Asphericity of symmetric presentations

Publ. Mat. 50 (2006), 133–147 ASPHERICITY OF SYMMETRIC PRESENTATIONS Fulvia Spaggiari Abstract Using the notion of relative presentation due to Bogley and Pride, we give a new proof of a theorem of Prishchepov on the asphericity of certain symmetric presentations of groups. Then we obtain further results and applications to topology of low-dimensional manifolds. 1. Relative presentations This section is devoted to recall some definitions and results on the asphericity of relative presentations according to [2]. Arelative presentation is a triple P=hH, X :Risuch that: •His a group, •X={x1, x2,...}is a set of elements, •Ris a set of words in the alphabet H∪X∪X−1of the form xǫ1 1h1xǫ2 2h2···xǫn nhn where xi∈X,ǫi=±1 and hi∈H. We always assume that Rcontains no proper powers, and that the words are cyclically reduced in the following sense: if hi= 1 and xi= xi+1 (subscripts mod n), then ǫi=ǫi+1. The elements of X∪X−1are also called X-symbols. Let F(X) denote the free group on the set X. Then the group G(P)defined by the relative presentation Pis the quotient of the free product H∗F(X) by the normal closure of R. Let R∗be the set of all cyclic permutations of words from R∪R−1 which begin with X-symbols. Let us consider the bar operator on R∗ defined as follows. For any word w∈R∗, we write it in the form w=uh, where h∈Hand ubegins and ends with X-symbols. Then we set w=u−1h−1∈R∗. Note that w=w, and w=wif and only if whas the form uh1u−1h2, where ubegins and ends with X-symbols and h1,h2are elements of order 2 in H. The relative presentation P=hH, X :Riis 2000 Mathematics Subject Classification. 20F05, 20E22, 57M07, 57N13. Key words. Relative presentations, labelled pictures, dipoles, asphericity, symmetric presentations, manifolds. 134 F. Spaggiari slender if {w}∗∩R={w}, for any w∈R. The relative presentation Pis orientable if it is slender and no element of Rhas a cyclic permutation fixed under the bar operator (i.e., no element of Ris a cyclic permutation of its inverse). Apicture Pis a finite collection of pairwise disjoint discs {∆1,...,∆m} in the interior of a disc D2, together with a finite collection of pairwise disjoint simple arcs {α1,...,αn}properly embedded in the closure of D2\Sm i=1 ∆i. For any i= 1, . . . , m, the corners of ∆iare the closures of the connected components of ∂∆i\Sn j=1 αj. The regions of Pare the closures of the connected components of D2\(Sm i=1 ∆i∪Sn j=1 αj). An inner region of Pis a simply connected region of Pwhich does not meet ∂D2. The picture Pis connected if Sm i=1 ∆i∪Sn j=1 αjis connected, and is spherical if m≥1 and (Sn j=1 αj)∩∂D2=∅. A picture Pis said to be labelled if: •Each arc is equipped with a normal orientation, indicated by a short arrow meeting the arc transversely, and labelled by an X-symbol. •Each corner of Pis oriented anticlockwise with respect to the disk ∆iin whose boundary it is contained, and labelled by an element of the group H. Let cbe a corner of a disc ∆iin the labelled picture P. Then we denote by w(c) the word obtained by reading in anticlockwise order the labels on the arcs and corners meeting ∂∆ibeginning with the label on the arc which follows c. A label xon an arc gives the generator xor x−1 if its normal orientation agrees or not with the reading sense. A connected spherical labelled picture Pis said to be a picture over the relative presentation P=hH, X :Riif the following conditions are satisfied: •For any corner cof P, the word w(c) belongs to R∗. •If h1, h2,...,hγ(i)is the sequence of the corner labels encountered in a clockwise traversal of the boundary of an inner region of P, then h1h2···hγ(i)= 1 in H. Remark. An ordinary group presentation can be considered as the particular case of a relative presentation P=hH, X :Rifor which H= 1 (hence, there are no labels at corners of a picture over P). Adipole in a picture Pover a relative presentation Pconsists of a pair of corners cand c′with an arc αconnecting the beginning of one corner with the end of the other such that cand c′belong to the same region of Pand w(c′) = w(c). Asphericity of Symmetric Presentations 135 A relative presentation Pis said to be (combinatorially)aspherical if every nonempty connected spherical picture Pover Pcontains a dipole. To complete the section, we illustrate a connection between the notion of aspherical relative presentation and the concept of topological asphericity. Let P=hH, X :Ribe a relative presentation for a group G. If K(H, 1) is a Eilenberg-MacLane space for the group H, then consider the wedge K′=K(H, 1) ∨(∨x∈XS1 x). For each w∈R, let φw:S1 w→K′be an attaching map which represents the word w∈H∗F(X)∼ =π1(K′). Then the canonical complex K(P) associated to Pis the CW-complex K(P) = K′∪([ w∈R D2 w) where D2 wis a 2-cell attached to K′via φw. By construction, we have the isomorphism G∼ =π1(K(P)). Theorem 1. If P=hH, X :Riis an orientable (combinatorially) aspherical relative presentation for a group G, then the canonical complex K(P)is topologically aspherical, that is, K(P) = K(G, 1). 2. A family of symmetric presentations Prishchepov [17] considered a family of symmetric presentations of groups depending on a finite number of positive integers: P(r, n, k, s, q) = hx1,...,xn: r Y j=1 xi+(j−1)q = s Y j=1 xi+k−1+(j−1)q(i= 1,...,n)i where the subscripts are taken modulo n,r≥2, and 1 ≤q < n. He gave arithmetic conditions on the parameters (r, n, k, s, q) which imply the asphericity of the presentations P(r, n, k, s, q) (see Section 3). Further results on the groups defined by these presentations and their generalizations can be found in [8]. The family P(r, n, k, s, q) is very interesting from a topological point of view, and contains many classes of symmetric presentations, previously considered by several authors. 136 F. Spaggiari •The presentations P(r, n, r + 1,1,1) define the Fibonacci groups F(r, n), r≥2 and n≥3 (see for example [14]). The group F(2,2m), m≥2, is the fundamental group of the m-fold cyclic covering of the 3-sphere branched over the figure-eight knot, as proved in [11]. The groups F(n−1, n), n≥3, are fundamental groups of Seifert fibered 3-manifolds [5]. •The presentations P(r, n, 2, r−1,2) define the generalized Sieradski groups S(r, n), r≥2, n≥2, introduced and geometrically studied in [6]. The group S(r, n) is the fundamental group of the n-fold cyclic covering of the 3-sphere branched over the torus knot of type (2r−1,2), as shown in the quoted paper. •The presentations P(r, n, k +r, 1,1) and P(r, n, r + 1, s, 1) define the groups F(r, n, k) and H(r, n, k), respectively, for any r≥2, n≥3, and k, s ≥1. These groups were introduced in [4] as natural generalizations of the Fibonacci groups F(r, n). A lot of topological and algebraic results on these classes of groups can be found in the quoted paper and in [18]. •The presentations P(2, n, 2,1, t) define the groups H(n, t) studied in [16] and [10]. The group H(n, t) has infinite abelianization if and only if n≡0 (mod 6) and t≡2 (mod 6). The group H(n, t) is perfect if and only if either t= 1 or nis coprime to 6 and t≡2 (mod 6). The following theorem, due to Gilbert and Howie, gives arithmetic conditions for the asphericity of groups H(n, t). Theorem 2. Suppose that (n, t)/∈ {(8,3),(9,4),(9,7)}. Then the group H(n, t)is aspherical, except for the values of (n, t)listed below: (1) (n, 0), for n≥2, (2) (n, 2), for n≥3, (3) (n, n −1), for n≥3, (4) (2t−1, t), for t≥3, (5) (2t−2, t), for t≥3, and (6) (n, t) = (6,3),(7,3),(7,5),(9,3), or (9,6). •The presentations P(2, n, k + 1,1, m) define the groups Gn(m, k), introduced in [7], and successively studied in [1]. They are natural generalizations of the Gilbert-Howie groups as Gn(m, 1) = H(n, m). The group Gn(m, k) is said to be strongly irreducible if the parameters satisfy the following conditions: 0 < m < k < n, gcd(n, m, k) = 1, gcd(n, k)>1, and gcd(n, k −m)>1; otherwise, Gn(m, k) is proved to be cyclic, a non-trivial free product, or a Gilbert-Howie group. Asphericity of Symmetric Presentations 137 The following theorem, due to Bardakov and Vesnin, gives arithmetic conditions for the asphericity of strongly irreducible groups Gn(m, k). Theorem 3. Let Gn(m,k)be a strongly irreducible group. Then Gn(m,k) is aspherical if all the following conditions are not satisfied: (1) There exists an integer ℓ≥1such that ndivides ℓ(2k−m)and 1 ℓ+gcd(n, k) n+gcd(n, k −m) n>1. (2) n=k+m. (3) n= 2(k−m)and gcd(n, k)≤n 2. (4) n= 2kand gcd(n, k −m)<n 2. 3. Asphericity The following theorem, due to Prishchepov, gives arithmetic conditions for the asphericity of the presentations P(r, n, k, s, q). Theorem 4. Let P(r, n, k, s, q)be the symmetric presentation defined in Section 2, where either r > 2s > 0or s > 2r > 0. Let A=k−1, B=k−1−(r−s)q, and suppose that one of conditions (i), (ii) and (iii) holds: (i) ndoes not divide any of 3A,4A,5A,2B,B±A,B±2A,B+3A, 2B+A. (ii) ndoes not divide any of 3B,4B,5B,2A,A±B,A±2B,A+ 3B, 2A+B. (iii) ndoes not divide any of 2A,3A,2B,3B,A±B,2B+A,2A+B. Then the presentation P(r, n, k, s, q)is aspherical. In this case, the group defined by P(r, n, k, s, q)is torsion-free and infinite. We now give a new proof of Theorem 4 by using the concept of relative presentation. We shall proceed as follows. Extending a symmetrically presented group by a finite cyclic group which cyclically permutes the set of generators and the set of relators, one obtains a group defined by a one-relator relative presentation over the finite cyclic group in question. The theory of aspherical relative group presentations, as developed by Bogley and Pride [2], applies to this set-up, there being an equivalence between relative asphericity of the relative presentation and asphericity of the original symmetric presentation. Let θdenote the automorphism of P(r, n, k, s, q) which permutes cyclically the generators, i.e., θ(xi) = xi+1 (subscripts mod n). Let us consider the split extension of P(r, n, k, s, q) by Zn=hθ:θn= 1i. If we substitute relations θ−ix1θi=xi+1 into those of P(r, n, k, s, q) and set y−1=x1θ−q, 138 F. Spaggiari then the split extension is generated by θand yand has a presentation Q(r, n, k, s, q) = hθ, y :θn= 1, ysθk−1=θk−1−(r−s)qyri. We can regard Q(r, n, k, s, q) as a relative presentation in the sense of Bogley and Pride, that is, Q(r, n, k, s, q) = hH, y :ysθA=θByri where H=hθ:θn= 1i,A=k−1 and B=k−1−(r−s)q. Lemma 5. If the relative presentation Q(r, n, k, s, q)is aspherical, then the ordinary presentation P(r, n, k, s, q)is aspherical. Proof: Let Pbe a spherical picture over the ordinary presentation P(r, n, k, s, q). Then Pcontains discs ∆icorresponding to relations   r Y j=1 xi+(j−1)q   s Y j=1 x−1 i+k−1+(s−j)q  = 1 as shown in Figure 1. ∆i xi+k−1+q xi+k−1 xi xi+q xi+q(r−1) xi+k−1+q(s−1) . . . . . . Figure 1. An inner disc in a spherical picture Pover P(r, n, k, s, q). Asphericity of Symmetric Presentations 139 Here we have no labels at the corners since we regard the ordinary presentation P(r, n, k, s, q) as a relative presentation with H= 1. Let us replace each inner disc ∆iby a picture Σiover Q(r, n, k, s, q) considered as an ordinary presentation (see Figure 2). Here we have replaced arcs labelled by xi+jq (and similarly for x−1 i+k−1+jq ) by sequences of arcs using relations xi+jq =θ−(i+jq−1)x1θi+jq−1=θ−(i+jq−1)y−1θi+(j+1)q−1. anticlockwise order θi+k−1+qs−1 θi+k−1+qs−1 . . . . . . . . . θB θi+rq−1 . . . . . . θi+rq−1 . . . . . . yy θi+k−1+q−1 y θi+q−1 θi+q−1 θi+k−1+q−1 θA . . . yy . . . . . . . . . . . . Figure 2. The picture Σiover the ordinary presentation Q(r, n, k, s, q). Along the boundary of Σiwe get the relation r Y j=1 y−1θi+jq−1θ−(i+jq−1)θ−B s Y j=1 θi+k−1+(s+1−j)q−1θ−(i+k−1+(s+1−j)q−1)yθA= 1 140 F. Spaggiari which is equivalent to the i-th relation of P(r, n, k, s, q). Along the boundary of the interior disc in Σiwe get the relation y−rθ−BysθA= 1 which is a relation of Q(r, n, k, s, q). The arcs of Σihaving both ends on ∂Σican be made into floating circles. These circles can be removed from the resulting picture. Furthermore, we will replace all other arcs with θ-labels by corner labels on the disc as shown in Figure 3. We get again the relation y−rθ−BysθA= 1. anticlockwise order θ−B θA yr ys 1 . . . 1 11 . . . Figure 3. A picture Qover the relative presentation Q(r, n, k, s, q). Repeating the same construction for each disc ∆iof Pyields a picture Qover the relative presentation Q(r, n, k, s, q). By the assumption of asphericity for Q(r, n, k, s, q), the picture Qmust contain a dipole, i.e., a pair of opposite oriented discs connected by an arc which define pairwise inverse words (see Figure 4). Asphericity of Symmetric Presentations 141 . . . . . . . . . . . . θ−B θ+B θA θ−A ys y yr−1 1 ys corner ccorner c′ yr−1 1 11 Figure 4. A dipole in the picture Qover the relative presentation Q(r, n, k, s, q). It is easy to see that any such dipole in Qarises from a pair of identical but oppositely oriented discs in Pconnected by an arc with label xifor some i. Moreover, two bridge moves in Pproduce a cancelling pair of discs. This means that if Qhas a pair of cancelling discs, then Phas a pair of cancelling discs, too. Thus, the initial picture Pmust contain a dipole. Therefore, any nonempty spherical picture over P(r, n, k, s, q) is equivalent to one having two fewer discs, hence this presentation is aspherical by induction. To study the asphericity of the relative presentation Q(r, n, k, s, q) = hH, y :ysθA=θByri where H=hθ:θn= 1i, we use the following algebraic criterion, due to Prishchepov, which is stated here in terms of relative presentations.