scholarly journals Relation modules of amalgamated free products and HNN extensions

1989 ◽  
Vol 31 (3) ◽  
pp. 263-270 ◽  
Author(s):  
Torsten Hannebauer
Keyword(s):  
Normal Subgroup ◽  
Exact Sequence ◽  
Short Exact Sequence ◽  
Free Products ◽  
Relation Module ◽  
Amalgamated Free Products ◽  
Hnn Extensions ◽  
Image Position

Let G be a group anda free presentation of G, i.e. a short exact sequence of groups with F free. Conjugation in F induces on = R/R', the abelianized normal subgroup R, the structure of a right G-module (if r∈ R, x∈ F then (r)(xπ) = x-1rxR'). The G-module is called the relation module determined by the presentation (1). For a detailed discussion of this subject we refer to Gruenberg [3].

scholarly journals The growth of certain amalgamated free products and HNN-extensions

1992 ◽  
Vol 52 (3) ◽  
pp. 285-298 ◽  
Author(s):  
M. Edjvet ◽  
D. L. Johnson
Keyword(s):  
Rational Function ◽  
Normal Forms ◽  
Free Products ◽  
Growth Series ◽  
Amalgamated Free Products ◽  
Hnn Extensions ◽  
Rewrite Rules ◽  
Image Position

AbstractHere we mean growth in the sense of Milnor and Gromov. After a brief survey of known results, we compute the growth series of the groups, with respect to generators {x, y}. This is done using minimal normal forms obtained by informal use of judiciously chosen rewrite rules. In both of these examples the growth series is a rational function, and we suspect that this is not the case for the Baumslag-Solitar group.

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.

scholarly journals Non-abelian exterior products of groups and exact sequences in the homology of groups

1987 ◽  
Vol 29 (1) ◽  
pp. 13-19 ◽  
Author(s):  
G. J. Ellis
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Type Theorem ◽  
Exterior Product ◽  
Products Of Groups ◽  
Hopf Formula ◽  
Definition Of ◽  
Image Position ◽  
Exact Sequences

Various authors have obtained an eight term exact sequence in homologyfrom a short exact sequence of groups,the term V varying from author to author (see [7] and [2]; see also [5] for the simpler case where N is central in G, and [6] for the case where N is central and N ⊂ [G, G]). The most satisfying version of the sequence is obtained by Brown and Loday [2] (full details of [2] are in [3]) as a corollary to their van Kampen type theorem for squares of spaces: they give the term V as the kernel of a map G ∧ N → N from a “non-abelian exterior product” of G and N to the group N (the definition of G ∧ N, first published in [2], is recalled below). The two short exact sequencesandwhere F is free, together with the fact that H2(F) = 0 and H3(F) = 0, imply isomorphisms..The isomorphism (2) is essentially the description of H2(G) proved algebraically in [11]. As noted in [2], the isomorphism (3) is the analogue for H3(G) of the Hopf formula for H2(G).

scholarly journals C*-SIMPLE GROUPS: AMALGAMATED FREE PRODUCTS, HNN EXTENSIONS, AND FUNDAMENTAL GROUPS OF 3-MANIFOLDS

2011 ◽  
Vol 03 (04) ◽  
pp. 451-489 ◽  
Author(s):  
PIERRE DE LA HARPE ◽  
JEAN-PHILIPPE PRÉAUX
Keyword(s):  
Sufficient Conditions ◽  
Simple Groups ◽  
Fundamental Groups ◽  
Normal Subgroups ◽  
Free Products ◽  
Amalgamated Free Products ◽  
Hnn Extensions ◽  
Groups Acting On Trees ◽  
Jsj Decompositions ◽  
Fixed Point Sets

We establish sufficient conditions for the C*-simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNN-extensions, and their nontrivial subnormal subgroups; for example normal subgroups of Baumslag–Solitar groups. The second class is that of fundamental groups of compact 3-manifolds, related to the first class by their Kneser–Milnor and JSJ decompositions. Much of our analysis deals with conditions on an action of a group Γ on a tree T which imply the following three properties: abundance of hyperbolic elements, better called strong hyperbolicity, minimality, both on the tree T and on its boundary ∂T, and faithfulness in a strong sense. An important step in this analysis is to identify automorphisms of T which are slender, namely such that their fixed-point sets in ∂T are nowhere dense for the shadow topology.

scholarly journals Projective characters of degree one and the inflation-restriction sequence

1989 ◽  
Vol 46 (2) ◽  
pp. 272-280 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Exact Sequence ◽  
Schur Multiplier ◽  
Original Result ◽  
Image Position ◽  
A Finite Group ◽  
Invariant Element

AbstractLet G be a finite group, α be a fixed cocycle of G and Proj (G, α) denote the set of irreducible projective characters of G lying over the cocycle α.Suppose N is a normal subgroup of G. Then the author shows that there exists a G- invariant element of Proj(N, αN) of degree 1 if and only if [α] is an element of the image of the inflation homomorphism from M(G/N) into M(G), where M(G) denotes the Schur multiplier of G. However in many situations one can produce such G-invariant characters where it is not intrinsically obvious that the cocycle could be inflated. Because of this the author obtains a restatement of his original result using the Lyndon-Hochschild-Serre exact sequence of cohomology. This restatement not only resolves the apparent anomalies, but also yields as a corollary the well-known fact that the inflation-restriction sequence is exact when N is perfect.

On Nonabelian H2 for Profinite Groups

1992 ◽  
Vol 44 (2) ◽  
pp. 388-399
Author(s):  
K.-H. Ulbrich
Keyword(s):  
Normal Subgroup ◽  
Exact Sequence ◽  
Profinite Group ◽  
Profinite Groups ◽  
Image Position

Let G be a profinite group. We define an extension (E, J) of G by a group A to consist of an exact sequence of groups together with a section j : G → E of K satisfying: for some open normal subgroup Sof G, and the map is continuous (A being discrete).

Multiplicities in graded rings II: integral equivalence and the Buchsbaum–Rim multiplicity

1996 ◽  
Vol 119 (3) ◽  
pp. 425-445 ◽  
Author(s):  
D. Kirby ◽  
D. Rees
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Noetherian Ring ◽  
Free Module ◽  
Graded Rings ◽  
Finitely Generated ◽  
Graded Ring ◽  
The Subject ◽  
Integral Equivalence ◽  
Image Position

While this paper is principally a continuation of [5], with as its object the application of sections 6 and 7 of that paper to obtain results related to the Buchsbaum–Rim multiplicity, it also has connections with [8] which are the subject of the first of the four sections. These concern integral equivalence of finitely generated R-modules. where R is an arbitrary noetherian ring. We therefore introduce a finitely generated R-module M and relate to it a short exact sequence (s.e.s.),where F is a free module generated by m elements u1,…, um, and L is generated by elements yj, (j = 1, …, n), of F. We identify the elements u1, …, um with a set of indeterminates X1, …, Xm, and F with the R-module S1 of elements of degree 1 of the graded ring S = R[X1, …, Xm].

Obstructions to Liftings in Commutative Squares

1971 ◽  
Vol 23 (6) ◽  
pp. 977-982 ◽  
Author(s):  
Irwin S. Pressman
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Abelian Category ◽  
One To One ◽  
Image Position ◽  
Zero Class

A commutative square (1) of morphisms is said to have a lifting if there is a morphism λ: B1 → A2 such that λϕ1 = α and ϕ2λ = β1Let us assume that we are working in a fixed abelian category . Therefore, ϕi will have a kernel “Ki” and a cokernel “Ci” for i = 1, 2. Let k : K1 → K2 and c: C1 → C2 denote the canonical morphisms induced by α and β.We shall construct a short exact sequence (s.e.s.)2using the data of (1). We shall prove that (1) has a lifting if and only if k = 0, c = 0, and (2) represents the zero class in Ext1(C1, K2). Furthermore, if (1) has one lifting, then the liftings will be in one-to-one correspondence with the elements of the set |Hom(G1, K2)|.

scholarly journals RATIONAL SUBSETS IN HNN-EXTENSIONS AND AMALGAMATED PRODUCTS

2008 ◽  
Vol 18 (01) ◽  
pp. 111-163 ◽  
Author(s):  
MARKUS LOHREY ◽  
GÉRAUD SÉNIZERGUES
Keyword(s):  
Structural Properties ◽  
Finitely Generated ◽  
Free Products ◽  
Amalgamated Free Products ◽  
Hnn Extensions ◽  
Amalgamated Products

Several transfer results for rational subsets and finitely generated subgroups of HNN-extensions G = 〈 H,t; t-1 at = φ(a) (a ∈ A) 〉 and amalgamated free products G = H *A J such that the associated subgroup A is finite. These transfer results allow to transfer decidability properties or structural properties from the subgroup H (resp. the subgroups H and J) to the group G.

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