Groups which are not properly 3-realizable

arXiv:0709.1576v3 [math.GT] 31 May 2012 Groups which are not properly 3-realizable Louis Funar1, Francisco F. Lasheras2and Duˇsan Repovˇs3∗ 1Institut Fourier BP 74, UFR Math´ematiques, Univ.Grenoble I 38402 Saint-Martin-d’H`eres Cedex, France 2Departamento de Geometria y Topologia, Universidad de Sevilla, Apdo 1160, 41080 Sevilla, Spain 3Faculty of Mathematics and Physics, University of Ljubljana, P.O. Box 2964, Ljubljana 1001, Slovenia February 11, 2013 Abstract A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some 3-manifold. We prove that when such a group is also quasi-simply filtered then it has pro-(finitely generated free) fundamental group at infinity and semi-stable ends. Conjecturally the quasi-simply filtration assumption is superfluous. Using these restrictions we provide the first examples of finitely presented groups which are not properly 3-realizable, for instance large families of Coxeter groups. AMS Math. Subj. Classification(2000): 57 M 50, 57 M 10, 57 M 30. Keywords and phrases: Properly 3-realizable, geometric simple connectivity, quasi-simple filtered group, Coxeter group. 1 Introduction The aim of this paper is to obtain necessary conditions for a finitely presented group to be properly 3-realizable, which lead conjecturally to a complete characterization. Lasheras introduced and studied this class of groups in [8, 9, 21]. Recall that: Definition 1.1. A finitely presented group Γis said to be properly 3-realizable (abbreviated P3R from now on) if there exists a compact 2-dimensional polyhedron Xwith fundamental group Γsuch that the universal covering e Xis proper homotopy equivalent to a 3-manifold W3. ∗Emails: [email protected] (L.Funar), [email protected] (F.F.Lasheras), [email protected] (D.Repovˇs) 1 Hereafter we will consider only infinite groups Γ and thus the associated 3manifolds W3appearing in the definition above will be non-compact. Notice that, in general, the 3-manifolds W3will also have non-compact boundary. Remark 1.1. In the definition of a P3R group one does not claim that the universal covering of any compact 2-dimensional polyhedron Xwith fundamental group Γis proper homotopy equivalent to a 3-manifold. However it was proved in ([1], Proposition 1.3) that given a P3R group Gthen for any 2-dimensional compact polyhedron Xwith fundamental group G, the universal covering of the wedge XWS2is proper homotopy equivalent to a 3-manifold. Recall the following classical theorem about embeddings up to homotopy, due to Stallings. Let Pbe a finite CW-complex of dimension k, let Mbe a PL-manifold of dimension mand let f:P→Mbe a c-connected map. If m−k≥3 and if c≥2k−m+ 1 then there exist a compact subpolyhedron j:Q ֒→Mand a homotopy equivalence h:P→Qsuch that jh is homotopic to f. This was generalized to the non-compact situation in [7] by replacing the connectivity with the proper connectivity. Recall that a locally finite CW complex is said to be properly c-connected (c≥1) if its proper homotopy type can be represented by a CW complex whose c-skeleton is reduced to an end-faithful tree (see [7] for details). Thus the proper homotopy type of a locally finite CW-complex Xof dimension nis represented by a closed subpolyhedron of R2n−cif Xis properly c-connected. In particular, the universal covering e Xof an arbitrary compact 2-polyhedron X2is proper homotopy equivalent to a 4-manifold, because any 2-polyhedron embeds, up to proper homotopy, into R4. Therefore P3R groups are singled out among the set of all finitely presented groups by the fact that the universal covering e Xof some compact polyhedron Xwith given π1(X) is proper homotopy equivalent to a particular 4-manifold, namely the product of a 3-manifold with an interval. Remark 1.2. Fundamental groups of compact 3-manifolds are obviously P3R, but there also exist P3R groups which are not 3-manifold groups. For instance, any ascending HNN extension of a finitely presented group is P3R ([21], see also other explicit examples in [9]). Moreover, given any infinite finitely presented groups Gand H, their direct product G×His P3R (according to [8]). Further amalgamated products of P3R groups (and HNN extensions) over finite groups yield P3R groups (see [10]). Let us introduce very briefly, for the sake of completeness, some end invariants of non-compact spaces which will be used in the sequel. Standard references where these notions are studied in detail are [2, 23]. 2 Given the sequence of homomorphisms Ai−1←Ai, called bonding morphisms, one builds the tower of groups A0←A1← · · · . A pro-isomorphism between the towers A0←A1← · · · and B0←B1← · · · is given by two sequences of morphisms Bj2n+1 →Ai2n+1 and Ai2n→Bj2nwhere 0 = i1< j1< j2< i2< i3< j3< j4< i4<···, which commute with the respective compositions of bonding morphisms in the two towers. A pro-isomorphism class of towers of groups is called a pro-group. Definition 1.2. A pro-group is said to be pro-(finitely generated free) if it has a representative tower in which all groups involved are finitely generated free groups. It was shown in [21] that if a pro-group is pro-(finitely generated free) and has a representative tower with surjective bonding maps, then it has a representative telescopic tower (i.e., a tower in which both conditions hold simultaneously). Pro-groups arise in topology by means of towers associated to exhaustions of non-compact spaces. Definition 1.3. If Xis a polyhedron then a proper map ω: [0,∞)→Xis called a proper ray. Two proper rays define the same end if their restrictions to the subset of natural numbers are properly homotopic. An end is called semi-stable if every two proper rays defining this end are actually properly homotopic; one also says that the two rays define the same strong end. A finitely presented group has semi-stable ends if there exists a compact polyhedron Xwith the given fundamental group whose universal covering has semi-stable ends. Given now a proper base ray ωin Xand an exhaustion C1⊂C2⊂ · · · ⊂ X=∪∞ i=1Ciby compact subpolyhedra, we can associate a tower of groups π1(X, ω(0)) ←π1(X−C1, ω(1)) ← · · · where the bonding morphisms are induced, on the one hand, by the inclusions of spaces and on the other hand, by the change of base points which are slid along the ray ωrestricted to integral intervals. Definition 1.4. The (fundamental) pro-group at infinity of Xbased at ω, denoted π∞ 1(X, ω), is the pro-group associated to the tower of groups π1(X, ω(0)) ←π1(X−C1, ω(1)) ← · · · 3 Two rays defining the same strong end yield isomorphic pro-groups. In particular, if the end is semi-stable, the pro-group at infinity is an invariant of the end, and called the (fundamental) pro-group of the end. The end is called simply connected at infinity (or π1-trivial) if the associated pro-group is pro-isomorphic to a tower of trivial groups. The (fundamental) pro-group at infinity of a finitely presented group is the pro-group at infinity of the universal covering of a compact polyhedron with the given fundamental group. This depends of course, on the base ray (and thus only on the end if it is semi-stable), but not on the the particular compact polyhedron we chose. Remark 1.3. There are alternative equivalent definitions of the semi-stability, in particular the one used in Siebenmann’s thesis: an end is called semistable if its fundamental pro-group has a representative tower with surjective bonding morphisms (see also [20]). For the sake of completeness we recall that an end is called stable if there exist some representative tower in which all bonding morphisms are isomorphisms. Examples of Davis (see [12]) show that the ends of universal coverings of finite complexes might be not stable, although it is not known whether they should be always semistable. Notice that sometimes in the literature one uses the terms π1-stable, π1-semi-stable etc. for the corresponding notions introduced above. As already observed above, we can infer from [21] that a semi-stable end having pro-(finitely generated free) fundamental pro-group at infinity admits a representative telescopic tower for that fundamental pro-group at infinity. If a group has semi-stable ends then the universal covering of any compact polyhedron with the given fundamental group has semi-stable ends. Although there exist spaces whose ends are not semi-stable, there are still no known examples of finitely presented groups (i.e. universal coverings of compact polyhedra) without semi-stable ends (see also [19, 24]). The main source of examples of P3R groups is the paper of Lasheras ([21]) where it is proved that a one-ended finitely presented group which is semistable and whose fundamental pro-group at infinity is pro-(finitely generated free) is P3R. In particular, any one-ended finitely presented group Γ which is simply connected at infinity (and hence automatically semi-stable at infinity) is P3R. We expect the following to be a complete characterization of this class of groups: Conjecture 1 (3-dimensional homotopy covering conjecture).A finitely presented group is P3R iff each one of its ends is semi-stable and has pro- (finitely generated free) fundamental pro-group. 4 Remark 1.4. In [22] the authors proved the sufficient part of the conjecture, namley that a finitely presented group whose ends are semi-stable and have pro-(finitely generated free) fundamental pro-groups is P3R. In this paper we give evidence in the favor of this conjecture, by proving it in the case when the group under consideration satisfies an additional hypothesis related to the geometric simple connectivity. In order to explain this we have to introduce, following Brick - Mihalik ([5]) and Stallings ([33]), the following tameness condition for groups and spaces. Definition 1.5. A space Xis called quasi-simply filtered (abbreviated qsf) if for any compact C⊂Xthere exists a connected and simply connected compact Ktogether with a map f:K→Xsuch that f(K)⊃Cand f|f−1(C):f−1(C)→Cis a homeomorphism. A finitely presented group Γis called qsf if there exists a (equivalently, for every) compact polyhedron Pwith fundamental group Γsuch that the universal covering ˜ Pis qsf. The condition qsf is a rather mild assumption on finitely presented groups. There are still no known examples of groups which do not have the qsf property and most classes of known groups, as hyperbolic, semi-hyperbolic, automatic, tame combable etc., are qsf (see [16, 25]). We can now state our main result: Theorem 1.1. If a finitely presented group is P3R and qsf then all of its ends are semi-stable and have pro-(finitely generated free) fundamental group at infinity. Remark 1.5. We do not know whether all finitely presented groups which have semi-stable ends and pro-(finitely generated free) fundamental groups at each end are actually qsf. Notice that by a theorem of Wright (see [18], Theorem 16.5.6), one-ended groups with stable end having an element of infinite order must be either simply connected at infinity or pro-Zat infinity. Thus they are P3R by the result of Lasheras cited above. Remark 1.6. 1. The homotopy covering conjecture implies the well-known covering conjecture in dimension 3 which states that the universal covering of an irreducible closed 3-manifold M3with infinite fundamental group is simply connected at infinity. In fact, the universal covering f M is an open contractible 3-manifold (thus one-ended) which is semi-stable and has pro-(finitely generated free) fundamental pro-group at infinity. This implies that there exists an exhaustion by compact submanifolds Ci 5 such that π1(f M−Ci)are finitely generated and free. Tucker’s criterion from [34] implies that the manifold f Mis a missing boundary manifold and thus it is homeomorphic to int(N3), for a suitable compact 3-manifold N3with boundary. By the contractibility of the universal covering, each component of ∂N3is homeomorphic to a 2-sphere and this implies that int(N3)(and hence f M) is simply connected at infinity. 2. Conversely, it is obvious that the universal covering conjecture implies the homotopy covering conjecture for closed 3-manifold groups because open 3-manifolds which are simply connected at infinity are semi-stable and have pro-(finitely generated free) pro-group at infinity (in fact a trivial pro-group!). 3. Notice that the universal covering e Xof a compact 2-polyhedron Xcan never be proper homotopy equivalent to an open (simply connected) 3-manifold M3. In fact, the Poincar´e duality would give us that the third cohomology group with compact support H3 c(e X)is isomorphic to H3 c(M) = H0(M) = Z, which is impossible, as dim( e X) = 2. Remark 1.7. Let us consider the universal covering f M3, of a 3-manifold M3with boundary. If the boundary is a union of spheres then f Mis obtained from the universal covering of a closed 3-manifold (obtained by capping off boundary spheres by balls) by deleting a collection of disjoint balls. Assume that the boundary is non-trivial i.e. not a union of 2-spheres. Then M3is Haken and thus, by Thurston’s theorem, it is a geometric 3-manifold. Let us moreover assume that M3is atoroidal, i.e. there are no Z⊕Zembedded in π1(M)other than peripheral subgroups coming from the boundary torus components. Then Thurston’s geometrization theorem tells us that M3is hyperbolic. Therefore the universal covering f Mis obtained geometrically by deleting a collection of horoballs from the hyperbolic 3-space. In particular the pro-group at infinity of f Mis pro-(finitely generated free) and its ends are semi-stable. Thus the conjecture holds for fundamental groups of atoroidal 3manifolds with non-trivial boundary. A similar but more involved discussion shows that it also holds for all 3-manifolds with non-trivial boundary (since they are geometric). Remark 1.8. The homotopy covering conjecture implies that all 1-relator groups are P3R. This is already known for 1-relator finitely ended groups (see [11]). In fact, 1-relator groups are semi-stable at infinity (see [26]) and it was proved in ([11], Proposition 2.7) that their pro-groups at infinity are pro-(finitely generated free). Notice that 1-relator groups are also qsf (see [25]). Recently, Lasheras and Roy ([22]) have extended the results of [11] to a class of groups which contains all 1-relator groups. 6 It is presently unknown (but quite plausible) that any finitely presented group which is qsf, semi-stable and has pro-(finitely generated free) pro-groups at infinity is P3R. As an application of Theorem 1.1 we will obtain explicit examples of groups which are not P3R, as follows. Theorem 1.2. Let Γbe one of the following: 1. the fundamental group of a finite non-positively curved complex which is a homology n-manifold (n≥3), but not a topological manifold. We further assume that the link of every vertex is a topological manifold. 2. the right angled Coxeter group associated to a flag complex Lwhose geometric realization is a closed combinatorial n-manifold (n≥3) and π1(L)is not a free group. Then Γis not P3R. In particular many Coxeter groups are not P3R. Similar examples were announced by Cardenas. Acknowledgements. The authors are indebted to Ross Geoghegan and Valentin Poenaru for useful discussions and comments and to an anonymous referee for simplifying the proofs. The first author was supported by the Proteus program (2005-2006), no 08677YJ and the ANR Repsurf: ANR06-BLAN-0311. The second author was supported by the project MTM 2007-65726 and the third author was supported by the Proteus program (2005-2006), no 08677YJ. 2 Proofs 2.1 Tameness criterion for non-compact 3-manifolds Recall that a polyhedron Pis called weakly geometrically simply connected (wgsc) if it admits an exhaustion by compact connected subpolyhedra P1⊂ P2⊂ · · · such that π1(Pn) = 0, for all n. The wgsc property for polyhedra is the piecewise-linear analogue of the geometric simple connectivity of open manifolds, namely the existence of a proper handlebody decomposition without index one handles. It is proved in [14, 17] that an open 3-manifold proper homotopy equivalent to a weakly geometrically simply connected polyhedron is simply connected 7 at infinity. In this section we will extend this result to non-compact 3manifolds. In the realm of manifolds with boundary the relevant tameness condition that will replace the simple connectivity at infinity is the following: Definition 2.1. A manifold Wis called a missing boundary manifold (also called almost compact) if there exists a compact manifold with boundary M and a closed subset A⊂∂M of the boundary (not necessarily a subcomplex) such that Wis homeomorphic to M−A. Interesting examples of manifolds which are not missing boundary manifolds can be found in [32, 35]. We first introduce a family of 3-manifolds which is, in some sense, the smallest one containing the missing boundary 3-manifolds and allowing manifolds to have infinitely many boundary components. These manifolds will be the proper analog of the open manifolds which are simply connected at infinity in the non-compact case. Before we proceed, let us recall that a compact 0-dimensional subset Cis said to be tame (or tamely embedded) in Rnif there exists a homeomorphism of Rnsending Cinto a subset of R×{0} ⊂ Rn. It is well-known that perfect (i.e. without isolated points) compact 0-dimensional separable topological spaces are homeomorphic to the Cantor space. Hence the tameness condition above is mostly relevant for Cantor subsets of Rn. Notice that there exist wild Cantor sets in any Rn, with n≥3, while Cantor sets in R2are tame, by a classical theorem of Bing ([3]). Definition 2.2. Astandard model is a 3-manifold with boundary Vconstructed as follows. Let {Bi}i∈Ibe a collection of pairwise disjoint 3-balls in the interior int(B)of the 3-ball whose radii go to 0and whose limit set Lis a tame 0-dimensional subset disjoint from ∂B. Let X⊃Lbe a tame 0-dimensional subset of int(B)which is disjoint from int(Bi), for all i∈I, and T⊂∂B ∪ ∪i∈I∂Bi. Then we put V=B−(X∪T∪i∈Iint(Bi)). Manifolds of this form, where T∩∂B =∅, were called ragged cells by Brin and Thickstun in ([6], pp.9-10). In order to simplify some arguments we will use in the sequel the fact that there are no fake homotopy disks in dimension 3, as the Poincar´e conjecture has been settled by Perelman in [29, 30] (see a detailed and self-contained exposition of Perelman’s proof in [27]). Remark 2.1. 1. Open simply connected 3-manifolds Vwhich are simply connected at infinity can be described as the manifolds of the form 8 S3−X, where Xis a tame 0-dimensional compact subset of B3. Alternatively, Vcan be written as an ascending union of compact simply connected submanifolds, i.e. disks - with - holes, by the Poincar´e Conjecture (see [17, 36]). 2. A simply connected missing boundary 3-manifold Vis homeomorphic to M−T, where Mis a simply connected compact 3-manifold and Tis a closed subset of ∂M (see e.g. [36]). By the Poincar´e Conjecture there is a finite set of pairwise disjoint balls Bi,i∈Isuch that V=B−(∪i∈Iint(Bi)∪T)and Tis a closed subset of ∂B ∪ ∪i∈I∂Bi. Thus standard models Vwith finite Icorrespond precisely to simply connected missing boundary manifolds. Actually, any standard model can be obtained by making connected sums of (possibly infinitely many) simply connected missing boundary manifolds. Remark 2.2. 1. Another characterization of standard models was given by Brin and Thickstun (see[6], Full End Description Theorem (b), p.10), as follows. Modulo the Poincar´e Conjecture, the set of simply connected end 1-movable 3-manifolds coincides with that of standard models. In particular, 3-manifolds with semi-stable ends are homeomorphic to standard models. 2. Cardenas announced as an application of the Brin-Thickstun structure theorem ([6]), that 1-ended groups which are P3R and semi-stable have actually pro-(finitely generated free) pro-group at infinity. Remark 2.3. The boundary of a standard model consists of 2-spheres and open planar surfaces. Each end has pro-(finitely generated free) fundamental group at infinity. In fact, the complement of an unknotted ball in a 1-ended standard model is homotopy equivalent to the complement of a finite graph, namely a holed handlebody. Thus its fundamental group is a finitely generated free group. Moreover, each end of a standard model is semi-stable. The homotopy covering conjecture admits an (`a priori stronger) restatement as follows: Conjecture 2. Given a finitely presented P3R group, the universal covering of some compact 2-dimensional polyhedron with this fundamental group is proper homotopy equivalent to a standard model. Remark 2.4. The equivalence between the two conjectures stated in this paper is a consequence of the Brin-Thickstun structure theorem ([6]). Details are left to the reader. 9 [23] S. Mardeˇsi´c and J.Segal, Shape theory. The inverse system approach, NorthHolland Mathematical Library 26, North-Holland Publishing Co., AmsterdamNew York, 1982. [24] M.Mihalik, Semistability at the end of a group extension, Trans. Amer. Math. 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