In this paper, we show that the class of all properly 3-realizable groups is
closed under amalgamated free products (and HNN-extensions) over finite groups.
We recall that $G$ is said to be properly 3-realizable if there exists a
compact 2-polyhedron $K$ with $\pi_1(K) \cong G$ and whose universal cover
$\tilde{K}$ has the proper homotopy type of a 3-manifold (with boundary).Comment: J. Pure Appl. Alg., to appea

Cardenas, M.

Lasheras, F. F.

Quintero, A.

Repovš, D.

English

arXiv

M. Cardenas

F.F. Lasheras

A. Quintero

D. Repovš

Crossref

Amalgamated products and properly 3-realizable groups

In this paper, we show that the class of all properly 3-realizable groups is closed under amalgamated free products (and HNN-extensions) over finite groups. We recall that G is said to be properly 3-realizable if there exists a compact 2-polyhedron K with π1(K)≅G and whose universal cover View the MathML source has the proper homotopy type of a 3-manifold (with boundary).Ministerio de Educación y Cienci

Cárdenas Escudero, Manuel Enrique

Fernández Lasheras, Francisco Jesús

Quintero Toscano, Antonio Rafael

Repovš, Dušan

idUS. Depósito de Investigación Universidad de Sevilla

arXiv:math/0609289v1  [math.GT]  11 Sep 2006AMALGAMATED PRODUCTS AND PROPERLY 3-REALIZABLEGROUPSM. CARDENAS, F. F. LASHERAS, A. QUINTERO AND D. REPOVSˇAbstract. In this paper, we show that the class of all properly 3-realizablegroups is closed under amalgamated free products (and HNN-extensions) overfinite groups. We recall that G is said to be properly 3-realizable if there existsa compact 2-polyhedron K with pi1(K) ∼= G and whose universal cover K˜ hasthe proper homotopy type of a 3-manifold (with boundary).1. IntroductionWe are concerned about the behavior of the property of being properly 3-realizable (for finitely presented groups) with respect to the basic constructionsin Combinatorial Group Theory; namely, amalgamated free products and HNN-extensions. Recall that a finitely presented group G is said to be properly 3-realizable if there exists a compact 2-polyhedron K with π1(K) ∼= G and whoseuniversal cover K˜ has the proper homotopy type of a 3-manifold. It is worth men-tioning that the property of being properly 3-realizable has implications in thetheory of cohomology of groups, in the sense that if G is properly 3-realizable thenfor some (equivalently any) compact 2-polyhedron K with π1(K) ∼= G we haveH2c (K˜;Z) free abelian (by manifold duality arguments), and hence so is H2(G;ZG)(see [9]). It is a long standing conjecture that H2(G;ZG) be free abelian for everyfinitely presented group G. In [1] it was shown that the property of being properly3-realizable is preserved under amalgamated free products (HNN-extensions) overfinite cyclic groups. See also [3, 4, 7] to learn more about properly 3-realizablegroups and related topics. In this paper, we continue in the line of [1]. Our mainresult is :Theorem 1.1. The class of all properly 3-realizable groups is closed under amal-gamated free products (and HNN-extensions) over finite groups.This generalizes to show that the fundamental group of a finite graph of groupswith properly 3-realizable vertex groups and finite edge groups is properly 3-realizable,since such a group can be expressed as a combination of amalgamated free productsand HNN-extensions of the vertex groups over the edge groups.Recall that, given a finitely presented group G and a compact 2-polyhedron Kwith π1(K) ∼= G and K˜ as universal cover, the number of ends of G is the number ofends of K˜ which equals 0, 1, 2 or∞ [6] (see also [8, 13]). The 0-ended groups are thefinite groups and the 2-ended groups are those having an infinite cyclic subgroup offinite index, and they are all known to be properly 3-realizable (see [1]). Note thatStallings’ Structure Theorem [12] characterizes those groups G with more than oneend as those which split as an amalgamated free product (or an HNN-extension)2000 Mathematics Subject Classification. Primary 57M07; Secondary 57M10, 57M20.12 M. CARDENAS, F. F. LASHERAS, A. QUINTERO AND D. REPOVSˇover a finite group (see also [13, 8]). In addition, Dunwoody [5] showed that thisprocess of further splitting G must terminate after finitely many steps.Corollary 1.2. In order to show whether or not all finitely presented groups areproperly 3-realizable it suffices to look among those groups which are 1-ended.2. Main resultThe purpose of this section is to prove Theorem 1.1. We will make use of thefollowing result :Proposition 2.1 ([1], Prop. 3.1). Let M be a manifold of the same proper homo-topy type of a locally compact polyhedron K with dim(K) < dim(M). Then, anyFreudenthal end ǫ ∈ F(M) can be represented by a sequence of points in ∂M .Proof of Theorem 1.1. Let G0, G1 be properly 3-realizable groups and F be a finitegroup with presentation 〈a1, . . . , aN ; r1, . . . , rM 〉. Consider monomorphisms ϕi :F −→ Gi(i = 0, 1), and denote by G0 ∗F G1 = 〈G0, G1;ϕ0(ai) = ϕ1(ai), 1 ≤ i ≤ N〉the corresponding amalgamated free product. Let X0, X1 be compact 2-polyhedrawith π1(Xi) ∼= Gi and such that their universal covers have the proper homotopytype of 3-manifolds M0,M1 respectively. Let L = ∨Ni=1S1 and fi : L −→ Xi(i = 0, 1) be cellular maps such that Im fi∗ ⊆ π1(Xi) corresponds to the subgroupIm ϕi ⊆ Gi. We take the standard 2-dimensional CW-complex Y′ associated to theabove presentation of F , i.e., Y ′ has one 1-cell ei for each generator ai (1 ≤ i ≤ N),all of them sharing the only vertex in Y ′, and one 2-cell dj for each relation rj(1 ≤ j ≤ M) attached via a map S1 −→ ∨Ni=1ei which “‘spells” the relation rj .Consider the adjunction spaces Y = (∨Ni=1ei)× I ∪(∨Ni=1ei)×{ 12} Y′ (homotopy equi-valent to Y ′) and Z = Y ∪f0×{0}∪f1×{1} (X0⊔X1). By van Kampen’s Theorem, Z isa compact 2-polyhedron with π1(Z) ∼= G0 ∗F G1. Let Z˜ be the universal cover of Zwith covering map p : Z˜ −→ Z. Then, p−1(Xi) consists of a disjoint union of copiesof the universal cover X˜i of Xi, since the inclusion Xi →֒ Z induces a monomor-phism Gi →֒ G0 ∗F G1 between the fundamental groups, i = 0, 1 (see [10]). On theother hand, let Γ be a connected component of p−1(∨Ni=1ei) ⊂ p−1(Y ′) and Y˜ ′ bethe connected component of p−1(Y ′) containing Γ. Observe that Y˜ ′ is a copy ofthe universal cover of Y ′ (which is compact), as the inclusion Y ′ →֒ Z induces amonomorphism F →֒ G0 ∗F G1. Then, it is easy to see that p−1(Y ) consists of adisjoint union of copies of the compact CW-complex K = (Γ× I)∪Γ×{ 12} Y˜′. Thus,Z˜ comes together with the following data (see [13]) :(a) The disjoint unions⊔p∈NX˜0,p and⊔r∈NX˜1,r of copies of X˜0 and X˜1 respectively;(b) a disjoint union⊔p,q∈NKp,q of copies of K; and(c) a bijective function ϕ : N × N −→ N × N, (p, q) 7→ (r, s) (given by the groupaction of G0 ∗F G1 on Z˜), so that for each p, q ∈ N, Γ× {0} ⊂ Kp,q is being gluedto X˜0,p via a lift f˜0p,q : Γ×{0} −→ X˜0,p of the map f0, and Γ×{1} ⊂ Kp,q is beingglued to X˜1,r via a lift f˜1r,s : Γ× {1} −→ X˜1,r of the map f1.Next, for each copy of X˜i, i = 0, 1, in Z˜ (written as X˜0,p or X˜1,r), we take oneof the maps f˜ iλ,µ : Γ × {i} −→ X˜i and observe that this map is nullhomotopic soAMALGAMATED PRODUCTS AND PROPERLY 3-REALIZABLE GROUPS 3we can replace it (up to homotopy) with a constant map giλ,µ : Γ×{i} −→ X˜i withIm giλ,µ ⊂ Im f˜iλ,µ, and we do this equivariantly using the group action ofGi on X˜i.Since this action is properly discontinuous, the collection of all these homotopiesgives rise to a proper homotopy equivalence between Z˜ and a new 2-dimensionalCW-complexW obtained from a collection of copies of K and a collection of copiesof X˜0 and X˜1 by gluing each copy of Γ × {i} to the corresponding copy of X˜i viathe bijection ϕ and the new maps giλ,µ, i = 0, 1.We will now manipulate the CW-complex K as follows. First, let K ′ be theCW-complex obtained from K by shrinking to a point v × {i} each copy T × {i}(i ∈ I) of a maximal tree T ⊂ Y˜ ′ ⊂ K. Next, we take K ′′ to be the CW-complexobtained from K ′ by identifying the subcomplexes Γ × {i}/T × {i}, i = 0, 1, toa (different) point which we will denote by [v × {0}] and [v × {1}]. Note thatK ′′ has a copy of Y˜ ′/T as a subcomplex. Since Y˜ ′/T is compact and simply con-nected, it follows from ([14], Prop. 3.3) that Y˜ ′/T is homotopy equivalent to a finitebouquet of 2-spheres ∨α∈AS2 (which we may regard as a connected 2-dimensionalCW-complex with no 1-cells). Moreover, we may assume that this homotopy equi-valence is given by a cellular map Y˜ ′/T −→ ∨α∈AS2 so that the 1-skeleton Γ/T ofY˜ ′/T is mapped to the wedge point. Finally, taking into account this homotopyequivalence, it is not difficult to see that K ′′ is homotopy equivalent to the CW-complex K̂ obtained from the disjoint union of a finite bouquet ∨α∈A∪BS2 (whereCard(B) = 2 rank(π1(Γ)) and the unit interval I by identifying12 ∈ I with thewedge point, so that I ⊂ K̂ would correspond to the subcomplex v × I ⊂ K ′ and0, 1 ∈ I would correspond to [v×{0}], [v×{1}] ∈ K ′′. Notice that K̂ thickens to a3-manifold P ց K̂ containing 3-dimensional 1-handles H and H ′ (with a free endface each of them) corresponding to the edges [0, 12 ], [12 , 1] ⊂ I ⊂ K̂ respectively.According to the above, one can see that the CW-complexW (proper homotopyequivalent to Z˜) is in turn proper homotopy equivalent to the quotient space ob-tained from the following data :(a) A disjoint union⊔p∈NX˜0,p of copies of X˜0 together with a locally finite sequenceof points {xpq}q∈N ⊂ X˜0,p, for each p ∈ N, corresponding to the images of the con-stant maps g0p,q : Γ× {0} −→ X˜0,p considered above in the construction of W ;(b) a disjoint union⊔r∈NX˜1,r of copies of X˜1 together with a locally finite sequence ofpoints {yrs}s∈N ⊂ X˜1,r, for each r ∈ N, corresponding to the images of the constantmaps g1r,s : Γ× {1} −→ X˜1,r from the construction of W ;(c) a disjoint union⊔p,q∈NK̂p,q of copies of K̂; and(d) the bijective function ϕ : N×N −→ N×N, (p, q) 7→ (r, s), so that 0 ∈ I ⊂ K̂p,q isbeing identified with xpq ∈ X˜0,p and 1 ∈ I ⊂ K̂p,q is being identified with yrs ∈ X˜1,r((r, s) = ϕ(p, q)), for each p, q ∈ N.We now follow an argument similar to the proof of ([1], Lemma 3.2). Fix properhomotopy equivalences h : X˜0 −→ M and h′ : X˜1 −→ N , where we now denoteM0 by M and M1 by N . Given the above data, we set A = N × N and consider4 M. CARDENAS, F. F. LASHERAS, A. QUINTERO AND D. REPOVSˇmaps i : A −→⊔p∈NX˜0,p , i′ : A −→⊔r∈NX˜1,r given by i(p, q) = xpq and i′(p, q) = yrs ,where (r, s) = ϕ(p, q). It is easy to check that i and i′ are proper cofibrations, asthe corresponding sequences of points are locally finite. Next, we take exhaustivesequences {Apm}m∈N and {Brn}n∈N of copies Mp and Nr of the 3-manifolds Mand N respectively by compact submanifolds, and define proper cofibrations j :A −→⊔p∈NMp , j′ : A −→⊔r∈NNr as follows. Given (p, q) ∈ A and the properhomotopy equivalences hp = h : X˜0,p −→ Mp , h′r = h′ : X˜1,r −→ Nr (with(r, s) = ϕ(p, q)), we take m(q), n(s) ∈ N to be the least natural numbers such thathp ◦ i(p, q) /∈ Apm(q) ⊂ Mp and h′r ◦ i′(p, q) /∈ Brn(s) ⊂ Nr. Then, using Proposition2.1, we define j(p, q) and j′(p, q) to be points j(p, q) = ap,q ∈ ∂Mp − Apm(q) andj′(p, q) = br,s ∈ ∂Nr − Brn(s) so that (i) j, j′ are one-to-one maps (note that h, h′need not be one-to-one); and (ii) ap,q and hp ◦ i(p, q) (resp. br,s and h′r ◦ i′(p, q))are in the same path component of Mp − Apm(q) (resp. Nr − Brn(s)). Notice that jand j′ are proper maps by construction. Consider now mapsG :⊔p∈NX˜0,p× {0} ∪ (i(A)× I) −→ ⊔p∈NMpH :(⊔r∈NX˜1,r)× {0} ∪ (i′(A)× I) −→⊔r∈NNrwith G|X˜0,p×{0} = hp = h and H |X˜1,r×{0} = h′r = h′ (p, r ∈ N), and so thatαp,q = G|i(p,q)×I (resp. βr,s = H |i′(p,q)×I) is a path in Mp−Apm(q) from hp ◦ i(p, q)to ap,q (resp. a path in Nr − Brn(s) from h′r ◦ i′(p, q) to br,s). Observe that G andH are proper maps, since h, h′, j and j′ are proper. By the Homotopy ExtensionProperty, the maps G,H extend to proper mapsĜ :⊔p∈NX˜0,p× I −→ ⊔p∈NMp , Ĥ :(⊔r∈NX˜1,r)× I −→⊔r∈NNrwhich yield commutative diagramsAi{{vvvvvvvvvj##GGGGGGGGG⊔p∈N X˜0,phˆ//⊔p∈N MpAi′zzvvvvvvvvvj′##GGGGGGGGG⊔r∈N X˜1,rhˆ′//⊔r∈N Nrwhere hˆ = Ĝ|(⊔p∈N X˜0,p)×{1}and hˆ′ = Ĥ |(⊔r∈N X˜1,r)×{1}are proper homotopy equiv-alences. Moreover, hˆ and hˆ′ are proper homotopy equivalences under A, by ([2],Prop. 4.16) (compare with [11], Chap. 6, § 5). Hence, they induce a proper ho-motopy equivalence between the quotient space described above (proper homotopyequivalent to W ) and the following 3-manifold obtained as the quotient space givenby the data :(a) The disjoint unions⊔p∈NMp and⊔r∈NNr of copies of the 3-manifolds M and Nrespectively;AMALGAMATED PRODUCTS AND PROPERLY 3-REALIZABLE GROUPS 5(b) a disjoint union⊔p,q∈NPp,q of copies of the compact 3-manifold P ց K̂; and(c) the bijective function ϕ : N×N −→ N×N, (p, q) 7→ (r, s), so that for each p, q ∈N, the free ends of the corresponding 3-dimensional 1-handlesHp,q, H′p,q ⊂ Pp,q con-sidered above are being identified homeomorphically with small disks Dp,q ⊂ ∂Mpand D′r,s ⊂ ∂Nr about the points ap,q and br,s respectively.In the case of an HNN-extension G∗F = 〈G, t; t−1ψ0(ai)t = ψ1(ai), 1 ≤ i ≤ N〉(with monomorphisms ψi : F −→ G, i = 0, 1), let X be a compact 2-polyhedronwith π1(X) ∼= G and whose universal cover has the proper homotopy type ofa 3-manifold, and let fi : ∨Ni=1S1 −→ X (i = 0, 1) be cellular maps so thatIm fi∗ ⊆ π1(X) corresponds to the subgroup Im ψi ⊆ G. Let Y be the 2-dimensional CW-complex constructed as above and consider the adjunction spaceZ = Y ∪f0×{0}∪f1×{1} X , with π1(Z)∼= G∗F . Then, the proof goes just as the onegiven above for the amalgamated free product. AcknowledgementsThe first three authors were supported by the project MTM 2004-01865. Thisresearch was also supported by Slovenian-Spanish research grant BI-ES/04-05-014.References[1] R. Ayala, M. Ca´rdenas, F.F. Lasheras, A. Quintero. Properly 3-realizable groups. Proc. Amer.Math. Soc. 133(2005), 1527-1535.[2] H-J. Baues, A. Quintero. Infinite Homotopy Theory. K-monographs in Mathematics, KluverAcademic Publishers, 2001.[3] M. Ca´rdenas, F.F. Lasheras. Properly 3-realizable groups : a survey. Proceedings of theConference on Geometric Group Theory and Geometric Methods in Group Theory (Seville2003), Contemp. Math., to appear.[4] M. Ca´rdenas, F.F. Lasheras, R. Roy. Direct products and properly 3-realizable groups. Bull.Australian Math. Soc. 70(2004), 199-206.[5] M. J. Dunwoody. The accessibility of finitely presented groups. Invent. Math. 81(1985), 449-457.[6] H. Freudenthal. U¨ber die Enden topologischer Raume und Gruppen. Math. Z. 33(1931), 692-713.[7] D. J. Garity, F. F. Lasheras, D. Repovsˇ. Topology of 2-dimensional complexes. Preprint.[8] R. Geoghegan. Topological Methods in Group Theory. Book in preparation.[9] R. Geoghegan, M. Mihalik. Free abelian cohomology of groups and ends of universal covers.J. Pure and Appl. Algebra, 36(1985), 123-137.[10] R. C. Lyndon, P. E. Schupp. Combinatorial Group Theory. Springer-Verlag, 1977.[11] P.J. May. A Concise Course in Algebraic Topology. Chicago Lectures in Mathematics. Uni-versity of Chicago Press, 1999.[12] J. Stallings. Group theory and three dimensional manifolds. Yale Math. Monographs, no. 4,Yale Univ. Press, New Haven, Conn. 1971.[13] P. Scott, C.T.C. Wall. Topological methods in group theory. Homological Group Theory,London Math. Soc. Lecture Notes, Cambridge Univ. Press, Cambridge (1979), 137-204.[14] C.T.C. Wall. Finiteness conditions for CW-complexes. Ann. of Math. 81(1965), 56-69.Departamento de Geometr´ıa y Topolog´ıa, Universidad de Sevilla, Apdo 1160, 41080-Sevilla, SpainE-mail address: mcard@us.es, lasheras@us.es, quintero@us.esInstitute of Mathematics, Physics and Mechanics, University of Ljubljana, P.O. Box2964, Ljubljana 1001, SloveniaE-mail address: dusan.repovs@fmf.uni-lj.si

Journal of Pure and Applied Algebra

https://idus.us.es/xmlui/bitstream/11441/43364/1/Amalgamated%20products%20and%20properly%203-realizable%20groups.pdf

Amalgamated products and properly 3-realizable groups

Abstract

Similar works

Full text

Available Versions

Crossref

idUS. Depósito de Investigación Universidad de Sevilla