On subgroup conjugacy separabilityin the class of virtually free groups

On subgroup conjugacy separability
in the class of virtually free groups

O. Bogopolski
Institute of Mathematics of
Siberian Branch of Russian Academy of Sciences,
Novosibirsk, Russia
and Düsseldorf University, Germany
e-mail: OlegBogopolski@yahoo.com

F. Grunewald 111During the process of writing this paper Fritz Grunewald died.
Düsseldorf University, Germany
Abstract

A group is called subgroup conjugacy separable (abbreviated as SCS), if any two finitely generated and non-conjugate subgroups of remain non-conjugate in some finite quotient of . We prove that the free groups and the fundamental groups of finite trees of finite groups with some normalizer condition are SCS. We also introduce the subgroup into-conjugacy separability property and prove that the above groups have this property too.


1 Introduction

The subgroup conjugacy separability (see Definition 1.2) is a residual property of groups, which logically continues the following series of well known properties of groups: the residual finiteness, the conjugacy separability, and the subgroup separability (LERF). These properties help to solve some algorithmic problems in groups and they are also important in the theory of 3-manifolds.

Recall that a group is called separable (or residually finite, abbreviated as RF), if for any two elements , there exists a homomorphism from to a finite group such that in .

Similarly, is called conjugacy separable (abbreviated as CS), if for any two non-conjugate elements , there exists a homomorphism from to a finite group such that is not conjugate to in .

If in these definitions we replace the words “elements” by the words “finitely generated subgroups”, we obtain the following two definitions (the first one is well known, and the second is new).

Definition 1.1

A group is called subgroup separable (or LERF, for locally extended residually finite), if for any two finitely generated subgroups , there exists a homomorphism from to a finite group such that in .

Note that this definition is equivalent to the usual one: is called subgroup separable if for any finitely generated subgroup of and for any element , there exists a subgroup of finite index in such that , but .

The LERF property is useful in 3-manifold topology: if is a LERF group and is an immersion of an incompressible surface, then there is an embedding in a finite cover of . For more information, see the inspiring paper of P. Scott [36] and an overview of D. Wise in [45]. It would be interesting to find applications of the following property in topology.

Definition 1.2

A group is called subgroup conjugacy separable (abbreviated as SCS), if for any two finitely generated non-conjugate subgroups , there exists a homomorphism from to a finite group such that is not conjugate to in .

A.I. Mal’cev was the first, who noticed, that finitely presented residually finite (resp. conjugacy separable) groups have solvable word problem (resp. conjugacy problem) [26]. Arguing in a similar way, one can show that finitely presented LERF groups have solvable membership problem and that finitely presented SCS groups have solvable conjugacy problem for finitely generated subgroups. The last means, that there is an algorithm, which given a finitely presented SCS group and two finite sets of elements and , decides whether the subgroups and are conjugate in .

Clearly, any group with the property CS, LERF, or SCS is residually finite. We conjecture, that SCS does not imply CS or LERF and conversely.

There is a lot of papers devoted to the properties RF, CS, and LERF. We cite here some positive results about the CS and LERF properties. It would be interesting to establish, which of the listed below groups have the SCS property.

The conjugacy separability was established for

  1. virtually polycyclic groups (V. Remeslennikov [31] and E. Formanek [15])

  2. finitely generated virtually free groups (see J.L. Dyer [12] combined with P.F. Stebe [39])

  3. groups which can be obtained from these groups by repeatedly forming free products with cyclic amalgamations (L. Ribes, D. Segal und P. Zalesskii [33])

  4. virtually surface groups222For free products of two free groups, amalgamated over a cyclic group, in particular for surface groups see the paper of J.L. Dyer [13]. For Fuchsian groups see the paper of B. Fine and G. Rosenberger [14]. and the fundamental groups of Seifert 3-manifolds (A. Martino [27])

  5. fundamental groups of finite, 1-acylindrical graphs of free groups with finitely generated edge groups (O. Cotton-Barratt und H. Wilton [10])

  6. virtually limit groups (S. Chagas and P. Zalesskii [7, 9])

  7. finitely presented residually free groups (S. Chagas and P. Zalesskii [8])

  8. right angled Artin groups and their finite index subgroups (A. Minasyan [30]).

  9. free products of CS groups (P.F. Stebe [39] and V.N. Remeslennikov [32])

  10. non-uniform arithmetic lattices of and consequently the Bianchi groups (S. Chagas and P. Zalesskii [7]; see also the paper of I. Agol, D.D. Long, and A.W. Reid [1])

The subgroup separability was established for

  1. polycyclic groups (A.I. Mal’cev [26])

  2. free groups (M. Hall [21]

  3. surface groups (P. Scott [36])

  4. limit groups (H. Wilton [43])

  5. free products of LERF groups (R.G. Burns [5] and N.S. Romanovskii [35])

  6. free products of two free groups amalgamated along a cyclic group (A.M. Brunner, R.G. Burns and D. Solitar [4]; see also a generalization of M. Tretkoff [41])

  7. free products of a LERF group and a free group amalgamated along a maximal cyclic subgroup in (R. Gitik [17]) (Note, that the free product of two LERF groups amalgamated along a cyclic subgroup is not necessarily a LERF group (see [34] and [18])

If splits as a finite graph of free groups with cyclic edge groups, then is LERF if and only if does not contain a non-trivial element , such that is conjugate to for some (D. Wise [44]).

In [29], V. Metaftsis and E. Raptis are proved that a right-angled Artin group with associated graph is subgroup separable if and only if does not contain a subgraph homeomorphic to either a square or a path of length three.

P. Scott in [36] showed, that LERF is inherited by subgroups and finite extensions, in particular it is invariant under commensurability. In contrast, CS is not invariant under commensurability: in [28], A. Martino and A. Minasyan constructed a finitely presented CS-group, which has an index 2 subgroup without the CS property. An example of a finitely generated (but not finitely presented) non-CS-group , containing a CS-subgroup of index 2, was constructed by A. Gorjaga in [19].

The subgroup conjugacy separability property. It seems that the SCS property is much harder to establish than CS and LERF. One of the reasons is that this property has no an evident reformulation in terms of the profinite topology on .

Recall that the profinite topology on a group is the topology, having the family of all cosets of subgroups of finite index in as a base of open sets. Clearly, a finitely generated group is residually finite, respectively conjugacy separable or subgroup separable, if and only if one-element subsets of , respectively conjugacy classes of one-element subsets, or finitely generated subgroups are closed in the profinite topology. We conjecture, that the subgroup conjugacy separability for is not the same as the closeness, in the profinite topology, of the union of the conjugacy classes of any finitely generated subgroup of .

We know only one paper on SCS (without restrictions on subgroups): in [20], F. Grunewald and D. Segal proved that all virtually polycyclic groups are subgroup conjugacy separable (see also Theorem 7 in Chapter 4 of [37]).

In this paper we consider finitely generated virtually free groups. These groups are subgroup separable (since they are commensurable with free groups) and they are conjugacy separable (J.L. Dyer [12] and P.F. Stebe [39]). Therefore it is natural to ask, whether all finitely generated virtually free groups are subgroup conjugacy separable.

Recall that every finitely generated virtually free group is the fundamental group of a finite graph of finite groups (A. Karrass, A. Pietrowski and D. Solitar [23]). The main results of this paper are Theorems 1.31.5, and 1.81.9.

Theorem 1.3

Free groups are subgroup conjugacy separable.

In the following definition we use the notations of Section 5.1.

Definition 1.4

We say that a finite graph of finite groups (and its fundamental group) satisfies the normalizer condition, if for each nontrivial subgroup of every edge group of .

For instance, satisfies the normalizer condition, if are finite and is malnormal in , i.e. for every . Note that the normalizer condition for is equivalent to the condition that any finitely generated subgroup of has finite index in its normalizer. Moreover, the normalizer condition for a finite graph of finite groups can be verified algorithmically (see [2]).

Theorem 1.5

Let be a finite tree of finite groups, which satisfies the normalizer condition. Then its fundamental group is subgroup conjugacy separable.

We deduce these theorems from Theorems 1.8 and 1.9, and Proposition 1.7, where the following variation of Definition 1.2 is used.

Definition 1.6

1) For two subgroups and of a group , we say that is conjugate into , if there is an element such that is a subgroup of .

2) A group is called subgroup into-conjugacy separable (abbreviated as SICS) if for any two finitely generated subgroups such that is not conjugate into , there exists a homomorphism from to a finite group such that is not conjugate into in .

Proposition 1.7

Let be a virtually free group. Suppose that is subgroup into-conjugacy separable. Then is subgroup conjugacy separable.

Theorem 1.8

Free groups are subgroup into-conjugacy separable.

Theorem 1.9

Let be a finite tree of finite groups, which satisfies the normalizer condition. Then its fundamental group is subgroup into-conjugacy separable.

Methods. In the proof of Theorem 1.8 we use coverings of graphs, while in the proof of Theorem 1.9, we use a 3-dimensional topological realization of the graph of groups. Any covering of this realization can be obtained by gluing of some elementary spaces, which we call covering pieces. This technique is similar to that, which was developed by the first author in the papers [2] and [3] for a classification of groups with the M. Hall property.

To construct certain coverings (and so certain subgroups), we use a gluing schema, which comes from Theorem 3.3 of Füredi, Lazebnik, Seress, Ustimenko, and Woldar on the existence of -bipartite graphs without short cycles.

The structure of the paper is the following. In Section 2 we prove Proposition 1.7, in Section 3 we give some auxiliary statements. In Sections 4 and 5 we prove our main Theorems 1.8 and 1.9.

2 The SICS-property implies the SCS-property
for virtually free groups

Lemma 2.1

Let be two finitely generated subgroups of a virtually free group , such that is conjugate into and is conjugate into . Then is conjugate to .

Proof. It is sufficient to prove this theorem for finitely generated .

Let and for some . Then for . Moreover, if and only if and .

Suppose that strictly less than . Then, for , we have the strictly ascending chain of subgroups: . Let be a free normal subgroup of finite index in . We compare this chain with the chain .

Since the indices are finite and independent of , and since the indices are increasing with , the second chain is also strictly ascending: . This contradicts to the theorem of M. Takahasi (see [40], or [22, Theorem 14.1]), which claims, that a free group of a finite rank (in our case ) does not contain a strictly ascending chain of subgroups of a finite bounded rank. Thus, and so .

Proof of Proposition 1.7. Let be two non-conjugate, finitely generated subgroups of . By Lemma 2.1, w.l.o.g. we may assume that is not conjugate into . Since is a SICS-group, there exists a homomorphism from to a finite group , such that is not conjugate into in . In particular, is not conjugate to in . So, is subgroup conjugacy separable.

3 Auxiliary statements

Lemma 3.1

Let be subgroups of a group . Then the following conditions are equivalent:

(1) is conjugate into every finite index subgroup of , containing ;

(2) For every finite quotient of , the image of is conjugate into the image of .

Proof. : Suppose that (2) holds and let be a finite index subgroup of , containing . Then contains a finite index subgroup , which is normal in . By (2), the image of in is conjugate into the image of in . This implies that is conjugate into , and so into .

: Suppose that (1) holds and let be a finite quotient of . By (1), is conjugate into . Then the image of in is conjugate into the image of in .

Lemma 3.2

Let be a free product: , and let be a finitely generated subgroup of . Suppose that each and each is conjugate into a factor , where depends on (on i,j). Then the whole is conjugate into some .

Proof. We may assume that . By the Bass-Serre theory (see [38]), acts on a simplicial tree without inversions of edges so that the stabilizers of vertices of are conjugate to . So, each and each stabilize a vertex of . By Corollary 3 in [38, Chapter I, Section 6.5] of Serre, stabilizes a vertex of and hence is conjugate into some or into . The last cannot happen, since contains a non-trivial generator , which is conjugate into some .

A graph is called bipartite if the set of its vertices is a disjoint union of two nonempty sets and , such that every edge of connects a vertex from to a vertex from . A bipartite graph is said to be bi-regular if there exist integers such that for all and for all . In this case is called bi-degree of . Note that the lengths of cycles in a bipartite graph are always even.

Theorem 3.3

[16]. For any natural , there exists a finite connected bipartite graph of bi-degree , with length of smallest cycle exactly .

An -star is a tree with vertices and edges, outgoing from one common vertex. This vertex will be called central and the other ones peripherical. It is convenient to reformulate a weaker version of this theorem.

Theorem 3.4

For any natural , one can glue several -stars to several -stars, so that all peripherical vertices of -stars will be identified (by some bijection) with all peripherical vertices of -stars, the resulting graph will be connected, and it will not have cycles of length smaller than .

4 SICS-property for free groups

4.1 Notations

Our proof of Theorem 1.3 uses coverings of labeled graphs. Here we define a core of a covering, an outer edge, and an outer vertex of a core.

Let be a graph. By we denote the set of its vertices and by the set of its edges. The inverse of an edge is denoted by , the initial and the terminal vertices of are denoted by and .

Let be a free group with finite basis . Let be the graph consisting of one vertex and oriented edges . We label by and by . We will identify with by identifying with the homotopy class .

To every subgroup corresponds a covering map , such that is the image of the induced map . We lift the labeling of to . So, an edge of is labeled by if its image is labeled by .

If is finitely generated, then has a finite core, , that is a finite connected subgraph, which is homotopy equivalent to . We can enlarge if necessary and assume that is a vertex of and that every vertex of has valency 1 or . The vertices of valency 1 and the edges incident to these vertices are called outer. All other vertices and edges of are called inner. Let be an oriented outer edge of , which starts at an outer vertex. Then there is a unique oriented path in , such that , the labels of edges are coincide, and the last edge is outer. We will write . Clearly . Thus, we get a free involution on the set of outer edges of .

4.2 Proof of Theorem 1.8

Let be a free group with finite basis . Let be two nonconjugate finitely generated subgroups of such that is not conjugate into . By Lemma 3.1, it is sufficient to construct a finite index subgroup of , that contains and does not contain a conjugate of .

Let and let . Since is a residually finite, there exists a normal subgroup of finite index in , such that does not contain nontrivial elements of of length or smaller. Since is normal, does not contain any conjugate to these elements. This means that the covering graph is finite, every its vertex has valency , and

Without loss of generality, we assume that the vertices of have valency 1 or . Now we will embed into a finite labeled graph without outer edges. Let be the set of all edges of , that start at outer vertices of . For every edge we choose an edge in with the same label. Let be the labeled graph, obtained from the disjoint union of graphs

by identifying the vertices with and with for every .

Figure 1

Since every vertex of has valency , there is a finitely sheeted covering map , respecting the labeling. Thus for some finite index subgroup of . Since is a subgraph of , the subgroup contains as a free factor: .

We show, that is not conjugate into . Suppose the contrary: for some , Then every element , can be represented by a closed path in based at . By definition of the constant , every path can be freely homotopic to a closed path in of length at most . By Condition (1) and by Construction (2), every such path is freely homotopic to a closed path in . This means that every element , can be conjugated into by an element . By Lemma 3.2, can be conjugated into by an element . This contradicts to the assumption, that cannot be conjugated into in .

5 SICS-property for virtually free groups

By [6], every finitely generated virtually free group is the fundamental group of a finite graph of finite groups (see also [11, Chapter IV, Theorem 1.6] and historical comments on page 133 of [11]). We will also represent these groups as fundamental groups of some graphs of spaces (3-dimensional complexes). Below we introduce notations and recall some definitions. In Subsection 5.6 we prove Theorem 1.9.

5.1 Graphs of groups

A graph of groups is a system consisting of a connected graph , of vertex groups , , of edge groups , , and of boundary monomorphisms and , , which satisfy and .

A path in the graph of groups is a sequence of the form , where is a path in , and for . This path is closed, if ; in this case we say that it is based at the vertex . There is a usual (partial) multiplication of paths in .

Now we define three types of elementary transformations of a path :

1) replace a subpath of of the form , where , and , by the path , where and for some ;

2) replace a subpath of of the form , where and , by the element ;

3) this is the transformation inverse to 2).

Two paths and in are called equivalent, if can be obtained from by a finite number of elementary transformations. The equivalence class of is denoted by .

The fundamental group of the graph of groups with respect to a vertex , denoted , is the set of equivalence classes of all closed paths in based at with respect to the multiplication .

Denote . Every element can be represented by a closed path with minimal . We call such the length of and denote it by .

Note, that every vertex group of the graph of groups can be embedded into by the following rule. Choose a path is from to . The map , determines an embedding of into . If we choose another path from to , the resulting subgroup will be conjugate to the first one. Thus, canonically determines the conjugacy class of a subgroup of . Any subgroup of this class will be called a vertex subgroup of , corresponding to .

5.2 Graph of spaces

Below all spaces are assumed to be path connected topological spaces. In particular, their fundamental groups are well defined (up to isomorphism).

A graph of spaces is a system consisting of a connected graph , of vertex spaces , , of edge spaces , , and of -injective continuous boundary maps and , , which satisfy and . For the later it is convenient to think, that and are two copies of the same space.

The topological realization of the graph of spaces , denoted , is defined to be the quotient obtained from

by gluing to and to for every and , and by identifying the spaces and through the map , , . Denote .

The space                                      The pieces of

Figure 2                                    Figure 3

A body of is a subspace of of the form , . The piece associated with the body , denoted , is defined to be the quotient obtained from the topological space

by identifying with for each edge outgoing from and for every . The subspaces are called the handles of this piece. The subspaces are called the faces of this piece.

Any covering of a piece in is called a covering piece. Lifts of the body, of the handles, and of the faces of the piece are called the body, the handles, and the faces of the covering piece. Note that if is finite and is finite for every , then there is only a finite number of covering pieces, up to homeomorphism.

Clearly, the space can be obtained by an appropriate gluing of all pieces , , along their free faces. Every covering space of can be obtained by gluing of (may be infinitely many) copies of covering pieces along their faces.

A topological space is called a pre-covering of , if is a connected subspace of some covering of , which can be presented as a result of gluing of some covering pieces along their faces. A face is called a free face of , if it is a face of exactly one handle of . A handle of which contains a free face is called a free handle of .

With every pre-covering of we can naturally associate a graph by collapsing its bodies to vertices and its handles to “half”-edges. Let be the collapsing map for . We equip with the pseudometric induced by the usual path metric on  (where the “half”-edges have length ). In particular, the distance between any two points of a body of is zero and the maximal distance between two points of a handle is .

Let be the set of middle points of edges of length 1 in . A curve is called regular, if has endpoints in and is locally injective on . The -length of a regular curve in , denoted , is the sum of lengths of the “half”-edges which passes.

A curve is called regular, if the curve is regular. The -length of a regular curve in , denoted , is defined to be . Roughly speaking, is the number of handles which passes, divided by 2.

5.3 From graphs of groups to graphs of spaces

With every group we associate the 2-dimensional CW-complex , consisting of the unique vertex , the edge set , and the set of 2-cells , where the boundary of is glued along the path . We identify the groups and through the canonical isomorphism .

With every embedding of groups we associate the embedding of complexes , such that the induced homomorphism of fundamental groups coincides with .

Now, with any graph of groups we associate the graph of spaces , such that for and the embeddings of spaces, , , correspond to the embeddings of groups , . For any vertex , the groups and are canonically isomorphic and we will identify them through this isomorphism. Every element can be realized by a regular closed path in based at , such that the number of subspaces it crosses is equal to ; in our notations we have .

5.4 Subgroups and coverings, cores of coverings

Simplifying notations in the previous section, we denote , , and . We may assume .

For every subgroup of there exists a covering map , such that . The space can be presented as the topological realization of a graph of spaces . The vertex spaces and the edge spaces of are connected components of , , and of , respectively.

If is finitely generated, then there is a pre-covering of , such that
1) can be obtained by gluing of finitely many covering pieces along some of their faces;
2) every loop in can be freely homotoped into .

Any such pre-covering will be called a core of . We choose one of them and denote it by . The closure of every connected component of will be called a thick tree. Every thick tree grows from a free face of the core and it can be deformationally retracted onto this face.

5.5 Trivial handles

A covering handle will be called trivial if its fundamental group is trivial. The following is a preparation to the proof of Theorem 1.9.

5.5.1 A linear order on the set of trivial covering handles

From now on we assume that is a tree. For every edge , let denote the connected component of which contains . For every we set

Now we choose an arbitrary linear order on each and extend these orders to a linear order on by saying that edges in are smaller than edges in if and only if .

We will consider covering spaces up to equivalence of coverings. Let be two trivial covering handles, which cover the handles and of . We say that is smaller than and write , if .

5.5.2 Extensions of pre-coverings through their free trivial handles

We explain a construction, which enables to extend pre-coverings of through their trivial free handles keeping the fundamental group unchanged.

Let be a pre-covering of and let be a trivial free handle of . By definition, the face of is free in and is the universal covering of a handle in , say . Let be the other handle in which has a common face with , and let be the piece in which contains . Consider the universal covering and some lift of in . We can extend by gluing and along the free faces of and . So we obtain a new pre-covering of with the same fundamental group as .

Figure 4

As a preparation to the proof of Theorem 1.9, we explain a more complicated gluing. Let be all lifts of in . Now we take copies of , say (with the handles corresponding to ) and glue them to so that the free face of will be glued to the free face of for . The resulting space is again a pre-covering of and its fundamental group is isomorphic to the free product of copies of . We call the tuple ) the tuple associated with the handle .

The following lemma allows to prove Theorem 1.9 by induction.

Lemma 5.1

Let be a handle of , different from . Then .

Proof. The proof is straightforward and uses the assumption, that is a tree.

5.6 Proof of Theorem 1.9

Let be the fundamental group of a finite graph of finite groups: , where is a tree and suppose that satisfies the normalizer condition. By Section 5.3, we can write , where is the topological realization of the graph of spaces associated with the graph of groups .

Let be two finitely generated subgroups of and suppose that is not conjugate into in . We will construct a finite index subgroup of , such that is contained in and is not conjugate into . Then Theorem 1.9 will immediately follow from Lemma 3.1.

Suppose that and let , where is the length function on (see Section 5.1).

Consider the covering which corresponds to , i.e. . By enlarging the core of if necessary, we may assume that . We will complete to a finite sheeted covering space and put then . We will do that in several steps.

1) Since satisfies the normalizer condition, there is a compact pre-covering that contains and whose free faces are trivial.

2) Consider the compact pre-covering that contains and whose free faces are at distance from . The components of are parts of thick trees, which grow from the free faces of . By 1) these componens are contractible.

Figure 5

Note for step 4), that .

3) Let be the free handles of that are maximal, with respect to , among all free handles of . Let be the tuple associated with the handle (and so with each ). Taking into account only these handles, it is convenient to think that has the form of the -star and has the form of the -star.

We take several copies of and several copies of and glue them according to Theorem 3.4, where we put .

Figure 6

The resulting space has the following properties.

(a) The free handles of are trivial.

(b) The maximal (with respect to ) free handles of are smaller than that of .

Property (a) follows from 1). Property (b) follows from the fact, that the maximal free handles of the copies of became non-free after gluing them with the free handles of the copies of . Moreover, the handles of copies of , which remain free in are smaller than with respect to by Lemma 5.1.

4) We construct by fulfilling Step 3) for instead of . Continuing in this way, we obtain the sequence of regular spaces , with the property that the maximal free handles of are smaller than those of . Since the order is finite, the sequence is finite and the last space, denote it by , has no free handles. Then is a finite sheeted covering of . Let be the corresponding finite index subgroup of .

Note, that is the result of gluing of several copies of , say , and several contractible spaces, say , along their free faces. So, we have for some free group . This means, that

for some . After renumbering, we may assume that , and so .

Lemma 5.2

Every regular loop in , which crosses a face of some and has the e-length smaller than is contractible.

Proof. Let be the minimal natural number, such that lies in a copy of , which was used in the construction of . For simplicity, we assume that this copy coincides with . Note that the -length of any closed regular curve in is a nonnegative integer number (see Section 5.2).

We will proceed by induction on , assuming that the pairs are lexicographically ordered. If , then entirely lies in a face of , and hence is contractible. So, we assume that .

Case 1. Suppose that . Then lies in and crosses a free face of . Recall that by Step 2), the distance between any free face of and is and that every component of is contractible. Since , the curve lies in some component of and so is contractible.

Case 2. Suppose that . Recall that is the result of gluing of several copies of and several -spaces. We will say that these -spaces have level .

Since is minimal, crosses some -space of level , say . If completely lies in , it is contractible. So, we assume that crosses a free face of , say , and enters into a copy of , say . After running inside it must again cross a free face of .

Subcase 2.1. Suppose that . We write , where is a subcurve of , which lies in , has endpoints in and . If , then lies in and so lies in , that contradicts to the minimality of . Hence .

Let be a path in from the terminal point of to the initial one. Then and . Since both and meet the face and lie in , they are contractible by induction. So, is contractible.

Subcase 2.2. Suppose that . Denote by the -space, which is adjacent to through the common face . Clearly it has level . The curve must leave through a face . Since is contractible, we may assume that . Continuing, we obtain that passes through a cyclic sequence of subspaces

where we may assume that for every three consecutive subspaces the faces and are different. So, , that contradicts to the construction of according to Theorem 3.4.

Lemma 5.3

Every regular loop in , which has length smaller than , is either contractible or lies in some .

Proof. This follows from the previous lemma in view of the facts that is the disjoint union of the interiors of and that each is contractible.

Lemma 5.4

is not conjugate into in the group .

Proof. Assume the contrary, say for some , and represent the elements