## Diagram groups and directed 2-complexes: homotopy and homology (2003)

### BibTeX

@MISC{Guba03diagramgroups,

author = {V. S. Guba and et al.},

title = {Diagram groups and directed 2-complexes: homotopy and homology},

year = {2003}

}

### OpenURL

### Abstract

### Citations

425 |
Combinatorial group theory
- Lyndon, Schupp
- 1977
(Show Context)
Citation Context ...1). Its cohomological dimension, cd(G), is the length of the shortest projective resolution of the trivial ZG-module Z. It is easy to see [7] that cd(G) ≤ gd(G). (17) By the Eilenberg – Ganea theorem =-=[23]-=-, cd(G) = gd(G) provided cd(G) ̸= 2 or gd(G) ̸= 3. Theorems 7.2 and 7.7 immediately imply that for diagram groups over complete directed 2-complexes these two dimensions coincide. Theorem 7.12. For ev... |

329 |
Cohomology of groups, Graduate Texts
- Brown
- 1982
(Show Context)
Citation Context ...bed in the language of group homomorphisms: H is a retract of G if and only if there are two homomorphisms φ:G → H and ψ:H → G such that φψ = idH (ψ acts first). Since Hn(−, Z) is a covariant functor =-=[7]-=-, this implies that Hn(H; Z) is a retract of Hn(G; Z). In particular, Hn(H; Z) is also free Abelian and its rank does not exceed the rank of Hn(G; Z). So we proved Theorem 7.9. For any n ≥ 0 and for a... |

174 |
Introductory notes on Richard Thompson’s groups
- Cannon, Floyd, et al.
- 1996
(Show Context)
Citation Context ...ections. It has one vertex, one edge, and one positive 2-cell. The remarkable feature of the Dunce hat is that the famous R. Thompson’s group F is its diagram group (see Section 6 below). (The survey =-=[13]-=- collects some known results about F. See also [9, 5, 6, 16, 17, 4, 15] for other results about this group.) � �����✒❅ ❅ x ❅❅❅❅❘✲ x x Figure 1. There exists a natural way to assign a directed 2-comple... |

124 | The algebra of cubes
- Brown, Higgins
- 1981
(Show Context)
Citation Context ...a directed 2-complex K can be considered as the space of certain positive paths in K. First we define a multidimensional version of the Squier complex Sq(K) from [16] as a semicubical complex. Recall =-=[29, 10]-=- that a semi-cubical complex is a family of sets {Mn,n ≥ 0} (elements of Mn are called n-cubes) with face maps λ k i :Mn → Mn−1 (1 ≤ i ≤ n, k = 0,1) satisfying the semi-cubical relations: λ k i λk′ j ... |

123 |
The algebraic theory of context-free languages
- Chomsky, Schützenberger
- 1963
(Show Context)
Citation Context ...uage. There exists a one-to-one correspondence between words of length n + 1 in L and essential n-cubes in Sq(K,s). Since the generating function of a rational language L is rational (see for example =-=[11]-=-), the Poincaré series of Sq(K,s) is a rational function as well. Recall [7] that for any group G, its geometric dimension, gd(G), is the smallest dimension of a K(G,1). Its cohomological dimension, c... |

81 |
Finiteness properties of groups
- Brown
- 1987
(Show Context)
Citation Context ...ive 2-cell. The remarkable feature of the Dunce hat is that the famous R. Thompson’s group F is its diagram group (see Section 6 below). (The survey [13] collects some known results about F. See also =-=[9, 5, 6, 16, 17, 4, 15]-=- for other results about this group.) � �����✒❅ ❅ x ❅❅❅❅❘✲ x x Figure 1. There exists a natural way to assign a directed 2-complex to every semigroup presentation (to a string rewriting system). It is... |

57 |
Groups of piecewise linear homeomorphisms of the real line
- Brin, Squier
- 1985
(Show Context)
Citation Context ...ive 2-cell. The remarkable feature of the Dunce hat is that the famous R. Thompson’s group F is its diagram group (see Section 6 below). (The survey [13] collects some known results about F. See also =-=[9, 5, 6, 16, 17, 4, 15]-=- for other results about this group.) � �����✒❅ ❅ x ❅❅❅❅❘✲ x x Figure 1. There exists a natural way to assign a directed 2-complex to every semigroup presentation (to a string rewriting system). It is... |

40 | Diagram groups
- Guba, Sapir
- 1997
(Show Context)
Citation Context ...resentations). Some results about diagram groups were obtained by Meakin’s student Vesna Kilibarda (see [21, 22]). Further results about diagram groups have been obtained by the authors of this paper =-=[16, 17, 18]-=-, D. Farley [14], and B.Wiest [32]. The definition of diagram groups in terms of string rewriting systems does not reflect the geometry of diagram groups and geometrical nature of the constructions th... |

25 |
The geometry of rewriting systems: a proof of the Anick-GrovesSquier theorem
- Brown
- 1989
(Show Context)
Citation Context ... Homology Theorem 3.6 shows that the components Sq(K,w) are K(G,1) spaces for diagram groups G = D(K,w). In fact in most cases Sq(K) is too large. Here we will use the technique of collapsing schemes =-=[9, 8, 12]-=- to find a “smaller” CW complex, which is homotopy equivalent to Sq(K) (at least in the case when K is complete). We recall the concept of collapsing scheme from [8, 12, 9]. Let X be a semi-cubical co... |

23 |
An infinite-dimensional torsion-free FP∞ group
- Brown, Geoghegan
- 1984
(Show Context)
Citation Context ...ive 2-cell. The remarkable feature of the Dunce hat is that the famous R. Thompson’s group F is its diagram group (see Section 6 below). (The survey [13] collects some known results about F. See also =-=[9, 5, 6, 16, 17, 4, 15]-=- for other results about this group.) � �����✒❅ ❅ x ❅❅❅❅❘✲ x x Figure 1. There exists a natural way to assign a directed 2-complex to every semigroup presentation (to a string rewriting system). It is... |

23 |
Homologie singuliére des espaces fibrés. Applications
- Serre
- 1951
(Show Context)
Citation Context ...a directed 2-complex K can be considered as the space of certain positive paths in K. First we define a multidimensional version of the Squier complex Sq(K) from [16] as a semicubical complex. Recall =-=[29, 10]-=- that a semi-cubical complex is a family of sets {Mn,n ≥ 0} (elements of Mn are called n-cubes) with face maps λ k i :Mn → Mn−1 (1 ≤ i ≤ n, k = 0,1) satisfying the semi-cubical relations: λ k i λk′ j ... |

22 |
A counterexample to a conjecture of Serre
- Anick
- 1982
(Show Context)
Citation Context ...complex Sq(K). We expand the set of paths in K allowing paths that go “inside” 2-cells. Let K be a directed 2-complex. Attaching a 2-cell f ∈ F + with p = ⌈f⌉, q = ⌊f⌋ can be done as follows. Let D = =-=[0,1]-=- × [0,1] be a unit square. For any t ∈ [0,1] we have the path dt in D defined by dt(s) = (s,t) ∈ D (s ∈ [0,1]). We attach this square to K in such a way that d0 is identified with p, d1 is identified ... |

22 |
Finiteness and CAT(0) properties of diagram groups
- Farley
(Show Context)
Citation Context ...lts about diagram groups were obtained by Meakin’s student Vesna Kilibarda (see [21, 22]). Further results about diagram groups have been obtained by the authors of this paper [16, 17, 18], D. Farley =-=[14]-=-, and B.Wiest [32]. The definition of diagram groups in terms of string rewriting systems does not reflect the geometry of diagram groups and geometrical nature of the constructions that can be applie... |

20 |
The ubiquity of Thompson’s group F in groups of piecewise linear homeomorphisms of the unit interval
- Brin
- 1999
(Show Context)
Citation Context |

19 |
On the geometry of semigroup presentations
- Remmers
- 1980
(Show Context)
Citation Context ...oup presentation P. It is convenient to define diagrams over a directed 2-complex K in an “abstract” way, without referring to 2-paths of K. Such a definition was given by Kashintsev [20] and Remmers =-=[27]-=- in the case of semigroup presentations. Here we basically repeat their definition and result. Definition 2.3. A diagram over K = 〈E | ⌈f⌉ = ⌊f⌋, f ∈ F + 〉 is a finite plane directed and connected gra... |

17 | The Dehn function of Richard Thompson’s group F is quadratic - Guba |

11 |
rewriting, a survey for group theorists
- Cohen
- 1993
(Show Context)
Citation Context ... Homology Theorem 3.6 shows that the components Sq(K,w) are K(G,1) spaces for diagram groups G = D(K,w). In fact in most cases Sq(K) is too large. Here we will use the technique of collapsing schemes =-=[9, 8, 12]-=- to find a “smaller” CW complex, which is homotopy equivalent to Sq(K) (at least in the case when K is complete). We recall the concept of collapsing scheme from [8, 12, 9]. Let X be a semi-cubical co... |

11 |
On the algebra of semigroup diagrams
- Kilibarda
- 1997
(Show Context)
Citation Context ...f diagram groups was given by Meakin and Sapir in terms of string rewriting systems (semigroup presentations). Some results about diagram groups were obtained by Meakin’s student Vesna Kilibarda (see =-=[21, 22]-=-). Further results about diagram groups have been obtained by the authors of this paper [16, 17, 18], D. Farley [14], and B.Wiest [32]. The definition of diagram groups in terms of string rewriting sy... |

10 |
Semigroups and combinatorial applications. Pure and Applied Mathematics. A Wiley-Interscience Publication
- Lallement
- 1979
(Show Context)
Citation Context ...be explained by the following Theorem 7.11. Let K be a complete finite almost 2-path connected directed 2-complex. Then the Poincaré series of any of its diagram group is rational. Proof. We refer to =-=[24]-=- for the well-known properties of rational languages. Let A = P∪(P×F − ), where P consists of all irreducible 1-paths in K, including the empty 1-paths, F − consists of all negative 2-cells of K. Let ... |

10 |
The algebra of directed complexes
- Steiner
- 1993
(Show Context)
Citation Context ...ure of the constructions that can be applied to diagram groups. So in this paper we introduce a more geometric definition of diagram groups in terms of directed 2-complexes. A directed 2-complex (see =-=[31, 26]-=-) is a directed graph equipped with 2-cells each of which is bounded by two directed paths (the top path and the bottom path). With any directed 2-complex one can associate the set of (directed) homot... |

8 | Calculating generators of π2, in: Two-dimensional Homotopy and Combinatorial Group Theory - Bogley, Pride - 1993 |

8 | Rigidity properties of diagram groups - Guba, Sapir |

7 |
Diagram groups are totally orderable
- Guba, Sapir
(Show Context)
Citation Context ...in terms of directed 2-complexes. In particular, we show that the class of diagram groups is closed under arbitrary (countable) direct products. 2The results of this paper are used in our next paper =-=[19]-=- to show that all diagram groups are totally orderable. 2 Combinatorial definition We start by giving a precise definition of directed 2-complexes. Our definition differs insignificantly from the orig... |

7 |
Algebraic topology for two dimensional complexes
- Sieradski
- 1993
(Show Context)
Citation Context ...h the top of δ ′′ , then one can define a concatenation (product) of them denoted by δ ′ ◦ δ ′′ (formally, this is just the sequence δ ′ ,δ ′′ ). As in the standard homotopy theory (see, for example, =-=[30]-=-), we need to identify homotopic 2-paths and then define a diagram group D(K,p) based at a 1-path p as the group of classes of equivalent 2-paths connecting p with itself. To do this, we choose a comp... |

6 |
On subgroups of R.Thompson’s group F and other diagram groups
- Guba, Sapir
- 1999
(Show Context)
Citation Context ..., three edges x, y, z and three positive cells of the forms xy = x, y = y, yz = z on Figure 14 (to obtain the complex from the diagram , we identify all edges having the same labels). It is proved in =-=[17]-=- that the diagram group D(Q,xyz) is isomorphic to the restricted wreath product Z wr Z. Theorem 24 of [17] shows that the triple (Z wr Z, Q,xyz) is rigid. x y � � � � x y z Figure 14. The free Abelian... |

5 | Geometric methods in combinatorial group theory - Pride - 1995 |

4 |
Subgroups of finite index in groups with finite complete rewriting systems
- Pride, Wang
(Show Context)
Citation Context ...ure of the constructions that can be applied to diagram groups. So in this paper we introduce a more geometric definition of diagram groups in terms of directed 2-complexes. A directed 2-complex (see =-=[31, 26]-=-) is a directed graph equipped with 2-cells each of which is bounded by two directed paths (the top path and the bottom path). With any directed 2-complex one can associate the set of (directed) homot... |

4 |
Diagram groups are left-orderable (submitted). Victor Guba
- Wiest
(Show Context)
Citation Context ...groups were obtained by Meakin’s student Vesna Kilibarda (see [21, 22]). Further results about diagram groups have been obtained by the authors of this paper [16, 17, 18], D. Farley [14], and B.Wiest =-=[32]-=-. The definition of diagram groups in terms of string rewriting systems does not reflect the geometry of diagram groups and geometrical nature of the constructions that can be applied to diagram group... |