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Factorable Groups and their Homology [Elektronische Ressource] / Balázs Visy. Mathematisch-Naturwissenschaftliche Fakultät

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55 pages
Factorable Groups and their HomologyDissertationzurErlangung des Doktorgrades (Dr. rer. nat.)derMathematisch-Naturwissenschaftlichen Fakult¨atderRheinischen Friedrich-Wilhelms-Universit¨at Bonnvorgelegt vonBal´azs VisyausHatvan, UngarnBonn, November 2010Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at derRheinischen Friedrich-Wilhelms-Universit¨at Bonn1. Gutachter: Prof. Dr. Carl-Friedrich B¨odigheimer2. Gutachter: Prof. Dr. Catharina StroppelTag der Promotion: 15. April 2011Erscheinungsjahr: 2011ContentsIntroduction 21 Normed Groups 71.1 Basic Notions and Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Norms from Word Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Direct and Semi-direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Norm Filtration 122.1 of the Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Filtration of the Bar Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 of the Bar Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Non-trivial Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Filtration of the Classifying Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6 The Rips Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Factorable Groups and their Homology
Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-NaturwissenschaftlichenFakultat der RheinischenFriedrich-Wilhelms-UniversitatBonn
vorgelegt von BalazsVisy aus Hatvan, Ungarn
Bonn, November 2010
AngefertigtmitGenehmigungderMathematisch-NaturwissenschaftlichenFakultatder RheinischenFriedrich-Wilhelms-UniversitatBonn
1.Gutachter:Prof.Dr.Carl-FriedrichBodigheimer 2. Gutachter: Prof. Dr. Catharina Stroppel Tag der Promotion: 15. April 2011 Erscheinungsjahr: 2011
Contents
Introduction 2 1 Normed Groups 7 1.1 Basic Notions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Norms from Word Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Direct and Semi-direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Norm Filtration 12 2.1 Filtration of the Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Filtration of the Bar Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Filtration of the Bar Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Non-trivial Coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 Filtration of the Classifying Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6 The Rips Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Factorable Groups 20 3.1 Factoring Group Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Direct, Semi-direct and Free Products . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4 The Homology of Factorable Groups 30 4.1 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 The partition-type ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.4 Non-trivial Coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5 Symmetric Groups 43 5.1 The Word Length as a Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2 Factorability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.3 The Alternating Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.4 Homology of Symmetric Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Bibliography 52
1
Introduction
Group homology plays an important role in many areas of pure mathematics. The homology of a groupGwith coecients in aG-moduleMis dened to be the homology of the complexCZGM, whereCis a projective resolution ofZoverZG is well known, that the homology we obtain is. It independent of the choice of the resolution. The (normalized) bar resolutionE(G) provides a free resolution, but the chain groups ofE(G) are usually too big for computations. In this work we introduce a family of normed groups, called factorable groups. Such a groupG admits for each of its elementga factorizationg=gg0into two factorsg, called remainder and g0, called prex, such that the norms ofg andg0add up to the norm ofg, and the norm ofg0is minimal. Furthermore, there are axioms for the remainder and prex of a product. The factorization leads to a normal form, to geodesics on the Cayley graph, to a combing ofG, to an automatic structure onG, and many more things. the normal form one can obtain From a collapsing scheme on the normalized bar resolution ofGand therefore there are connections to discrete Morse theory and to re-writing systems (see [7], [8]). These topics are not subject of this thesis, but will be developed in the works of A. Hess, M. Rodenhausen, V. Ozornova, L. Stein and others. We use the structure of a factorable groupGto nd a free resolution ofZoverZGwhich has fewer generators than the bar resolution, making computations easier. Endowing the groupG with a normN:GN0gives a ltration of the bar resolutionE(G) and of the bar complex B(G) =E(G)ZGZG, which we call thenorm ltration. More precisely, ifX= (gq|. . .|g1) is aq-dimensional generator ofB(G) in the inhomogeneous bar notation, then we set its norm or ltration degree equal toN(gq) +.....+N(g1). We study the spectral sequence associated to the norm ltration. The ltration quotients of degree hnorm ltration of the bar complex are denoted byof the N(G)[h general, if the coe-]. More cient moduleMhas an appropriate ltration itself, we obtain a spectral sequence converging to the homology ofGwith coecients inM. The complexN(G)[hthe thesis, Theorem 4.1.1., is that for a] is bounded and the main result of factorable groupGthe homology ofN(G)[h] is concentrated in the top degree=h. Consequently, the spectral sequence associated to the norm ltration collapses after theE2-term and the homology of the groupGcan be computed as the homology of theE1-term . . .→Hh(N(G)[h])→Hh1(N(G)[h1])→. . .
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Moreover, we can identify generators of the homology groupsHh(N(G)[h]) by introducing a new chain complexR(G) and mapsh:Rh(G)Hh(N(G)[h]) for eachh0. TheZ-module Rq(G) is freely generated byq-tuples of elements ofGwith minimal non-zero norm and satisfying a monotonicity property. In contrast, the bar complexBq(G) is generated by allq-tuples of ele-ments ofG. We show that the mapshare injective homomorphisms ofZ-modules and thath is an isomorphism if the groupGis the symmetric group pfor anyp1. In fact, using similar arguments, Rui Wang proves in [14] thathis an isomorphism for any groupG, provided thatG has nitely many elements with minimal non-zero norm. There are many examples of normed groups which are factorable. Indeed, factorability is a property with respect to the given normNon the groupG turns out that any group. ItGis factorable with respect to the constant norm, i.e. the norm withN(g) = 1 forg6= 1; namely we simply set g = 1 andg0=g. But for this factorization the norm ltration coincides with the skeletal ltration and thus nothing new is gained. We show that direct-, semi-direct- and free-products of factorable groups are factorable, implying for example that free groups are factorable with respect to the word length norm and that dihedral groups are factorable. Our most important examples of factorable groups are the symmetric groups pwith the cycle norm, i.e. the word length norm with respect to all transpositions. The factorability of the sym-metric groups has important applications in computation of the homology of moduli spaces of Riemann surfaces. In fact, the complexesN(p)[h] for a xed value ofhand for varying values of pappeared as the columns of a certain double-complex and the theory of factorable groups arose from the investigations of this double-complex, which we describe now in more detail. LetMgmn,denote the moduli space of compact, connected, oriented Riemann surfacesFn,gmof genus g0 withn1 boundary curves and withm moduli space is a non-compact This0 punctures. manifold obtained as the classifying space of the mapping class group ,nmg, the group of isotopy classes of orientation-preserving dieomorphisms xing the boundary pointwise and permuting the punctures. In [3] C.-F. Bdigheimer introduces a multi-simplicial spacePar(g m n), the space of parallel slit o mov domains. He shows thatPar(g m n) is a vector bundlePar(g m n)Mg,ner the moduli space, in particular this is a homotopy equivalence. More precisely, he considers a certain con-guration space of 2h= 4g4 + 4n+ 2mhorizontal parallel slits inndisjoint complex planes. Cutting the planes along the slits and re-gluing these pieces according to certain gluing-rules one obtains a surface. These spaces appeared later for example in the works [2], [4], [5], [9], [10] and [13]. Thereisanequivalent,morecombinatorialdescriptionofthemodulispace(see[10],[13]).To describe a Riemann surface, one considers a (q (+ 1)-tupleq . . .  0) of permutations in the sym-metric group psatisfying certain conditions; hereqvaries from 1 toh, andpfrom 1 to 2h. Some important conditions are: (a)0has a special form, in particular it hasncycles, (b) the sum of the norms ofkk11ish, (c)qhasm cycles+ 1 This data determines a nite bi-graded complexP(h m n). The vertical face operator0kis the bar-face for the group p, the horizontal face operators0k0are the simultaneous application of
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certain functionsDk: pp1, i.e.0k0((q . . .  0)) = (Dk(q) . . .  Dk(0 functions)). TheDk are not homomorphisms, but they satisfy the simplicial identities. If a face does not satisfy the conditions above it is in a subcomplexP0(h m n) of ”degenerate” surfaces. quotient complex The P /P0of some vector bundle over moduli spaceis homotopy equivalent to the Thom space Mg,mnif h= 2g2 + 2n+m. ThusP /P0is the Spanier-Whitehead dual toMn,gm, and the homology of one is Poincare dual to the cohomology of the other. The simplicial chain complex of this pair is denoted byS,(h m n is a complex of nite type,). It with top degree equal toh. Note that we are working with two sets of parameters: the genusg, the numbernof boundary curves, and the numbermof punctures determine the topological type of the surface, and therefore the moduli space;pdetermines the symmetric group p, andqis the homological degree in the bar resolution for the group (whereasp+qis the homological degree for the moduli space), andh(the negative Euler characteristic of the surface) is the ltration degree. What we do is to amalgamate the bar resolutions of all symmetric groups to a bi-complex, and then select a specic ltration quotient. R.Ehrenfriedin[10]andJ.Abhauin[1],andbothauthorstogetherwithBodigheimerin[2],used the complexS,(h m n) to compute the homology ofMnmg,in the casen= 1, andh= 2g+m5. However, the number of cells in these complexes grows very fast withh. In those computations they have encountered an interesting phenomenon. Namely, the columns of the spectral sequence associated to this double complex have all homology concentrated in their respective top degree. Thus the spectral sequence collapses after thed1 from-dierential. Starting this double complexS,(h m n) we investigate this phenomenon. We give now a more detailed description of the contents of the various chapters. InChapter 1: Groups Normedwe introduce the notion of normed groups. norm assigns The to each group elementgGa natural numberN(g) such that (1)N(gh)N(g) +N(h) for anyg hG, (2)N(g) = 0 if and only ifg= 1 and (3)N(g) =N(g1) for allgG. We give some examples of normed groups and constructions of norms, in particular norms on products and semi-direct products of normed groups. In NormChapter 2: Filtrationltrations of various objects and spaces associatedwe investigate to a group. The norm induces a ltration of the group itself, but more important of the bar resolutionE(G), of the bar complexB(G) and of the classyng spaceBGofG that. Note, the corresponding ltration on the classifying spaceBGis similiar, but dierent from the Rips ltration. We focus on the columnsN(G)[h] of the spectral sequence associated to the norm ltration. Similarly, if aG-moduleMhas anorm-admissibleltration, then the complexE(G M) is ltered and we obtain a spectral sequence converging to the homology ofGwith coecients in the module M columns of this spectral sequence are denoted by. TheN(G M)[h]. In Groups FactorableChapter 3: strategy Thewe develop the theory of factorable groups. here is to construct mapsNq(G)[h]→ Nq+1(G)[h], which behave like chain homotopies. They will
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be dened by means of a norm-preserving factorization map:GGG, that is a function (g) = (0) uch that g g, s (1)gg0=g, (2)N((g)) =N(g), that isN(g) +N(g0) =N(g), (3)N(g0minimal non-zero value of the norm) is the N:GN0, (4) Consider the following diagram: GGid//GGGid//GGid//GGG id ²G²//G²²G
where:GGGis the multiplication ofG. Denote the upper composition (id)(id)(id)(id) in the diagram byuand the lower compositionbyl. We require that for all pairs (g h)GGwe haveN(u((g h))) =N((g h)) if and only ifN(l((g h))) =N((g h)) and for pairs where both compositions are norm-preservingu((g h)) =l((g h)). We call a normed groupfactorable, ornorm-factorable, if it admits such a factorization map. The basic examples of factorable groups are discussed: groups with constant norm, symmetric groups, free groups, and direct-, semi-direct- or free-products of factorable groups. The main result of Homology of Factorable Groups TheChapter 4:is the following Theorem 4.1.1.:IfGis a factorable group with respect to the normN, then the homology of the complexN(G)[h]is concentrated in the top degreemh. Heremdenotes the minimal non-zero value of the normN. The next question which naturally arises is the following: can we nd generators for Hh(N(G)[h]) = ker{d:Nh(G)[h]→ Nh1(G)[h]}? We give a partial answer by identifying a setRh(G)[h] of certain generators of the top dimensional moduleNh(G)[h] and introducing a maph:Rh(G)[h]Hh(N(G)[h]), whereRh(G)[h] is the free module generated by the setRh(G)[h]. Our result is the following Proposition 4.3.5.:The maph:Rh(G)[h]Hh(N(G)[h])is a split injective homomorphism of modules. The case of non-trivial coecients is also investigated here and depending on the ltration of the coecient moduleMwe obtain generalizations of Theorem 4.1.1.. Chapter 5: Symmetric Groups specialize here to the symmetric groups . Wepwith the cycle norm mentioned earlier and we prove Theorem 5.2.1:The symmetric grouppis factorable with respect to the word length norm. We strengthen the general result about generators ofHh(N(p)[h]) for symmetric groups. Note that the generators ofNh(p)[h] and therefore elements ofRh(p)[h] areh-tuples of transpositions. We prove the following
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Theorem 5.4.1.:The map:Rh(p)[h]Hh(N(p)[h])is an isomorphism. As mentioned above, meanwhile R.Wang found a proof for any group G with a nete number of norm one elements. As an application we describe the complexR(p)[h] computing the homology of the symmetric group pin more detail.
AcknowledgmentsalofIwl,ldoukelihtotmknavdayrosiFrits-FriCarlchBedriehmidogire. He suggested this project and his continuous enouragement and interest in my work kept me moti-vated. I beneted a lot from the several discussions with him and his useful suggestions supported the completion of this thesis. During my studies in Bonn I learned a lot from the conversations with Elke Markert, Johannes Ebert, and with my fellow Ph.D. students Maria Guadalipe Castillo Perez, Juan Wang and Rui Wang. My warmest thanks go to my family for their support and patience. While this work was carried out, I was partially supported by the Mathematical Institute of the University of Bonn and the German Academic Exchange Service (DAAD). I would like to thank Prof. Dr. Matthias Kreck for his support and the Hausdor Research Institute for Mathematics in Bonn. I also thank the Graduiertenkolleg 1150 ”Homotopy and Cohomology” making it possible to attend to several conferences. A short summary of the results of this thesis has been published in: C.-F.Bodigheimer(joint work with B. Visy):Factorable groups and their homology, Oberwolfach Reports No. 32/2010, p. 11-13. DOI: 10.41717OWR/2010/32
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Chapter 1 Normed Groups 1.1 Basic Notions and Examples In this section we introduce the notion of normed groups. Denition 1.1.1.Anormed groupis a pair (G N), whereGis a group andNis anormonG, that is a functionN:GNsatisfying the following properties: N(g) = 0⇐⇒g= 1(N1) N(gh)N(g) +N(h) for anyg hG(N2) N(g1) =N(g) for anygG.(N3) The notion of normed groups is closely related to that of metric groups: a norm onGinduces a metric onGby settingd(g h) =N(gh1). Vice versa, a right translation-invariant, integer valued metric (i.e. a metric for whichd(gh gf) =d(h f) holds for anyg handfinG) determines a norm by settingN(g1h) =d(g h). Similarly, a left translation-invariant integer valued metric denes a norm byN(gh1) =d(g h). A metric, which is both left and right translation-invariant determines a conjugation-invariant norm, that is a normN, which satises N(ghg1) =N(h) for anyg hG.(N4) Note that relaxing the symmetry conditiond(g h) =d(h g) in the denition of the metric, one obtains the notion ofquasi-metricgroups. For the correspondingquasi-normcondition (N3) is not necessarily satised. In the following we consider some examples of normed groups. Example 1.1.2.Theconstant normNcon a groupGassigns to each element 16=gGa xed non-zero valuecN. The normNcclearly satises conditions (N1N3) and it is also conjugation-invariant. The metric which determinesNcis the discrete metric onG. Ifc= 1 we call the constant normN1thetrivial norm. Other basic examples of normed and quasi-normed groups are the cyclic groups. 7
Example 1.1.3.The innite cyclic groupZ, generated bytin a multiplicative notation, is a normed group with the normN(tn) =|n|for eachnZ. Example 1.1.4.Using a similar denitionN(tn) =nfor each 0np1 in the case of the nite cyclic groupZpwith generatort, one obtains a quasi-norm. Condition (N2) is clearly satised, since the (mod p) value ofn+mis less or equal thann+mbut this is only a quasi-norm, since, it is not symmetric:N(tn) =N(tpn)6=N(tn) if 2n6=p. However, we can also dene a norm for the groupZp. Example 1.1.5.The nite cyclic groupZpis a normed group with the normN(tn) =min{n pn} for each 0np follows from the denition, that1. ItN(tn) =N(tn) =N(tpn), hence condition (N3) is satised. The sumN1+N2of two norms onGis again a norm. More general, the linear combination 1N1+2N2of two norms is a norm for any1 2N. Moreover, given a normed group (G N) one obtains a norm on any subgroupHGby restricting NtoH.
1.2 Norms from Word Length
A large class of normed groups arises from the following construction, where we prescribe the values of the norm only for a given generating set of the group. The prescribed values are called weightsgroup using minimally weighted presentations of groupand they are extended to the entire elements. More precisely, the method is described in the following Construction 1.2.1.Assume thatSGis a generating set of the groupGandw:SNis weighting function, such that
w(s)>0 for each 16=sS(1.2.1) n w(s1s2...sn)Xw(si) for anys1 s2 ... snSfor whichs1s2...snS(1.2.2) i=1 w(s) =w(s1) for eachsSS1.(1.2.3) Then we can extend the functionwto a norm, denoted byNS,won the groupGby deningNS,w(g) to be the minimum among the natural numbers k X|j|w(sij) j=1 1   ... for each presentationg=si1sikk, wheresi1 . . .  sikSand1 . . .  kZ. The resulting functionNS,w:GNis a norm onG (: propertyN2) clearly follows from the construction ofNS,w and property (N3) follows from the fact thatg=si11...sikkis a presentation ofgwith minimal weight if and only ifg1=sikk...si11is a presentation ofg1with minimal weight. Note that ifShas minimal cardinality among the generating sets ofG, then any non-negative weighting
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function will induce a norm onG. In particular, an important special case of the construction above is when the weighting function w:SNis the constant functionw(s) = 1 for each 16=sS. The constant function satises the conditions (1.2.1-3) and the induced norm, which we denote byNSwlis called theword length norm, with respect to the generating setS. Denition 1.2.2.Theword lengthnormNSwlon a groupGwith respect to the generating setS ofGis dened for eachgGto be the minimal number (with multiplicity) of generators from the setSneeded to presentg. Remark 1.2.3.To obtain a normNSwlwhich is conjugation-invariant it is enough to assume that the generating setS if the minimal presentation ofis closed under conjugation. Indeed,hGis h=si1...sisk, thenghg1can be presented as (gsi1g1)1...(gsikg1)kand hence 1 condition (N4) is fullled:NSwl(ghg1) =NSwl(h). Examples of the word length norm include also some examples of Section 1.1. The constant norm Ncon a groupGis obtained by taking the generating setS=Gand the constant weighting functionw(g) =cfor each 16=gG. Norms for cyclic groups dened in Example 1.1.3. in and Example 1.1.5. are also word length norms with respect to the generating sets{t}and{t tp1}, respectively. We also obtain new and important examples: Example 1.2.4.The word length norm on the free groupFnof rankn, with respect to the set of free generatorsS={t1 ... tn}. The norm of a reduced word inFnis then the length of the word. Note that this norm is not conjugation-invariant. More generally, we obtain a norm on the free productGHof two normed group (G NG) and (H NH). Here we use the generating setS=GH of the groupGHwith the weighting functionw(s) =NG(s) ifsGandw(s) =NH(s) ifsH. SinceNGandNHare norms on the groupsGandHrespectively, it follows thatwsatises the conditions (1.2.1-3) and hence it induces a normNGH:=NGH,won the groupGH call. We this norm thefree product norm. Another important example is provided by the symmetric group p various generating. Among sets of pwe mention here two important cases. For our notions regarding the symmetric groups and for a more thorough discussion of Example 1.2.6. we refer to Section 5.1. Example 1.2.5.The symmetric group pis generated by the set of elementary transpositions Sel={i= (i i+ 1)|1ip1} relations between these generators are:. The i2= 1 for 1ip1 ii+1i=i+1ii+1for 1ip2 ij=jifor all 1i jp1 with|ij|>1 In the induced normNa transposition (i j)p, wherei < jhas normN((i j)) = 2(ji)1, since (i j) =i. . . j2j1j2. . . i Note that the setis a presentation with minimal length.Sel is not closed under conjugation, hence the resulting normNdoes not satisfy condition (N4). Example 1.2.6.We can take for the symmetric group also the larger generating setStr, the set of all transpositions in the symmetric group p induced conjugation invariant word length. The norm will play central role in Chapter 5. and we investigate the properties of this norm there.
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