Roger C. Lyndon and Paul E. Schupp. Combinatorial group theory. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....semigroup presentation h B j Q i which also de nes G and satis es jQj jBj = jRj jAj. The number jRj jAj for a nite (semigroup or group) presentation P = h A j R i is called the de ciency of P and is denoted by def(P) This de nition follows [13] and [2] some more classical sources such as [14] use jAj jRj. The semigroup de ciency def S (S) of a nitely presented semigroup S and the group de ciency def G (G) of a nitely presented group G are de ned as def S (S) minfdef(P) j P is a nite semigroup presentation for Sg def G (G) minfdef(P) j P is a nite group presentation for ....

R.C. Lyndon and P.E. Schupp. Combinatorial group theory. (Springer Verlag, Berlin, 1977).

....as the collapse of the deformation retraction, as illustrated by Example 2 in Section 8. In this section the focus is on maps (in the sense of small cancellation theory) R spheres. After the key concept of a map is defined, some examples are given, and a few results from Lyndon and Schupp ([9]) are quoted. In the few cases where the proofs in [9] are stronger than the statements of the lemmas to which they are attached, the statements given here have been altered to capture the full import of the proofs. The extra flexibility will be used in the general version of small cancellation ....

....by Example 2 in Section 8. In this section the focus is on maps (in the sense of small cancellation theory) R spheres. After the key concept of a map is defined, some examples are given, and a few results from Lyndon and Schupp ( 9] are quoted. In the few cases where the proofs in [9] are stronger than the statements of the lemmas to which they are attached, the statements given here have been altered to capture the full import of the proofs. The extra flexibility will be used in the general version of small cancellation theory. The section concludes with a general version of ....

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R. Lyndon and P. Schupp. Combinatorial Group Theory. Springer-Verlag, New York, 1977.

.... results may exist when one restricts to presentations of a special form: a typical example is the small cancellation theory, in which a number of properties are established for those groups or monoids defined by presentations satisfying some conditions about subword overlapping in the relations [24, 25, 30, 38]. Another example is Adyan s criterion [1, 37] which shows that a presented monoid embeds in the corresponding group if there is no cycle in some graph associated with the presentation. The aim of this paper is to study a combinatorial property of positive group presentations (i.e. of ....

R. C. Lyndon & P. E. Schupp, Combinatorial Group Theory, Springer (1977).

.... for us) then the fundamental group does not depend on v, thus when v does not matter then we shall note the fundamental group as (K) Let us recall that a finitely presentable group (or fp group for short) is a group that can be represented by a finite number of generators and relations (see [20, 13]) Theorem 6. If K is a finite connected complex then the fundamental group (K) is finitely presentable, moreover its finite presentation can be deduced effectively from K. Conversely, if G is a finitely presentable group then there exists a finite connected complex K, that can be constructed ....

Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer Verlag, 1977.

....i;j = s i;j ffl u: The notation Q(S) fS ffl u j u 2 Mg, is extended to matrices as well. Given n 1; m 1, and S 2 Bn;m hh M ii we denote by Q r (S) the set of row residuals of S: Q r (S) Q(S i; The ordering on B is extended componentwise to B n;m hh M ii. y We refer the reader to [LS77] for a definition of the free product of two monoids (or groups) We focus here on the monoid M = K W defined as the free product of a group K with a free monoid W . Definition 2.1 We denote by the binary relation over K W defined by: for every u; w 2 K W , u w , 9v 2 K W ; u ....

R.C. Lyndon and P.E. Schupp. Combinatorial Group Theory. Springer Verlag, 1977.

.... the study of Thue systems ( semigroup presentations) They were first formally introduced by Kashintsev [16] see also Remmers [29] Stallings [34] or Higgins [13] The role of semigroup diagrams in the study of semigroups is similar to the role of van Kampen diagrams in the study of groups (see [22] or [26] We shall give a precise definition of semigroup diagrams later. Here we need only that every semigroup diagram Delta is a labelled plane graph (a graph drawn on a plane) which has two distinguished paths top( Delta) and bot( Delta) These paths have common initial and terminal ....

....has two distinguished paths top( Delta) and bot( Delta) These paths have common initial and terminal vertices and the whole diagram is located between these two paths, so that the union of these two paths is the boundary of the diagram. By the result from [16] similar to the van Kampen lemma [22], two words are equal modulo a Thue system P = h Sigma j R i (here Sigma is an alphabet and R is a set of relations) if and only if there exists a semigroup diagram Delta such that the label of top( Delta) is u, the label of bot( Delta) is v, and the label of the boundary of each cell (face) of ....

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R. C. Lyndon and P. E. Schupp. Combinatorial group theory. Springer-Verlag, 1977.

.... than its analog (N1) This is again in order to be able to arrange a sequence of elementary transformations so that the complexity of a given element (in this case, just the lexicografic length of a cyclically reduced word) would decrease (or, at least, not increase) at every step see [14]. This arrangement still leaves us with a difficult problem to find out if one of two elements of the same complexity ( of the same length) can be taken to another by an automorphism of F . This is actually the most difficult part of Whitehead s algorithm. In one special case however this ....

R.Lyndon, P. Shupp, Combinatorial Group Theory, Series of Modern Studies in Math. 89. SpringerVerlag, 1977.

....(k 2) cubes of X share a common k cube, and pairwise share common (k 1) cubes, then they are contained in a (k 3) cube of X : In [51, 44] the CAT(0) cube complexes were called cubings. A rich class of CAT(0) 2 complexes is formed by PE complexes arising in small cancellation theory [40]. An (m; n) complex is a 2 complex X in which each face (2 cell) has at least m sides and for any vertex x every simple cycle in Link(x; X ) has at least n edges. If X is a simply connected (m; n) complex with mn 2(m n) and each face of X is a regular Euclidean polygon, then from Theorem ....

.... D with vertex v: If v is an interior vertex of D; define the curvature at v to be (v) 2 Gamma ff(v) When v is a vertex in the boundary D; define the turning angle at v to be (v) Gamma ff(v) A vertex v 2 D with (v) 0 is called a corner of D: The following Lyndon s Curvature Theorem [40] is a PE version of the Gauss Bonnet Theorem: X v2D Gamma D (v) X v2 D (v) 2: If a PE disk D is CAT(0) then (v) 0 for any interior vertex v: Consequently, D has at least two corners. 6 4. Weakly modular graphs In this section we briefly introduce some classes of graphs and recall ....

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R.C. Lyndon and P.E. Schupp, Combinatorial Group Theory, Springer--Verlag, Berlin, Heidelberg, New York, 1977.

....fu 2 Y j (w; u) 2 L(T ) for some w 2 Wg: A basic result on transducers states that T (W ) is rational whenever W 2 Rat X [1, Cor. III.4.2 and Th. III.6. 1] 3 Group languages For common concepts and results in group theory and combinatorial group theory, the reader is referred to [3] and [4], respectively. Most of the results included in this section are folklore, but we include some (simple) proofs. Given a subgroup H of a group G, we denote by [G : H] the index of H of G. We recall that a congruence on a group G is fully determined by the normal subgroup 1 : in fact, a b , ab ....

R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer-Verlag 1977. 17

....index subgroup in 1 ( Let K be a genus 1 knot, K] 6= 1 in 1 (M ) Then either contains a 1 injective closed surface, or K is tight. Indeed, if S is a punctured torus with S = K. Suppose : 1 (S) 1 ( is not injective. It its image is free of rank 2, then is an isomorphism [LS], a contradiction. If ( 1 (S) is free of rank one, then [K] 1 in 1 ( If ( 1 (S) is not free, then contains a 1 injective surface. 4.A recent theorem of Boyer Culler Shalen Zhang [BCSZ] asserts that if K 2 is not bered, then for m big enough , K becomes tight in a (m; n) surgery ....

....image of nite index, a contradiction to tightness. b) the image of 1 (S 0 [ S 0 ) 1 ( m ) is free. By the lemma above, this is a free group F with 2g generators. Now, F projects surjectively on 1 (S) F 2g , do F has exactly 2g generators and the projection is an isomorphism [LS]. But the restrictions of the projection to 1 (S 0 ) and 1 ( S 0 ) is obviously an isomorphism as well. It follows that j 1 (S 0 ) is identity. However, by a theorem of Conner [Co] for an action of a nite cyclic group on a K( 1) three manifold space, cannot have xed ....

R. Lyndon, Schupp, Combinatorial group theory.

.... exponential, multi exponential) The bridge groups that we consider in Section 6 are very special since they all have two generators and a 1 relator presentation where the subwords c 2 ; c Gamma2 do not occur in the relation (for an introduction to the theory of 1 relator presentations see [LSc77] and [Bau86] This leads to a second question Question 22 Given a uniform family of finitely presented groups fH n g 1 n=1 with 1 relator presentation, does there exist a uniform family of finitely presented groups fG n g 1 n=1 with two generators b; c and 1 relator presentation such that ....

R.C. Lyndon and P.E. Schupp. Combinatorial Group Theory. Springer, 1977.

....i let G i be the free group on countably many generators: x i;1 ; x i;k ; Consider the embeddings : G i Gamma G i 1 , x i;k 7 Gamma x i 1;k 2 . Define G1 = lim G i . 1 Obviously, G1 is not in C since G1 2 = G1 . We will need the next proposition which may be found in [1] p. 53. Proposition 3 Let N 1 and let u 1 ; um be elements of a free group F which satisfy u 1 N Delta Delta Delta um N = 1. Then the subgroup generated by u 1 ; um has a rank m=2. We are going to use a corollary of this result : Corollary 4 In the free group, F , on ....

Lyndon, R., Schupp, P.: Combinatorial group theory. New-York: Springer 1977.

....the concept of diagrams, which provide an intuitive and powerful geometrical representation for G grammars. These diagrams originate in the work of Van Kampen [17] and have found applications in combinatorial group theory and in studies of decidable subclasses of the word problem for groups [11]. 2 They have also been used to produce complete term rewriting specifications for certain classes of groups [2] Section 7 discusses in detail the conditions under which G grammars and the associated rewriting systems lead to equivalent definitions of the semantics phonology relation. These ....

....if and only if there exists a reduced diagram having boundary word w and whose regions are relator cells associated with the elements of R. 16 u 2 u n u 1 r 1 r 2 r n O Figure 9: Star diagram. The proof is not provided; it can easily be recovered from the property demonstrated in [11] (chapter 5, section 1) The proof involves the following remark. If one considers a product u 1 r 1 u 1 Gamma1 : un r nun Gamma1 with r Gamma1 2 R and u i arbitrary elements of F (V ) this product can be read as the boundary word of the star diagram represented in Fig. 9, ....

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Lyndon, R. and P. Schupp: 1977, Combinatorial Group Theory. SpringerVerlag.

....to the setting of finitely presented groups in [5] for subgroups in finitely generated free groups and coset enumeration for subgroups in finitely presented groups. 2. 1 Nielsen Reduction Let us start by giving a short description of Nielsen s method, which can be found in more detail e.g. in [13]. Let F be a free group with generating set Sigma. We call a word w j w1 : wk , w i 2 F , reduced, in case w = w1 ffi : ffi wk , i.e. jwj = P k i=1 jw i j. Subsets of F are written as U = fu i j i 2 Ng or U = fu1 ; ung depending on whether they are finite or not. The subgroup ....

....v1 ffi v2 6= implies jv 1 ffi v2 j maxfjv1 j; jv 2 jg; N2) v1 ffi v2 6= and v2 ffi v3 6= imply jv 1 ffi v2 ffi v3 j jv 1 j Gamma jv 2 j jv 3 j. Nielsen reduced sets play an important role, as they are free generating sets for the subgroup they generate. The following theorem (see e.g. [13]) states that freely reducing a product of elements of a Nielsen reduced set cannot result in arbitrary cancellations of the elements involved. Theorem 3 Let U be a Nielsen reduced set. Then for every u 2 U [U Gamma1 there are words a(u) and m(u) with m(u) 6= such that u j a(u)m(u) a(u ....

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R. C. Lyndon and P. E. Schupp. Combinatorial Group Theory. Springer, 1977.

....Hall. Theorem 1.2 [6] Every nitely generated subgroup of the free group is closed. The proof of Theorem 1.2 is based on a well known separation result: if G is a nitely generated subgroup of a free group, and if g 2 G, then there exists a subgroup of nite index containing G but not g. See [7, 17]. Given two subsets X and Y of the free group, recall that the product XY is the set XY = fxy j x 2 X and y 2 Y g: We propose the following conjecture, which obviously extends the theorem of Hall. Conjecture 1. If H 1 ; Hn is a nite sequence of nitely generated subgroups of the free ....

R.C. Lyndon and P.E. Schupp, Combinatorial Group Theory, Springer Verlag, 1977.

....than with the techniques of automata and inverse semigroup theory. With this in mind we will take for granted the (combinatorial) de nitions of 2 complexes, the fundamental group of a 2 complex, and coverings of 2 complexes. Also we will assume basic properties of such; the reader is referred to [32, 58] for more. While we will give the de nition of an immersion, we will use their basic properties, found in [50] without comment. If X is a 2 complex, X (1) will denote the 1 skeleton of X. We shall assume all morphisms of 2 complexes to be combinatorial. For a 2 complex X, 1 (X; v 0 ) will ....

R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer-Verlag, 1977.

....has a negative answer for infinite groups. This is related to the following Waldhausen s question: If G = hx 1 ; x n ; r 1 ; r m i can be generated by fewer than n elements, then does the normal closure of hr 1 ; r n i in F n contain a primitive element of F n (see [LyS], p. 92) The answer for this question is negative (see [No,Ev1] On the other hand, it was noted in [Du2] in fact, motivated his work) that this is equivalent to the question of whether a generating set for G with n elements can always be changed by Nielsen moves to a set containing the ....

....of the following automorphisms: R Sigma i;j (x i ) x i x Sigma1 j ; and R Sigma i;j (x l ) x l if l 6= i L Sigma i;j (x i ) x Sigma1 j x i ; and R Sigma i;j (x l ) x l if l 6= i These are exactly Nielsen moves when G = F k . A classical result of Nielsen [Ni] see [LyS,MKS]) shows that Aut(F k ) is generated by the Nielsen moves and elementary automorphisms of permutation and inversion of generators. Proposition 3.4.1 Let A = A (F k ) be the subgroup of Aut(F k ) generated by Nielsen moves. Then A (F k ) is a normal subgroup of index two in Aut(F k ) ....

R.C. Lyndon, P.E. Schupp, Combinatorial group theory, Springer, Berlin, 1977.

....indeed. Remark. It is interesting how relations on categories can be converted to cells in a double category. From this a picture of a proof that words are equal emerges. Then geometric arguments can be used to analyze the proof as we did in the previous proposition. Similar methods are used in [7] to great advantage. Their notion of van Kampen diagram is clearly related to our complexes, but our rectangular setting is more rigid and thus easier to use. 6 Examples Example: Tileorders (dissections of rectangles into rectangles) can be considered as rectangular complexes with vertex ....

R. Lyndon and P. Schupp, Combinatorial Group Theory, Springer-Verlag, 1977.

.... is a homomorphism of inverse (regular) semigroups. The above theorem applies, in particular, to the case of full amalgams. Assume (S; U; T ) is a regular semigroup amalgam satisfying the hypotheses of Theorem 9.11. If U has only trivial subgroups, then Corollary 9. 9 and the Kurosh Theorem [8] imply that the subgroups of S U T are isomorphic to free products of free groups and subgroups of the factors S and T . Work of Ordman [13, 14] gives normal forms for amalgamated products of unordered groupoids. Since the above theorem implies that the underlying groupoid of S U T is the ....

R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer-Verlag, New York, 1977.

....fix X G so that X [ feg is a set of coset representatives for the left cosets of H in G . Then every g 2 Gamma has a unique normal form g = x 1 x 2 Delta Delta Delta x n h (10) for h 2 H, n 2 N [ f0g and x j 2 X j where 1 6= 2 ; 2 6= 3 ; n Gamma1 6= n ; see [LS77]) Actually, this is a bit sloppy: the normal form is more correctly taken to be the list (x 1 ; x 2 ; x n ; h) but we trust no confusion will result. We define the length of g to be L(g) n and for 1 j n we let j (g) j where x j 2 X j . We now show that the condition in ....

R.D. Lyndon, P.E. Schupp, Combinatorial Group Theory, Springer--Verlag, 1977.

....and every simple loop in the link (see Section 2 for the definition of a link) of a vertex consists of at least n edges. The natural flat metric on X makes each face of X with k edges an isometrically immersed regular Euclidean k gon. Such spaces arise naturally in combinatorial group theory, see [LySc] and [BaBr] Let (X; Gamma) be a compact orbispace of nonpositive curvature. A geodesic oe : R X is called Gamma closed if there is an isometry OE 2 Gamma translating oe that is, OE(oe(t) oe(t t 0 ) for some t 0 6= 0 and all t 2 R. A Gamma closed geodesic oe and an isometry OE 2 Gamma ....

R. Lyndon & P. Schupp, Combinatorial Group Theory, Springer-Verlag, 1977.

....respectively groups, as examples of varieties, can be presented as quotients of free monoids respectively free groups, general rewriting is one technique to solve computational problems related to the respective structures. Such presentations in terms of generators and defining relations (see e.g. [Gi79, LySch77, MaKaSo76]) are closely related to so called string rewriting systems or semi Thue systems, which can be seen as special rewriting systems. Hence knowledge and procedures from this field, especially variations of the Knuth Bendix completion procedure [KnBe70] can be applied to solve monoid and group ....

R. C. Lyndon and P. E. Schupp. Combinatorial Group Theory. Springer Verlag(1977).

.... we obtain a natural variant of the problems which we call the bounded membership problem for the matrix group SL 2 (Z) and the bounded membership problem for the matrix monoid SL 2 (N ) The unbounded version of the problem is known as the Magnus problem for the unimodular group and monoid [10] [9]. In terms of complexity of the problems we expect that they become somewhat easier by letting all B i = I , and we call them simple versions of the MD and MC problems. More precisely, we define: Definition 1.3 Bounded membership problem for SL 2 (Z) denoted by BMZ, is the following decision ....

R. Lyndon, P. Schupp, Combinatorial group theory. Springer-Verlag, 1977.

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Roger Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer-Verlag, 1977.

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Roger C. Lyndon and Paul E. Schupp. Combinatorial group theory. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition.

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Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin--New York, 1977. 9

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R. Lyndon and P. Schupp, Combinatorial Group Theory, Springer-Verlag, 1977.

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R. Lyndon and P. Schupp, Combinatorial group theory, Series of Modern Studies in Math. 89. Springer-Verlag, 1977.

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Roger C. Lyndon, Paul E. Schupp, Combinatorial group theory, New York, Springer-Verlag (1977).

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R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Ergebnisse der Mathematik, band 89, Springer 1977.

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R.C. Lyndon, P.E. Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin/Heidelberg/New York, 1977.

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R.Lyndon, and P.Schupp (1977) Combinatorial group theory, SpringerVerlag.

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R. C. Lyndon, P. E. Schupp, Combinatorial group theory, (Springer-Verlag, BerlinHeidelberg -New York, 1977).

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R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin, 1977.

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Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer, 1977.

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R. C. Lyndon and P. E. Shupp. Combinatorial Group Theory. Springer-Verlag, New York/Berlin, 1977.

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R. Lyndon and P. Schupp (1977). Combinatorial group theory. Springer.

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R. C. Lyndon & P. E. Schupp,Com binatorialgrouptheory,Springer-Verlag(1977).

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R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, SpringerVerlag, 1977.

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R. Lyndon and P. Schupp. Combinatorial Group Theory. Springer-Verlag, New York, 1977.

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R. C. Lyndon & P. E. Schupp, Combinatorial group theory, Springer (1977).

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R.Lyndon, P.Schupp, Combinatorial Group Theory, Series of Modern Studies in Math. 89. Springer-Verlag, 1977.

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R. Lyndon and P. Schupp, Combinatorial Group Theory, Series of Modern Studies in Math. 89. Springer-Verlag, 1977.

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R. Lyndon and P. Schupp, Combinatorial Group Theory, Springer Verlag 1977.

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R. C. Lyndon and P. E. Schupp. Combinatorial Group Theory. Springer, 1977.

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R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer-Verlag 1977.

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R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, SpringerVerlag, New York, 1977.

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R.C. Lyndon, P.E. Schupp, "Combinatorial Group Theory", Berlin, Springer, 1977.

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R. C. Lyndon and P. E. Schupp. Combinatorial Group Theory. Springer Verlag, 1977.

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