On properly 3-realizable groups

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Topology and its Applications 153 (2005) 337–349

www.elsevier.com/locate/topo

On properly 3-realizable groups✩

Manuel Cárdenas, Francisco F. Lasheras∗

Departamento de Geometría y Topología, Universidad de Sevilla, Apdo 1160, 41080 Sevilla, Spain

Received 11 October 2002; accepted 28 October 2003

Abstract

We study the class of those finitely presented groups which are “properly 3-realizable”, i.e.groupsG for which there exists a compact 2-polyhedron havingG as fundamental group and whouniversal cover is proper homotopy equivalent to a 3-manifold (with boundary). We show that camalgamated free product of groupsG1 ∗F G2 (HNN-extensionsG∗F ), over a cyclic groupF , areproperly 3-realizable. As a consequence, we note that there are properly 3-realizable groupcannot be realized by a fake surface without points of typeIII . The question of whether or not evefinitely presented group is properly 3-realizable still remains open. 2004 Elsevier B.V. All rights reserved.

MSC:primary 57M07; secondary 57M10, 57M20

1. Introduction

In [2] it is shown that the proper homotopy type of anyd-dimensional locally finiteCW-complexP can be represented by a closed subpolyhedronQ in R2d ; moreover, ifP is properlyk-connected (i.e.,P has the proper homotopy type of a CW-complexY

with Y k = T an end-faithful tree) then the polyhedronQ can be taken inR2d−k (com-pare [13,3]). Thus, any finitely presented groupG is the fundamental group of a compa2-polyhedronK whose universal coverP = K has the proper homotopy type of ann-

✩ This work was partially supported by the project BFM 2001-3195-C02.* Corresponding author.

E-mail address:[email protected] (F.F. Lasheras).

0166-8641/$ – see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.topol.2003.10.011

338 M. Cárdenas, F.F. Lasheras / Topology and its Applications 153 (2005) 337–349

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manifold for eachn � 4, namely any regular neighborhood inRn of the correspondingsubpolyhedronQ in R4 ⊆ Rn, and we will refer to this property simply by saying thG is properlyn-realizable for eachn � 4. The question of whether or not every finitepresented groupG is properly 3-realizable remains open, i.e., does there exist a com2-polyhedronK with π1(K) ∼= G and whose universal coverK has the proper homotopy type of a 3-manifold (with boundary)? Observe that if the universal coverK isproperly 1-connected thenπ1(K) ∼= G is indeed properly 3-realizable. In case of a pitive answer, this property would allow us to use duality arguments in the study of clow-dimensional ((co)homological) proper invariants of the groupG, such as the seconcohomology groupH 2(G;ZG) (see [10]). See also [4,5] for a topological interpretatof H ∗(G;ZG).

For the study of this property it is convenient to restrict ourselves to a particular cl2-polyhedra introduced in [8], namely the class of (closed) fake surfaces, i.e., thosecompact 2-polyhedra in which each point has a neighborhood of one of the types in

It is worth noting that every compact 2-polyhedron is (simple) homotopy equivalea fake surface [15], and hence every finitely presented group appears as the fundgroup of a compact fake surface. Of course, one can check that all 3-manifold grouproperly 3-realizable, since every compact 3-manifoldM (with boundary) collapses to2-polyhedronK (which can be taken to be a fake surface), whose universal coverK is thenproper homotopy equivalent toM .

We denote byP the class of those compact fake surfaces with no points of typeIII .In [10] it was shown that ifG belongs to the classC of those finitely presented groupwhich are the fundamental group of a fake surface as inP , thenG is properly 3-realizableSee also [11] for an extension of this result in case each component of the subgraΓX

of all triple edges of a fake surfaceX contains at most one point of typeIII , and eachof the subgroups ofπ1(X) generated by the loopsγ0, γ1 ⊂ ΓX at such a point is of thform 〈γ0〉 ∗ 〈γ1〉 or 〈γ0〉 ⊕ 〈γ1〉, together with an extra condition wheneverγi represents aelement of (finite) odd order.

In this paper, we learn more about the classC (see Section 2) and continue the woin [10,11], dealing with certain amalgamated free products and HNN-extensions of gas inC. Recall that ifG0, G1 andF are groups andϕi :F → Gi (i = 0,1) are monomor-phisms one writesG0 ∗F G1 for the group〈G0,G1; ϕ0(a) = ϕ1(a), a ∈ F 〉, which is anamalgamated free product ofG0 andG1 along the “common” subgroupF . ForF = {1},we obtain the free productG0 ∗ G1. On the other hand, ifG and F are groups andψ0,ψ1 :F → G are monomorphisms one writesG∗F for the group〈G, t; t−1ψ0(a)t =ψ1(a), a ∈ F 〉, which is anHNN-extension ofG [7]. In Proposition 2.13 it is observe

Fig. 1.

M. Cárdenas, F.F. Lasheras / Topology and its Applications 153 (2005) 337–349 339

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thatC is closed under free products, but not under amalgamated free products in gThe main result of this paper is:

Definition 1.1. We say that a subcomplexJ ⊂ X ∈ P is “nice” if every componentCof ΓX intersectingJ has a trivialT -bundleN over it in X, andJ meets no more thatwo components ofN − C. A p.l. embeddingf :S1 → X is nice if f (S1) ⊂ X is a nicesubcomplex. Moreover, two p.l. embeddingsf , g :S1 → X form a nice pair of embeddingif the unionf (S1) ∪ g(S1) ⊂ X is a nice subcomplex.

Theorem 1.2. Let G,G0,G1 ∈ C andF be a cyclic group together with monomorphisϕi :F → Gi , ψi :F → G, i = 0,1.

(a) If F is finite, then the corresponding amalgamated free productG0 ∗F G1 (HNN-extensionG∗F ) is always properly3-realizable.

(b) If F is infinite and the monomorphismsϕi can be realized by nice p. l. embeddinfi :S1 → Xi into fake surfacesX0,X1 ∈P , then the corresponding amalgamated frproductG0 ∗F G1 is properly3-realizable.

(c) If F is infinite and the monomorphismsψi can be realized by a nice pair of disjoinp.l. embeddingsf0, fi :S1 → X into a fake surfaceX ∈ P , then the correspondinHNN-extensionG∗F is properly3-realizable.

In Remark 3.2 we see that Theorem 1.2 provides us with properly 3-realizable (nmanifold) groups which are not inC.

2. Fake surfaces

Let us recall some definitions and results from [9].

Definition 2.1. Let X be a 2-dimensional locally finite simplicial complex, and assumelink lk (v,X) is planar, for every vertexv of X. Let (v,w) be a 1-simplex ofX. Given anembeddingφv : lk(v,X) → R2, we denote byθφv (w) the cyclic ordering determined byφv

on lk((v,w),X) as we go aroundφv(w) following the orientation inR2. Note that if thecardinality| lk((v,w),X)| is � 2, then there is only one cyclic orderingθφv (w).

We denote byΓX ⊂ X the graph consisting of all the 1-simplexes ofX which are oforder> 2 (i.e., those which are the face of at least three 2-simplexes ofX). Consider thecochain complex ofΓX overZ2

0→ C0(ΓX;Z2)δ→ C1(ΓX;Z2) → 0.

Given a familyΦ = {φv : lk(v,X) → R2, v ∈ X0} of embeddings, we can associate tocochain (cocycle)ωΦ = ∑

σ⊂ΓXωΦ(σ) ·σ ∈ C1(ΓX;Z2) whereωΦ(σ) = 0 if θφo(σ)

(t (σ ))

andθφt(σ)(o(σ )) are opposite, andωΦ(σ) = 1 otherwise. Here,o(σ ) andt (σ ) are the ver-

tices ofσ . By extension, we defineωΦ(σ) = 0 for every 1-simplexσ of order� 2.


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We say that a simplicial complexX “thickens” to a polyhedronY if Y admits a CW-structure containing, as a subcomplex, a copy of a subdivision ofX onto whichY collapses.The following theorem is proved in [9] (see also [12]).

Theorem 2.2. LetX be a2-dimensional connected locally finite simplicial complex. ThX thickens to an orientable3-manifold if and only if:

(i) lk (v,X) is planar for every vertexv of X, and(ii) there exists a family of embeddingsΦ = {φv : lk(v,X) → R2, v ∈ X0} so that the

associated cochainωΦ is trivial.

Next, we recall the concept of a “fake surface” as well as some basic definitionresults.

Definition 2.3. A 2-dimensional locally compact polyhedronX is called a (closed) fakesurfaceif each point inX has a neighborhood of one of the p.l. types in Fig. 1.

In particular, lk(x,X) is planar for every pointx of X.

Proposition 2.4 [15, Theorem 1]. Every compact connected2-polyhedron is(simple) ho-motopy equivalent to a compact fake surface.

Given a fake surfaceX and a triangulationK of X, we denote byΓX ⊂ X the 1-dimen-sional subpolyhedron determined by the subgraph ofK consisting of all the 1-simplexes oorder 3, i.e., those which are the face of three 2-simplexes. Observe thatΓX is independenof the triangulationK of X, and it consists of all those points ofX of typesII or III . Wealso have that every point of typeIII is a vertex of the triangulation.

Remark 2.5. In the case of a fake surfaceX, Theorem 2.2 leads to a well defined obstrtion ξX = [ωΦ ] ∈ H 1(ΓX;Z2) to thickeningX to an orientable 3-manifold (see [9,12More geometrically, a fake surface thickens to an orientable 3-manifold if and onlydoes not contain a union of a Möbius band with an annulus (with one of its boucomponents attached to the middle circle of the Möbius band), see [12, Corollary 1.

Definition 2.6. A T ∗-bundleis a bundle whose fiberT ∗ consists of a collection of segmensharing a vertex. AT ∗-bundle over a circle is obtained fromT ∗ ×I by identifyingT ∗ ×{0}with T ∗ × {1} via a homeomorphism ofT ∗. In the caseT ∗ consists of three segments, walso writeT = T ∗.

Proposition 2.7 [8, Lemma 5]. By numbering the three segments ofT , the homeomorphisms ofT are classified(up to isotopy) by the elements of the symmetric groupS3.Thus, three types ofT -bundles over a circle occur, induced by the identity permutatthe 3-cycle (231), and the2-cycle (23), which we will denote by types(i), (ii) and (iii)respectively.


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Proposition 2.8 [16, Section 3]. LetX be a fake surface. LetC be a simple closed curve iΓX andN be a regular neighborhood ofC in X. Then, there is embedded inN a T -bundleoverC whose type is uniquely determined byC.

Remark 2.9. If X has no points of typeIII , then aT -bundle embedded in the regulneighborhoodN is itself a regular neighborhood ofC in X. Thus, since a regular neighborhood is a mapping cylinder [1], aT -bundle over a circleC is homeomorphic to one othe following

(a) The mapping cylinder of a mapf :Si ∪ S2 ∪ S3 → C with Si = S1 andf |Si of degree±1, if theT -bundle is of type (i).

(b) The mapping cylinder of a mapf :S1 → C of degree±3, if theT -bundle is of type (ii).(c) The mapping cylinder of a mapf :S1 ∪ S2 → C with Si = S1, f |S1 of degree±1 and

f |S2 of degree±2, if theT -bundle is of type (iii).

Definition 2.10. Let X be a fake surface andN(ΓX) be a regular neighborhood ofΓX inX. The connected components ofX − N(ΓX) are called 2-components ofX. Note thateach 2-component is an open surface.

Although [8, Lemmas 12 and 13] are for compact fake surfaces, their proof also sthe following:

Lemma 2.11. Let X be a1-acyclic fake surface, and letM be a component ofcl(X −N(ΓX)). Let M be the surface( possibly with boundary) obtained fromM by gluing adisk along the boundary of each compact component of∂M . ThenM − ∂M is eitherS2(∂M = ∅) or R2.

Lemma 2.12. Let X be a1-acyclic fake surface, and letM be a component ofcl(X −N(ΓX)). Then, no component ofcl(X − M) contains two distinct components of∂M .

Let us return to the classC defined in the introductory section, i.e., the class of thfinitely presented groups which are the fundamental group of a compact fake surfaceX ∈Pwith no points of typeIII . We have

Proposition 2.13. The classC satisfies the following properties:

(a) If G ∈ C andH � G with [G : H ] < ∞, thenH ∈ C.(b) No group inC contains non-trivial perfect subgroups of finite index.(c) C contains groups which are not3-manifold groups.(d) If G,H ∈ C, thenG ∗ H ∈ C.(e) In general,C is not closed under amalgamated free products.

Proof. For (a), consider a fake surfaceX ∈ P with π1(X) ∼= G, and letY be the cov-ering space ofX corresponding to the subgroupH � G. It is clear thatY ∈ P with


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π1(Y ) ∼= H ; andY is compact, since[G : H ] < ∞. (b) follows from (a) and [10, Proposition 1.1]. For (c), consider the standard 2-complexX ∈ P associated to the groupresentation〈a, b; ab−1a−1b2〉 (see [10, Remark 2.11]). To see (d), letX,Y ∈ P withπ1(X) ∼= G andπ1(Y ) ∼= H , and choose small closed disksD ⊂ X − ΓX , D′ ⊂ Y − ΓY .We construct a new fake surfaceZ with π1(Z) ∼= G ∗ H by gluing X andY along thedisksD andD′ via the identity map. It is clear thatZ ∈ P , sinceΓZ = ΓX � ΓY � S1

(disjoint union). In particular, all finitely generated free groups of rank� 0 are inC,sinceS2 ∈ P and the standard 2-dimensional CW-complex associated to the presen〈a, b;aba−1b−1, a〉 of the group of integers is also inP . Finally, Higman [6] showedthat the groupH = 〈a, b, c, d;a−2b−1ab, b−2c−1bc, c−2d−1cd, d−2a−1da〉 can be con-structed as a multiple amalgamated free product of the form(G∗Z G)∗Z∗Z (G∗Z G) whereG ∈ C is the Baumslag–Solitar group〈a, b;ab−1a−1b2) considered in (c).H /∈ C, sinceH

is perfect, and hence (e) follows. See Remark 3.2 for another example.�Theorem 2.14 [10, Theorem 1.2]. All groups inC are properly3-realizable.

Next, we want to take a closer look to those groups inC finding a generic presentatiofor them.

Let Yg,k (g � 0, k � 1) be a compact connected orientable surface of geng

(S2 #T 2 #(g)· · · #T 2) with k (open) disks removed, andZg,k (g � 1, k � 1) be a compac

connected non-orientable surface of genusg (RP 2 #(g)· · · #RP 2) with k (open) disks re

moved.Let Fm (m � 0) denote the free group onm generators (by conventionF0 = {1}). Let

x ∈ Yg,k (or Zg,k), andS1, . . . , Sk ⊂ Yg,k (or Zg,k) be the boundary components,xi ∈ Si ,andγi be a path fromx to xi , 1� i � k. Choose isomorphisms:

ρ :π1(Yg,k, x) → F2g+k−1, ζ :π1(Zg,k, x) → Fg+k−1

and take

αi = ρ([

γi.Si .γ−1i

]), βi = ζ

([γi.Si .γ

−1i

]), 1� i � k.

Notice that ifg = 0 andk = 1, thenα1 = 1; otherwise,αi �= 1. On the other hand,βi �= 1for g � 1, k � 1.

Let X ∈ P be a connected fake surface, and assumeX is not a surface. Then, for eve2-componentF of X, cl(F ) is homeomorphic to eitherYg,k or Zg,k for someg � 0, k � 1.Note thatX can be obtained as a quotient space from

⊔{cl(F ), F = 2-component} � ΓX

by gluing the boundary components of each cl(F ) to the corresponding component ofΓX

via a mapS1 → S1 of degree±1,±2,±3.We construct the following graphL associated withX: L has a vertexvF for each

2-componentF of X, and a vertexwC for each componentC of ΓX . Whenever a boundarcomponent of cl(F ) gets identified with a componentC ⊂ ΓX in X, we have an edgee inL joining vF andwC . This completes the description ofL. We orient this graph so that thinitial vertex of every edge is of the formvF , for some 2-componentF .

Next, we assign a group to each simplex ofL:

• G(e) = 〈θ〉 ∼= Z, e ⊂ L an edge,


groups

fbe

• G(wC) = 〈εC〉 ∼= Z,• G(vF ) = F2g+k−1 or Fg+k−1

(depending on whether cl(F ) is homeomorphic toYg,k or Zg,k) and morphisms:

• G(e) → G(wC), θ �→ εnC ,

• G(e) → G(vF ), θ �→ αi (or βi ),

wheree = (vF ,wC), cl(F ) ∼= Yg,k (orZg,k), andSi ⊂ cl(F ) is to be identified withC ⊂ ΓX

via a map of degreen (n = ±1,±2,±3).

Proposition 2.15. LetT ⊂ L be a maximal tree. Then, the fundamental group ofX admitsa presentation of the following type:Generators:

(i) G(u), for every vertexu ∈ L,(ii) te, for every edgee ⊂ L.

Relations:

(i) te = 1, for every edgee ⊂ T ,(ii) αi.te = te.ε

nC (or βi.te = te.ε

nC ),

wheree = (vF ,wC), cl(F ) ∼= Yg,k (or Zg,k), g � 0, k � 1, andSi ⊂ cl(F ) is to be identifiedwith C ⊂ ΓX via a map of degreen (n = ±1,±2,±2).

Proof. Observe that if no 2-component ofX is an open disk, then the graphL together withthe given data (vertex groups, edge groups and morphisms) constitute a graph of(G,L) (sinceαi,βi �= 1 if (g, k) �= (0,1)), and the fundamental group ofX is isomorphicto the fundamental group of the graph of groups(G,L) based atT which, by definition,has the presentation given in the statement of this proposition (see [14,4]).

Suppose now that one of the 2-componentsF ⊂ X is an open disk. LetS1 ⊂ cl(F ) beits boundary, andC be the component ofΓX thatS1 is to be identified with (via a map odegreen). Then, we should add a relationεn

C = 1 to the above presentation, which canwritten asα1.te = te.ε

nC , e = (vF ,wC), since cl(F ) ∼= Y0,1 andα1 = 1. This relation fits

into the set of relations (ii). �In view of this generic presentation, we have:

Proposition 2.16. The only torsion-free groupsG ∈ C with Gab∼= Z are Z and the

Baumslag–Solitar group〈a, b;ab−1a−1b2〉.

Proof. Let X ∈ P be a (compact) fake surface withπ1(X) ∼= G (and henceH1(X) ∼= Z).It is clear thatX is not a surface. Also, observe that all the generators ofπ1(X) of the form


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eck,

te, e ⊂ L−T , generate a free Abelian subgroup ofH1(X), and hence the graphL containsat most one cycle. We follow four steps:

Step1. No 2-componentF of X has its closure of the formYg,k (g � 1), since otherwiseone can check that all the generators of the corresponding tori would survive inH1(X)

giving us a group of rank� 2.Step2. No 2-componentF of X has its closure of the formZg,k . For if g > 1, the

generators of the corresponding projective planes would survive inH1(X) giving us agroup of rank> 1. On the other hand, if cl(F ) = Z1,k we denote byθ the generator othe corresponding projective plane, which survives inH1(X). Note that there must be ncycles in the graphL, since otherwise some generator ofπ1(X) of the form te wouldproduce another (independent) generator ofH1(X). Therefore,L is a tree and each of thboundary componentsS1, . . . , Sk of cl(F ) disconnectsX. Also, observe that no other 2component ofX is the formZ1,k (i.e., they all are 2-spheres with holes), since otherwthey would produce new generators forH1(X), namely the generators of the correspondprojective planes. Therefore, sinceH1(X) ∼= Z, one can check that, for each 1� i � k, thecomponent ofX − Si not containingF must be homotopy equivalent to a polyhedrobtained from a finite collection of 2-spheres (arranged in a row) with one or twoeach, and attached one to the next along a boundary component via a map of deg±1or ±2, and this polyhedron having finite fundamental group. Hence, each of theSis yieldsthe trivial element inπ1(X), for this group is torsion-free. Thus, the groupG also appearas the fundamental group of the projective plane obtained from cl(F ) by attaching a diskalong each of theSis, which is a contradiction. Hence, all 2-components ofX are 2-spherewith holes.

Step3. The graphL contains exactly one cycleγ . For this, supposeL was a tree, andchoose a 2-componentF of X. Then, as in the previous step, one can check that eathe boundary components of cl(F ) yields the trivial element inπ1(X) whenceG = {1},which is a contradiction.

Step4. We take a simple closed curveS ⊂ X realizing the cycleγ ⊂ L and whose intersection with each component of the graph of triple edgesΓX ⊂ X consists of at most onpoint. LetY ⊂ X be the subpolyhedron consisting of (the closure of) those 2-compoF of X with F ∩ S �= ∅. Arguing as in step 2, we get that every boundary componeSi

of the closure of each 2-componentF ⊂ Y with Si ∩ S = ∅ yields the trivial element inπ1(X), whenceG appears as the fundamental group of the fake surfaceZ obtained fromY by attaching a disk along eachSi ⊂ F ⊂ Y as above. Thus,Z can be obtained fromfinite collection of 2-spheres (distributed along the curveS) with one or two holes eachand attached one to the next along a boundary component via a map of degree±1 or ±2.Finally, sinceS generatesH1(X) ∼= H1(Z) and no component ofΓZ ⊂ Z is homologous toa multiple ofS in Z, it is then an exercise to check that the only possibilities forπ1(Z) (sothatH1(Z) ∼= Z) areZ (generated byS) or the group with presentation〈a, b;ab−1a−1b2〉(the generatora being represented byS), as desired. �Remark 2.17. Suppose nowG ∈ C is torsion-free withGab

∼= Z ⊕ Z andG � Z ⊕ Z, andlet X ∈ P with π1(X) ∼= G. We give a hint of what the groupG may look like using theabove presentation forπ1(X). First of all, arguing as in Proposition 2.16, it is easy to chthat no 2-componentF of X has its closure of the formYg,k (g > 1). On the other hand


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we cannot have cl(F ) = Y1,k , since otherwise the graphL would be a tree andX wouldhave the homotopy type of the torus obtained from cl(F ) by attaching a disk along eacboundary component, whenceG ∼= Z ⊕ Z which is a contradiction. Similarly, we rule outhe possibility that cl(F ) = Zg,k , g > 1, sinceH1(X) ∼= Z ⊕ Z. Next, assume that cl(F ) =Z1,k for some 2-componentF of X, and observe that the generator of the corresponprojective plane survives inH1(X). Then, either there is another 2-componentF ′ withcl(F ′) = Z1,k and the graphL has no cycles or any other 2-component ofX is a 2-spherewith holes andL contains exactly one cycle (realized by a simple closed curveS ⊂ X

as in Proposition 2.16). Therefore, arguing as in Proposition 2.16, the homotopy tyX is represented either by a fake surface obtained from cl(F ),cl(F ′) and a collection of2-spheres with holes (arranged in a row) connecting them and attached to each othea boundary component via maps of degree±1,±2 or a fake surface obtained from cl(F ),a collection of 2-spheres with holes distributed alongS and attached to each other as aboand a similar collection of 2-spheres with holes arranged in a row and connectincircuit alongS to cl(F ). The first case is ruled out, since in that situation one can cthatH1(X) � Z ⊕ Z, which is a contradiction.

The other possibility is that the graphL contained exactly two independent cycles (iL/T ∼= S1 ∨ S1) realized by simple closed curvesS,S′ ⊂ X whence the homotopy type oX would then be represented by a fake surface obtained from two collections of 2-swith holes distributed along the curvesS andS′ respectively, and attached one to the nalong a boundary component via maps of degree±1,±2. In case the two cycles inL weredisjoint then there would be a third collection of 2-spheres with holes arranged inand connecting the circuits along the curvesS,S′. Otherwise, the circuits alongS andS′share those 2-spheres with holes whose interior meets bothS andS′.

3. Proof of the main result

The purpose of this section is to prove Theorem 1.2.

Proof of Theorem 1.2. We start with part (a), sayF = 〈t; tn〉. Let G0,G1 ∈ C togetherwith monomorphismsϕi :F → Gi (i = 0,1), and denote byH = G0 ∗F G1 the corre-sponding amalgamated free product. LetX0,X1 ∈ P be (triangulated) fake surfaces wiπ1(Xi) ∼= Gi , andfi :S1 → Xi be p.l. maps such that Imfi∗ ⊆ π1(Xi) corresponds to thsubgroup Imϕi ⊆ Gi , i = 0,1. We take the 2-polyhedronY ′ = D2 ∪h S1, with h : ∂D2 →S1 a map of degreen, and consider the adjunction spacesY = (S1 × I ) ∪S1×{1/2} Y ′(homotopy equivalent toY ′) and Z = Y ∪f0×{0}∪f1×{1} (X0 � X1). Observe thatZ isa 2-polyhedron with fundamental groupπ1(Z) ∼= H (by Van Kampen’s theorem), anwhose subgraphΓZ of edges of order� 3 consists of the union of(ΓX0 ∪ Imf0) and(ΓX1 ∪ Imf1), together withS1 × {1/2} ⊂ Y in casen � 3. Let Z be the universal coveof Z with covering mapp : Z → Z. Then,p−1(Xi) consists of a disjoint union of copieof the universal cover Xi of Xi since the inclusionXi ↪→ Z induces a monomorphismGi ↪→ H between the fundamental groups,i = 0,1. On the other hand, the universal covY ′ of Y ′ consists of a collection ofn disks attached along their boundary via a map


ct 2-

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rst

d,

f

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opiesll

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heo

graph

.5.t each

ce

degree 1. Then,p−1(Y ) consists of a disjoint union of copies of(S1 × I ) ∪S1×{1/2} Y ′.Observe that each component ofp−1(Y ) contains a productS1 × I . Thus,Z is constructedfrom p−1(Y ) and a collection of copies ofX0 andX1, by gluing eachS1 × {i} ⊂ p−1(Y )

to a copy ofXi via liftings fi :S1 × {i} → Xi of the mapsfi , i = 0,1.Let i = 0 or 1. The proof of [10, Theorem 1.2] shows that there is a compa

polyhedronKi ⊃ Xi with π1(Ki) ∼= π1(X1), and whose universal coverKi is properhomotopy equivalent to a locally compact 2-polyhedronWi which thickens to a (noncompact) orientable 3-manifold, according to Theorem 2.2. The polyhedronKi is con-structed as follows. LetLj ⊂ ΓXi

⊂ Xi be a compact connected component, withproperty that it separates a regular neighborhoodNj of it in Xi into two components, ait happens in Remark 2.9(c). LetS1, S2 denote the boundary components ofNj . We buildKi from Xi by attaching 2-cells alongSk = p(Sk), k = 1,2, via attaching maps of degrenk , wherenk is the order of the element ofπ1(Xi) determined bySk . The polyhedronWi , proper homotopy equivalent toKi , may be seen as the polyhedron obtained fromKi

by replacing the regular neighborhoodsNj (which are mapping cylinders) with cylindeS1 × I . Thus, cl(Xi − ⋃

j Nj ) is still a subpolyhedron ofWi . Note that every componenof the subgraphΓWi

⊂ Wi of edges of order� 3 has now a trivialT ∗-bundle over it inWi .GivenK0 andK1 obtained fromX0 andX1 respectively by the method just describe

consider the adjunction complexK = Y ∪f0×{0}∪f1×{1} ∪ (K0 � K1) ⊃ Z with π1(K) ∼=H . The universal coverK of K is obtained fromp−1(Y ) and a collection of copies oK0 and K1, by gluing eachS1 × {i} ⊂ p−1(Y ) to a copy ofKi via the mapsf :S1 ×{i} → Xi ⊂ Ki , i = 0,1. For each copy ofKi in K (i = 0,1), we take one of the mapfi :S1 × {i} → Ki and homotope it (withinKi ) to an embeddinggi :S1 × {i} → Ki sothat Imgi is sufficiently small and it is contained in the “surface part”Ki − ΓKi

of thesimply connected 2-polyhedronKi , and we do this equivariantly using the action ofGi

on Ki . Since this action is properly discontinuous, the collection of all these homotgives rise to a proper homotopy equivalence betweenK and a new 2-polyhedron, stidenoted byK , obtained fromp1(Y ) and a collection of copies ofK0 andK1 by gluingeachS1 × {i} ⊂ p−1(Y ) to a copy ofKi via the new mapsgi , which are embeddingsatisfying the property that Imgi is contained in the surface part ofKi and it is not thecenterline of any Möbius band. This way, we can replace in the new polyhedronK eachcopy ofKi with the corresponding 2-polyhedronWi described above without changing timage of the mapsgi , so as to get a new 2-polyhedronW (proper homotopy equivalent tK) which is constructed fromp−1(Y ) and a collection of copies ofW0 andW1, by gluingeachS1 ×{i} ⊂ p−1(Y ) to a copy ofWi via the mapsgi . It will suffice then to show thatWthickens to an (orientable) 3-manifold. For this, note that each component of the subΓW ⊂ W of edges of order� 3 is either a circle or a line, and it has a trivialT ∗-bundle overit in W , by construction. Hence,W thickens to an orientable 3-manifold, by Remark 2Also, observe that the argument used in the proof of [10, Theorem 1.2] to show thaof theWis thickens to an orientable 3-manifold also applies toW .

In the case of an HNN-extensionH = G∗F (with monomorphismsψi :F → G,i = 0,1), let X ∈ P be a (triangulated) fake surface withπ1(X) ∼= G, andfi :S1 → X

(i = 0,1) be p.l. maps so that Imfi∗ ⊆ π1(X) corresponds to the subgroup Imψi ⊆ G.Let Y be the 2-polyhedron constructed as above and consider the adjunction spaZ =


he

--

sal

t

s

s

t

-

s,

f

lusand

Y ∪f0×{0}∪f1×{1} X with π1(Z) = H . Then, the proof of this case just mimics that of tabove.

For part (b), supposeF is infinite (i.e.,F ∼= Z). LetG0,G1 ∈ C together with monomorphismsϕi : Z → Gi (i = 0,1), and denote byH = G0 ∗Z G1 the corresponding amalgamated free product. LetX0,X1 ∈ P be (triangulated) fake surfaces withπ1(Xi) ∼= Gi ,andfi :S1 → Xi be nice p.l. embeddings inducing the monomorphismsϕi . Consider thepolyhedron obtained as the adjunction spaceZ = (S1 × I ) ∪f0×{0}∪f1×{1} (X0 � X1) withfundamental groupπ1(Z) ∼= H , and whose subgraphΓZ of edges of order� 3 consists ofthe union of(ΓX0 ∪ Imf0) and(ΓX ∪ Imf1). Let Z be the universal cover ofZ with cov-ering mapp : Z → Z. Then,p−1(Xi) consists of a disjoint union of copies of the univercoverXi of Xi , since the inclusionXi ↪→ Z induces a monomorphismGi ↪→ H betweenthe fundamental groups,i = 0,1. On the other hand,p−1(S1 × I ) consists of a disjoin

union of copies ofR × I , since the embeddingsS1 fi→ Xi ↪→ Z induce monomorphismbetween the fundamental groups. Thus,Z is constructed from a collection of copies ofX0

andX1, and a collection of copies ofR × I by gluingR × {i} to a copy ofXi via liftingsfi : R × {i} → Xi of the mapsfi , i = 0,1. As in part (a), we denote byK the adjunctionspaceK = (S1 × I ) ∪f0×{0}∪f1×{1} (K0 � K1) ⊃ Z with π1(K) ∼= π1(Z). The universalcoverK is again obtained from a collection of copies ofK0 andK1 and a collection ofcopies ofR×I , by gluing eachR×{i} to a copy ofKi via the mapsfi : R×{i} → Xi ⊂ Ki ,i = 0,1. We will manipulate the universal coverK within its proper homotopy type so ato get a new polyhedron which thickens to an (orientable) 3-manifold.

We can alter locally each of the mapsfi :S1 → Xi ⊂ Ki within its homotopy classso as to satisfy the property that Imfi is “transverse” toΓKi

, i.e., Imfi ∩ ΓKiis a finite

set of vertices, so that for every vertexx ∈ Imfi ∩ ΓKithere is a small arc inS1 about

f −1i (x) whose image inXi goes across a (small)T ∗-bundleNj in Ki over the componen

Lj ⊂ ΓKicontainingx and it intersects more than one component ofNj − Lj . Notice that

each copy ofKi in K can be replaced with a copy of the 2-polyhedronWi (described in(a)) without changing the image of the new liftingsfi .

Since the original embeddingsfi were nice (see Definition 1.1), the liftingsfi satisfythe property that for every componentC ⊂ ΓKi

with Im fi ∩ C �= ∅ and a regular neigh

borhoodN of C in Ki , Im fi always meets the same two components ofN − C. Next, wereplace each copy ofKi in K with a copy of the polyhedronWi (described above). Thuwe get a new polyhedronW (proper homotopy equivalent toK) which is constructed froma collection of copies ofW0 andW1 and a collection of copies ofR × I , by gluing eachR × {i} to a copy ofWi via the mapsfi .

Note that the subgraphΓW ⊂ W of edges of order� 3 consists of a disjoint union ocopies ofΓWi

⊂ Wi (i = 0,1) together with the image in eachWi of all the liftings fi ,where each componentC ⊂ ΓWi

is either a circle or a line always having a trivialT ∗-bundle over it inWi . As in part (a), it will suffice to prove the following

Lemma 3.1. The polyhedronW does not contain a union of a Möbius band with an annu(with one of its boundary components attached to the middle circle of the Möbius b),and henceW thickens to an orientable3-manifold.


dlet

g

s

at

which

p

Proof. Suppose there was a simple closed edge pathJ ⊂ ΓW so that there is embeddein W a T -bundle overJ consisting of a union of a Möbius band with an annulus, andΦ be a family of embeddings onW (as in Definition 2.1). Then,[ωΦ ] �= 0 in H 1(ΓW ;Z2)

by Theorem 2.2 and Remark 2.5; moreover, one has[ωΦ |J ] �= 0 in H 1(ΓW ;Z2) aswell. Observe thatJ must be contained in someWi0 and it must beJ ∩ ΓWi0

�= ∅,by Lemma 2.11. LetS1, . . . , Sk ⊂ ΓWi0

be those connected components intersectinJ

and choose edge pathsγi ⊂ Si so that if J meets a componentF of Wi0 − ΓWi0then

JF = (J ∩ cl(F )) ∪ ⋃γi⊂cl(F ) γi is a simple closed edge path contained in cl(F ) and

ωΦ |J = ∑J∩F �=∅ ωΦ |JF

∈ C1(ΓW ;Z2), whence[ωΦ |JF] �= 0 in H 1(ΓW ;Z2) for some

JF which must be the centerline of a Möbius band inW , which is not possible. For thiclaim, let fi0, f

′i0

: R × {i0} → Ki0 be liftings of fi0 and γk1 ⊂ Sk1, γk2 ⊂ Sk2 be edge

paths (as above) so thatJF ⊂ Im fi0 ∪ Im f ′i0

∪ γk1 ∪ γk2. Let Nk1, Nk2 beT -bundles over

Sk1, Sk2 ⊂ ΓWi0in Wi0. Since the original embeddingsfi :S1 → Xi (i = 0,1) were nice,

Im fi0 and Imf ′i0

intersect the same sheets ofNk1 andNk2, and hence one can check thJF cannot be the centerline of a Möbius band inW . �

In the case of an HNN-extensionH = G∗Z (with monomorphismsψi : Z → G, i =0,1), let X ∈ P be a (triangulated) fake surface withπ1(X) ∼= G, andf0, f1 :S1 → X bea nice pair of disjoint p.l. embeddings realizing the monomorphismsψ0,ψ1 (see Defini-tion 1.1). Consider the adjunction spaceZ = (S1 × I ) ∪f0×{0}∪f1×{1} X, with π1(Z) ∼= H .Then, the proof of Theorem 1.2(c) just mimics that of part (b) above.�Remark 3.2. Note that Theorem 1.2 already produces properly 3-realizable groupsare not inC. As an example, consider the groupH with presentation〈x, y, a, t;xyx−1y−1,

axa−1x−2, tyt−1y−2〉 which is an amalgamated free product of the formG0 ∗Z Gi ,whereG0,G1 ∈ C are described as follows. The groupG1 is the Baumslag–Solitar grou〈a, b;aba−1b−2〉 from Proposition 2.13(c), andG0 is an HNN-extension(Z ⊕ Z)∗Z withpresentation〈x, y, t;xyx−1y−1, tyt−1y−2〉. Fig. 2 shows a fake surface inP havingG0as fundamental group.

Fig. 2.


.2(b),p

n

hichraic

defini-houldncodeddown.

Colloq.

Math.

e Appl.

(1949)

902.

4.logy

ibrary

ndon

Finally, we get the groupH as an amalgamated free productG0∗Z G1 by identifying thegeneratorsx andb. It is easy to check that we are under the hypothesis of Theorem 1and henceH is properly 3-realizable. Moreover, one can also check that the grouH

cannot be obtained as the fundamental group of a fake surface inP , according to theinstructions given in Remark 2.17, whenceH /∈ C. On the other hand, sinceH contains theBaumslag–Solitar groupG1 as a subgroup and the latter is not a 3-manifold group, theH

itself is not a 3-manifold group, by covering space theory arguments.

The above remark provides with an example of a properly 3-realizable group for wpoints of typeIII cannot be avoided. This example partly justifies the mixture of algeband geometric hypothesis in Theorem 1.2. Given the purely geometric nature of thetion of properly 3-realizable groups (and hence of complicated algebraic nature), it snot be surprising that, for now, many properties of these groups seem to be better eas geometric statement rather than as a list of algebraic properties still to be pinned

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On properly 3-realizable groups