scholarly journals Gröbner-Shirshov Bases for Some One-relator Groups

2012 ◽  
Vol 19 (01) ◽  
pp. 99-116 ◽  
Author(s):  
Yuqun Chen ◽  
Chanyan Zhong
Keyword(s):  
Normal Form ◽  
Embedding Theorem ◽  
Hnn Extensions ◽  
Standard Normal ◽  
One Relator Groups

In this paper, we prove that two-generator one-relator groups with depth less than or equal to 3 can be effectively embedded into a tower of HNN-extensions in which each group has the effective standard normal form. We give an example to show how to deal with some general cases for one-relator groups. By using the Magnus method and Composition-Diamond Lemma, we reprove the Higman-Neumann-Neumann embedding theorem.

scholarly journals Separating conjugates in amalgamated free products and HNN extensions

1980 ◽  
Vol 29 (1) ◽  
pp. 35-51 ◽  
Author(s):  
Joan L. Dyer
Keyword(s):  
Nilpotent Groups ◽  
Conjugacy Classes ◽  
Free Products ◽  
Amalgamated Free Products ◽  
Hnn Extensions ◽  
Nontrivial Center ◽  
One Relator Groups ◽  
Amalgamated Subgroup ◽  
Conjugacy Separable ◽  
Base Group

AbstractA group G is termed conjugacy separable (c.s.) if any pair of distinct conjugacy classes may be mapped to distinct conjugacy classes in some finite epimorph of G. The free product of A and B with cyclic amalgamated subgroup H is shown to be c.s. if A and B are both free, or are both finitely generated nilpotent groups. Further, one-relator groups with nontrivial center and HNN extensions with c.s. base group and finite associated subgroups are also c.s.

QUASI-ISOMETRIES AND AMALGAMATIONS OF TAME COMBABLE GROUPS

1995 ◽  
Vol 05 (02) ◽  
pp. 199-204 ◽  
Author(s):  
STEPHEN G. BRICK
Keyword(s):  
Analogous Result ◽  
Finitely Generated ◽  
Hnn Extensions ◽  
One Relator Groups

We study the property of tame combability for groups. We show that quasi-isometries preserve this property. We prove that an amalgamation, A *C B, where C is finitely generated, is tame combable iff both A and B are. An analogous result is obtained for HNN extensions. And we show that all one-relator groups are tame combable.

scholarly journals Refinements and higher-order beliefs: a unified survey

2019 ◽  
Vol 71 (1) ◽  
pp. 7-34 ◽  
Author(s):  
Atsushi Kajii ◽  
Stephen Morris
Keyword(s):  
Game Theory ◽  
Normal Form ◽  
Econ Theory ◽  
Complete Information ◽  
Higher Order ◽  
Economic Modelling ◽  
The Game Theory ◽  
Standard Normal ◽  
Payoff Uncertainty ◽  
Theory Interpretation

AbstractThis paper presents a simple framework that allows us to survey and relate some different strands of the game theory literature. We describe a “canonical” way of adding incomplete information to a complete information game. This framework allows us to give a simple “complete theory” interpretation (Kreps in Game theory and economic modelling. Clarendon Press, Oxford, 1990) of standard normal form refinements such as perfection, and to relate refinements both to the “higher-order beliefs literature” (Rubinstein in Am Econ Rev 79:385–391, 1989; Monderer and Samet in Games Econ Behav 1:170–190, 1989; Morris et al. in Econ J Econ Soc 63:145–157, 1995; Kajii and Morris in Econ J Econ Soc 65:1283–1309, 1997a) and the “payoff uncertainty approach” (Fudenberg et al. in J Econ Theory 44:354–380, 1988; Dekel and Fudenberg in J Econ Theory 52:243–267, 1990).

scholarly journals On one relator groups and HNN extensions

1973 ◽  
Vol 16 (2) ◽  
pp. 249-256 ◽  
Author(s):  
James McCool ◽  
Paul E. Schupp
Keyword(s):  
Finite Order ◽  
Word Problem ◽  
Point Of View ◽  
Hnn Extensions ◽  
One Relator Groups

In his work [5] on subgroups of one relator groups, Moldavanski observed that if G is a one relator group whose defining relator R is cyclically reduced and has exponent sum zero on some generator occurring in it, then G is an HNN extension of a one relator group H whose defining relator is shorter than R. This observation, together with Britton's Lemma, can be used to give rather easy proofs of the basic results on one relator groups. To exposit this point of view, we give here a proof of the Freiheitssatz, the solvability of the word problem for one relator groups, and the theorem classifying elements of finite order in one relator groups. In particular, the solution obtained for the word problem is often easy to apply. We also give a proof of the “Spelling Theorem” of Newman [6].

VIRTUAL PROPERTIES OF CYCLICALLY PINCHED ONE-RELATOR GROUPS

2009 ◽  
Vol 19 (02) ◽  
pp. 213-227 ◽  
Author(s):  
GILBERT BAUMSLAG ◽  
BENJAMIN FINE ◽  
CHARLES F. MILLER ◽  
DOUGLAS TROEGER
Keyword(s):  
Free Group ◽  
Cyclic Subgroup ◽  
Free Groups ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
One Relator Groups ◽  
Special Case ◽  
Torsion Free ◽  
Amalgamated Product

We prove that the amalgamated product of free groups with cyclic amalgamations satisfying certain conditions are virtually free-by-cyclic. In case the cyclic amalgamated subgroups lie outside the derived group such groups are free-by-cyclic. Similarly a one-relator HNN-extension in which the conjugated elements either coincide or are independent modulo the derived group is shown to be free-by-cyclic. In general, the amalgamated product of free groups with cyclic amalgamations is free-by-(torsion-free nilpotent). The special case of the double of a free group amalgamating a cyclic subgroup is shown to be virtually free-by-abelian. Analagous results are obtained for certain one-relator HNN-extensions.

HNN extensions of inverse semigroups with zero

2007 ◽  
Vol 142 (1) ◽  
pp. 25-39 ◽  
Author(s):  
E. R. DOMBI ◽  
N. D. GILBERT
Keyword(s):  
Normal Form ◽  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Universal Group ◽  
Main Application ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
Natural Way

AbstractWe study a construction of an HNN extension for inverse semigroups with zero. We prove a normal form for the elements of the universal group of an inverse semigroup that is categorical at zero, and use it to establish structural results for the universal group of an HNN extension. Our main application of the HNN construction is to show that graph inverse semigroups –including the polycyclic monoids –admit HNN decompositions in a natural way, and that this leads to concise presentations for them.

scholarly journals HNN-extensions of algebras and applications

1984 ◽  
Vol 29 (2) ◽  
pp. 215-229
Author(s):  
Hans-Christian Mez
Keyword(s):  
Conjugacy Class ◽  
Additional Condition ◽  
Main Part ◽  
Regular Ring ◽  
Embedding Theorem ◽  
Closed Group ◽  
Existentially Closed ◽  
Hnn Extensions ◽  
Associative Rings ◽  
Order Sequence

The classic HNN-embedding theorem for groups does not transfer to associative rings or algebras. In its first part this paper presents constructions which provide such a theorem if an additional condition is put on the isomorphic subalgebras or if one restricts to algebras over fields and drops the associativity. The main part of the paper deals with applications of these results. For example, it is known that every existentially closed group is ω-homogeneous. It is shown that the corresponding is false for existentially closed associative Δ-algebras but true for existentially universal nonassociative K-algebras. Further-more, orthogonal sequences of idempotents in existentially closed associative Δ-algebras over a regular ring Δ are investigated. It is shown that the conjugacy class of such a sequence depends only on a corresponding order sequence. In particular, in every existentially closed K-algebra all idempotents different from 0 and 1 are conjugated.

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