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## RATIONAL SUBSETS IN HNN-EXTENSIONS AND AMALGAMATED PRODUCTS

Citations: | 4 - 3 self |

### Citations

821 | Rewrite Systems
- Dershowitz, Jouannaud
- 1990
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Citation Context ...v ∈ X ∗ we have (uℓ = ℓ ′ v) ⇒ |u| ≥ |ℓ ′ | and (uℓv = ℓ ′ ) ⇒ (u = v = ε ∧ r = r ′ ). 4sMay 30, 2007 15:16 WSPC/INSTRUCTION FILE 1 Then it can be easily shown that the relation →ST is confluent. See =-=[DJ91]-=- for more information about confluence and other properties of rewriting systems. 2.1. Finite automata over a free monoid Let X be an alphabet (that we do not assume to be finite, in general). A finit... |

619 |
Combinatorial group theory
- Lyndon, Schupp
- 1977
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Citation Context ...s dealing with algorithmic problems for equations over HNN-extensions [HNN49] and amalgamated free products of groups. These two operations are of fundamental importance in combinatorial group theory =-=[LS77]-=-. Recall that an amalgamated free product G = 〈H1 ∗ H2,a = ϕ(a)(a ∈ A)〉 (1) of two groups H1 and H2 with subgroups A ≤ H1, B ≤ H2 and an isomorphism ϕ : A → B, results from the free product H1∗H2 by i... |

187 |
Topology of finite graphs
- Stallings
- 1983
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Citation Context ...pecial case of an amalgamated free product of two finite groups, algorithmic problems for finitely generated subgroups were recently studied in [ME07b,ME07a] by extending Stalling’s folding technique =-=[Sta83]-=-. 2. Preliminaries We recall in this section all needed definitions and classical results concerning groups, monoids, finite automata, and rational subsets of a monoid. We also recall some classical a... |

168 |
Groups acting on graphs
- Dicks, Dunwoody
- 1989
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Citation Context ...ree subgroup of finite index) if and only if it can be built up from finite groups using amalgamated free products with finite identified subgroups and HNN-extensions with finite associated subgroups =-=[DD90]-=-. Finally, many important structural and algorithmic properties of groups are preserved by HNN-extensions (resp. amalgamated free products) with finite associated (resp. identified) subgroups, see e.g... |

152 | Subgroups of surface groups are almost geometric, - Scott - 1978 |

122 |
Embedding theorems for groups
- Higman, Neumann, et al.
- 1949
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Citation Context ...om the subgroup H (resp. the subgroups H and J) to the group G. 1. Introduction This paper is the first paper in a series of papers dealing with algorithmic problems for equations over HNN-extensions =-=[HNN49]-=- and amalgamated free products of groups. These two operations are of fundamental importance in combinatorial group theory [LS77]. Recall that an amalgamated free product G = 〈H1 ∗ H2,a = ϕ(a)(a ∈ A)〉... |

109 |
Group theory and three-dimensional manifolds
- Stallings
- 1971
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Citation Context ... of Stallings states that a group G has more than one end if and only if it is either an HNN-extension with finite associated subgroups or an amalgamated free product with finite identified subgroups =-=[Sta71]-=-. Moreover, a group is virtually-free (i.e., has a free subgroup of finite index) if and only if it can be built up from finite groups using amalgamated free products with finite identified subgroups ... |

72 | Monographs in Mathematics - Trees - 2003 |

49 |
Coset representations in free groups
- Hall
- 1949
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Citation Context ...2007 15:16 WSPC/INSTRUCTION FILE 1 Recall that a group G has the Howson property if the intersection of two finitely generated subgroups is again finitely generated. The group G has the LERF property =-=[Hal49]-=- if for every finitely generated subgroup H < G and every g ∈ G \ H there exists a subgroup Z of finite index in G with H ≤ Z < G and g ∈ G \ Z. Statement (a) is covered by [KWM05, Corollary 5.15], st... |

24 | Foldings, graphs of groups and the membership problem - Kapovich, Weidmann, et al. - 2005 |

23 | Graphs and separability properties of groups, - Gitik - 1997 |

23 | The ”last” decision problem for rational trace languages - Sakarovitch - 1992 |

21 | Monadic Thue systems, - Book, Jantzen, et al. - 1982 |

19 | Subgroup separability of graphs of free groups with cyclic edge groups - Wise |

17 | D Solitar, Subgroups of HNN groups and groups with one defining relator - Karrass - 1971 |

16 |
Characterizations of the decidability of some problems for regular trace languages
- Aalbersberg, Hoogeboom
- 1989
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Citation Context ...nal subsets is decidable [KSS06], whereas the inclusion problem for rational subsets (which is a special instance of the emptiness problem for boolean combinations of rational subsets) is undecidable =-=[AH89]-=-. Let us mention that more general notions of HNN-extensions for monoids were defined in the literature (see for instance [How63]) where A and B in (2) are not necessarily groups. It is not clear whet... |

16 |
Zur algebraischen Charakteristik der durch kontext-freie Sprachen definierten
- Anissimov, Seifert
- 1975
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Citation Context ...theless, we can always normalize finite t-automata in such a way that the map µ fulfils some nice algebraic property, see Definition 9 and Proposition 33. Anissimov and Seifert have shown: Theorem 1 (=-=[AS75]-=-). Let G be a group. The rational subgroups of G are exactly the finitely generated subgroups of G. We present a proof sketch for Theorem 1, because the underlying idea will be used later in a slightl... |

14 | On the rational subset problem for groups
- Kambites, Silva, et al.
(Show Context)
Citation Context ...ies decidability of the membership problem for rational subsets. On the other hand, for the group F × Z (where F is the free group of rank 2), the membership problem for rational subsets is decidable =-=[KSS06]-=-, whereas the inclusion problem for rational subsets (which is a special instance of the emptiness problem for boolean combinations of rational subsets) is undecidable [AH89]. Let us mention that more... |

13 | Algorithms in geometric group theory - Grunschlag - 1999 |

13 | On the free product of two groups with an amalgamated subgroup of finite index in each factor, - Karrass, Solitar - 1970 |

10 |
Parties rationnelles du groupe
- Benois
- 1969
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Citation Context ...star (which associates with a subset U ⊆ M the submonoid of M generated by U). By Kleene’s theorem, the rational subsets of a free monoid are a Boolean algebra. This result also holds for free groups =-=[Ben69]-=- and even virtually-free groups [Sén96]. Further results on rational subsets of groups can be found in [Gru90,KSS06,Ned00]. Some of our results are already known, others appear to be new. Apart from p... |

8 | On regular trace languages - Sakarovitch - 1987 |

7 |
On locally extended residually finite groups,
- Allenby, Gregorac
- 1972
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Citation Context ...3) If H1 and H2 have the LERF property, then also G has the LERF property. Statement (1) is covered by [KWM05, Corollary 5.15], statement (2) was shown in [Bez98,KS70], and statement (3) was shown in =-=[AG73]-=-. Remark 54. There is an alternative way to deduce Theorem 53: using the embedding (102) together with the Theorem 43 on HNN-extensions it suffices to prove that (i) decidability of the generalized wo... |

5 | Formal languages and word-rewriting - Sénizergues |

4 | Positive theories of HNN-extensions and amalgamated free products - Lohrey, Sénizergues - 2005 |

4 | Algorithmic problems in amalgams of finite groups: conjugacy and intersection properties - Markus-Epstein - 2007 |

4 |
The occurrence problem in a rational subset of the free product of groups. Vestn
- Nedbaj
(Show Context)
Citation Context ...n the boolean closure of Rat(H1) and Rat(H2), then the same is true for languages in the boolean closure of Rat(G). Statement (3) is covered by [KSS06, Theorem 5.1] (for free products it was shown in =-=[Ned00]-=-) for the case that H1 and H2 (and hence G) are groups, whereas (1), (2), and (4) are new to the knowledge of the authors. 45sMay 30, 2007 15:16 WSPC/INSTRUCTION FILE 1 6.4. Transfer results for finit... |

3 |
Embedding theorems for semigroups, Quarterly
- Howie
- 1963
(Show Context)
Citation Context ...ss problem for boolean combinations of rational subsets) is undecidable [AH89]. Let us mention that more general notions of HNN-extensions for monoids were defined in the literature (see for instance =-=[How63]-=-) where A and B in (2) are not necessarily groups. It is not clear whether our results hold for these HNN-extensions as well. In Section 4.6 we deal with finitely generated subgroups of HNN-extensions... |

3 | Equations in HNN-extensions
- Lohrey, Sénizergues
- 2006
(Show Context)
Citation Context ...〈L,Q,τ,δ,I,T〉 is said to be unitary if and only if for every type Θ ∈ T6 we have µA,0(1) ∩ (τ −1 (Θ) × τ −1 (Θ)) = IdQ ∩ (τ −1 (Θ) × τ −1 (Θ)). This last technical condition turns out to be useful in =-=[LS06a]-=-. It means that no edge between two different states of the same type is labelled with a set containing 1. Definition 12 (deterministic fta). The partitioned fta A = 〈L,Q,τ,δ,I,T〉 is said to be determ... |

3 | Stallings foldings and subgroups of amalgams of finite groups - Markus-Epstein |

3 |
On the rational subsets of the free group
- Sénizergues
- 1996
(Show Context)
Citation Context ...⊆ M the submonoid of M generated by U). By Kleene’s theorem, the rational subsets of a free monoid are a Boolean algebra. This result also holds for free groups [Ben69] and even virtually-free groups =-=[Sén96]-=-. Further results on rational subsets of groups can be found in [Gru90,KSS06,Ned00]. Some of our results are already known, others appear to be new. Apart from proving these new results, the purpose o... |

2 |
Subgroup separability of graphs of abelian groups
- Metaftsis, Raptis
(Show Context)
Citation Context ...ere exists a subgroup Z of finite index in G with H ≤ Z < G and g ∈ G \ Z. Statement (a) is covered by [KWM05, Corollary 5.15], statement (b) was shown in [Bez98,KS71], and statement (c) was shown in =-=[MR04]-=-. Finally, in Section 6 we briefly sketch the proof of analogous transfer results for amalgamated free products (Theorem 51 and 53). Since the treatment of amalgamated free products exactly follows ou... |

2 | Sur les semi-groupes vérifiant le théorème de Kleene - Reutenauer - 1985 |

1 | On the intersection of subgroups in HNN-groups - Bezverkhniĭ - 1998 |