On a Class of One-Relator Groups

In this paper, we consider the class of groups G(l, m; k) which are defined by the presentationwhere k, l, m are integers, and |l| > m > 0, k > 0. Groups in this class possess many properties which seem unusual, especially for one-relator groups. The basis for the results obtained below is the determination of endomorphisms.For certain of the groups, we are able to calculate their automorphism groups. One consequence of this is to produce examples of one-relator groups with infinitely generated automorphism groups. This answers a question raised by G. Baumslag (in a colloquium lecture at the University of Waterloo). Our examples are, perhaps, the simplest possible; J. Lewin [10] has found an example of a finitely presented group with an infinitely generated automorphism group.

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Automorphism groups of arithmetically saturated models

Journal of Symbolic Logic ◽

10.2178/jsl/1140641169 ◽

2006 ◽

Vol 71 (1) ◽

pp. 203-216 ◽

Cited By ~ 3

Author(s):

Ermek S. Nurkhaidarov

Keyword(s):

Automorphism Group ◽

Automorphism Groups ◽

Open Subgroup ◽

Peano Arithmetic ◽

Standard System ◽

Permutation Groups ◽

Saturated Model ◽

Saturated Models ◽

Homogeneous Set ◽

Image Position

In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that if M is a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2 be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then SSy(M1) = SSy(M2).We show that if M is a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if some maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. That fact is used to show the main theorem:Let M1, M2be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then for every n < ωHere RT2n is Infinite Ramsey's Theorem stating that every 2-coloring of [ω]n has an infinite homogeneous set. Theorem 0.2 shows that for models of a false arithmetic the converse of Kossak-Schmerl Theorem 0.1 is not true. Using the results of Reverse Mathematics we obtain the following corollary:There exist four countable arithmetically saturated models of Peano Arithmetic such that they have the same standard system but their automorphism groups are pairwise non-isomorphic.

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Rigid homogeneous chains

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100057881 ◽

1981 ◽

Vol 89 (1) ◽

pp. 07-17 ◽

Cited By ~ 7

Author(s):

A. M. W. Glass ◽

Yuri Gurevich ◽

W. Charles Holland ◽

Saharon Shelah

Keyword(s):

Automorphism Group ◽

General Problem ◽

Automorphism Groups ◽

Ordered Sets ◽

First Order ◽

Image Position

Classifying (unordered) sets by the elementary (first order) properties of their automorphism groups was undertaken in (7), (9) and (11). For example, if Ω is a set whose automorphism group, S(Ω), satisfiesthen Ω has cardinality at most ℵ0 and conversely (see (7)). We are interested in classifying homogeneous totally ordered sets (homogeneous chains, for short) by the elementary properties of their automorphism groups. (Note that we use ‘homogeneous’ here to mean that the automorphism group is transitive.) This study was begun in (4) and (5). For any set Ω, S(Ω) is primitive (i.e. has no congruences). However, the automorphism group of a homogeneous chain need not be o-primitive (i.e. it may have convex congruences). Fortunately, ‘o-primitive’ is a property that can be captured by a first order sentence for automorphisms of homogeneous chains. Hence our general problem falls naturally into two parts. The first is to classify (first order) the homogeneous chains whose automorphism groups are o-primitive; the second is to determine how the o-primitive components are related for arbitrary homogeneous chains whose automorphism groups are elementarily equivalent.

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Embeddings into groups with only a few defining relations

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700019066 ◽

1974 ◽

Vol 18 (1) ◽

pp. 1-7 ◽

Cited By ~ 3

Author(s):

W. W. Boone ◽

D. J. Collins

Keyword(s):

Word Problem ◽

Finitely Presented Groups ◽

Fixed Group ◽

Defining Relations ◽

Finitely Presented Group ◽

One Relator Groups ◽

Finitely Presented ◽

Unsolvable Word Problem ◽

Trivial Consequence

It is a trivial consequence of Magnus' solution to the word problem for one-relator groups [9] and the existence of finitely presented groups with unsolvable word problem [4] that not every finitely presented group can be embedded in a one-relator group. We modify a construction of Aanderaa [1] to show that any finitely presented group can be embedded in a group with twenty-six defining relations. It then follows from the well-known theorem of Higman [7] that there is a fixed group with twenty-six defining relations in which every recursively presented group is embedded.

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COHERENT EXTENSION OF PARTIAL AUTOMORPHISMS, FREE AMALGAMATION AND AUTOMORPHISM GROUPS

Journal of Symbolic Logic ◽

10.1017/jsl.2019.32 ◽

2019 ◽

Vol 85 (1) ◽

pp. 199-223 ◽

Cited By ~ 2

Author(s):

DAOUD SINIORA ◽

SŁAWOMIR SOLECKI

Keyword(s):

Automorphism Group ◽

Automorphism Groups ◽

Extension Property ◽

Finite Subgroup ◽

Homogeneous Structure ◽

Extension Theorems ◽

Locally Finite ◽

Finite Structures ◽

Image Position ◽

Fraïssé Class

AbstractWe give strengthened versions of the Herwig–Lascar and Hodkinson–Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraïssé classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraïssé limit of any Fraïssé class that is the class of all ${\cal F}$-free structures (in the Herwig–Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.

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Determination of torsion Abelian groups by their automorphism groups

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700037308 ◽

2003 ◽

Vol 67 (3) ◽

pp. 511-519

Author(s):

P. Schultz ◽

A. Sebeldin ◽

A. L. Sylla

Keyword(s):

Automorphism Group ◽

Automorphism Groups ◽

Abelian Groups ◽

Torsion Group ◽

Cyclic Component ◽

Cyclic Groups

An Abelian torsion group is determined by its automorphism group if and only if its locally cyclic component is determined by its automorphism group. We describe the locally cyclic groups that are determined by their automorphism groups.

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Groups acting freely on R-trees

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700006453 ◽

1991 ◽

Vol 11 (4) ◽

pp. 737-756 ◽

Cited By ~ 2

Author(s):

John W. Morgan ◽

Richard K. Skora

Keyword(s):

Euler Characteristic ◽

Cyclic Group ◽

Abelian Subgroup ◽

Closed Surface ◽

Infinite Cyclic Group ◽

Precise Result ◽

Finitely Presented Group ◽

Hnn Extension ◽

Image Position ◽

Finitely Presented

AbstractIn this paper we study the question of which groups act freely on R-trees. The paper has two parts. The first part concerns groups which contain a non-cyclic, abelian subgroup. The following is the main result in this case.Let the finitely presented group G act freely on an R-tree. If A is a non-cyclic, abelian subgroup of G, then A is contained in an abelian subgroup A′ which is a free factor of G.The second part of the paper concerns groups whch split as an HNN-extension along an infinite cyclic group. Here is one formulation of our main result in that case.Let the finitely presented group G act freely on an R-tree. If G has an HNN-decompositionwhere (s) is infinite cyclic, then there is a subgroup H′ ⊂ H such that either(a); or(b),where S is a closed surface of non-positive Euler characteristic.A slightly different, more precise result is also given.

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Serre’s Property (FA) for automorphism groups of free products

Journal of Group Theory ◽

10.1515/jgth-2019-0140 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Naomi Andrew

Keyword(s):

Automorphism Group ◽

Free Product ◽

Finite Groups ◽

Automorphism Groups ◽

Sufficient Conditions ◽

Free Products ◽

Cyclic Groups ◽

Link Type ◽

Open Case

AbstractWe provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.

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