U_i]. \]
We regard \(\hat\alpha\Pi_0\) in the group graph as a path with arbitrary initial vertex. After free reduction in \(\hat\alpha\Pi_0\) we obtain another path, which is called proper. \(\hat\alpha\Pi_0\) and its proper path are closed, but, generally speaking, not simply closed. If \(\Pi'\) is a proper part of \(\Pi_0\), then \(\hat\alpha\Pi'\) is not closed. The most important place in the proof of Theorem 1 is occupied by the following

Lemma 1. Suppose that all \(a_i=1\). If the proper path for \(\hat\alpha\Pi_0\) is not simply closed, then \(\alpha\) can be induced by such an endomorphism \(\hat\beta\) of the group \(\hat{\mathfrak G}\) that the length of the proper path will be smaller.

The idea of the proof is as follows: we consider the alternating product
\[ \{\hat\alpha\hat S_1,\ldots,\hat\alpha\hat U_g;\hat\alpha\Pi_0\} \]
and pass to the related reduced product
\[ \{\hat K_1,\ldots,\hat K_{m+2g};\Pi_{\hat K}\}, \]
where, for \(i\le m\),
\[ \hat K_i=\hat L_i\hat S_{p_i}\hat L_i^{-1}. \]
To every symbol from \(\hat L_i\) \((i\le m)\) and \(\hat K_j\) \((j>m)\) there belongs its formally inverse symbol from \(\hat L_i^{-1}\) or \(\hat K_j^{-1}\), which is called the inverse partner. In addition, the symbols that do not remain in the proper path for \(\Pi_{\hat K}\) have cancelling partners. We construct chains from inverse and cancelling partners. They are either closed or have ends on the proper path. Let \(\hat W\) be the reduced word for
\[ \Pi_{\hat K}(\hat K_1,\ldots,\hat K_{m+2g}) \]
and let \(\hat W=\hat W_1\hat W_2\) be a decomposition of the proper path into two loops. If there is a chain with ends \(\hat X_1,\hat X_2\) on \(\hat W_1=\hat V\hat X_1\hat V'\) and \(\hat W_2\), then we replace, in the words \(\hat K_1,\ldots,\hat K_{m+2g}\), the symbols of this chain by the elements \(\hat V^{-1}\hat V'^{-1}\) or \(\hat V'\hat V\). Since \(\hat W_1\) is a relation of the group \(\mathfrak G\), the new elements \(\hat K'_1,\ldots,\hat K'_{m+2g}\) induce in \(\mathfrak G\) the same elements that were induced by the elements \(\hat K_1,\ldots,\hat K_{m+2g}\). But the length of the proper path decreases by two or more.

A chain with ends on both loops exists if a double point separates on the proper path at least one pair \(\hat K_i,\hat K_i^{-1}\). The alternating product
\[ \{\hat\alpha\hat S_1,\ldots,\hat\alpha\hat U_g;\hat\alpha\Pi_0\} \]
has a related reduced ...

product with this property. Having reversed all the necessary elementary processes, we obtain an endomorphism \(\hat\beta\), which induces \(\alpha\).

Thus, one can induce \(\alpha\) by such an endomorphism \(\hat\gamma\) that the proper path for \(\hat\gamma\Pi_0\) is a simple closed contour. If we knew that it is the boundary of one disk in the planar group picture, then we would obtain the last equation of Theorem 1 and hence everything that is required.

§ 5. We shall consider the kernel \(\mathfrak R^*\) of the standard homomorphism.

Lemma 2. Let \(\mathfrak Z^{m+1}\) be a free commutative group with generators \(e_0,e_1,\ldots,e_m\). The mapping
\[ \hat L\hat S_i^{k_i}\hat L^{-1}\to e_i \quad\text{and}\quad \hat L\Pi_0\hat L^{-1}\to e_0 \]
for arbitrary \(\hat L\in\hat{\mathfrak G}\) determines a homomorphism \(\tau:\mathfrak R\to\mathfrak Z^{m+1}\).

This is proved by means of the Reidemeister–Schreier method \((^9)\) and the representation of the group \(\mathfrak G\) as a free product with amalgamated subgroups\(^*\).

Every endomorphism \(\hat\alpha:\hat{\mathfrak G}\to\hat{\mathfrak G}\) which induces an endomorphism of the group \(\mathfrak G\) determines an endomorphism of the group \(\mathfrak Z^{m+1}\) by the formulas
\[ e_0\to\tau\hat\alpha\Pi_0,\qquad e_j\to\tau\hat\alpha\hat S_j^{k_j}. \]
In its matrix \(M(\hat\alpha)\), every row except the first contains only one number different from zero.

Lemma 3. If the endomorphism \(\hat\varepsilon\) induces the identity automorphism of the group \(\mathfrak G\), then \(M(\hat\varepsilon)\) is the identity matrix.

In the proof one must distinguish the cases \(g=0\) \((^3)\), \(m=0\), and \(m,g>0\). It follows from the lemma that for every endomorphism over an automorphism of the group \(\mathfrak G\) the first row of the matrix has the form \((1,0,\ldots,0)\). If the proper path for \(\gamma\Pi_0\) is a simple closed contour, then it is the boundary of a subcomplex, and \(\tau\hat\gamma\Pi_0\) shows how many disks it contains with boundaries of the types \(\Pi_0,\hat S_1^{k_1},\ldots,\hat S_m^{k_m}\).

Thus,
\[ \hat\gamma\Pi_0=\hat M\Pi_0\hat M^{-1}, \]
as was required to prove.

Moscow State University
named after M. V. Lomonosov

Received
30 X 1963

REFERENCES

\(^1\) J. Nielsen, Acta Math., 50, 189 (1927).
\(^2\) H. Zieschang, Abhandl. math. Sem. Univ. Hamburg, 27, No. 1/2 (1964).
\(^3\) W. Vollmerhaus, Über die Automorphismen ebener Gruppen, Dissertation, Göttingen, 1963.
\(^4\) M. Greendlinger, Arch. Math., 12, 94 (1961).
\(^5\) H. Seifert, Abhandl. math. Sem. Univ. Hamburg, 12, 29 (1938).
\(^6\) W. Mangler, Math. Zs., 44, 541 (1939).
\(^7\) H. Hopf, J. reine u. angew. Math., 163, 71 (1930); 165, 225 (1931).
\(^8\) G. Burde, Math. Ann., 151, 101 (1963).
\(^9\) M. Hall, Theory of Groups, IL, 1962.

\(^*\) Let, for example, \(\mathfrak A\) be the group with generators \(S_1,\ldots,S_m\) and defining relations \(S_1^{k_1}=\cdots=S_m^{k_m}=1\), and let \(\mathfrak B\) be the free group with generators \(T_1,\ldots,U_g\). Then \(\mathfrak G\) is the free product of the groups \(\mathfrak A\) and \(\mathfrak B\) with amalgamated subgroups, which are generated by the elements \((S_1\cdots S_m)^{-1}\) and \(\Pi[T_i,U_i]\).