scholarly journals Tameness and geodesic cores of subgroups

2000 ◽  
Vol 69 (2) ◽  
pp. 153-161 ◽  
Author(s):  
Rita Gitik
Keyword(s):  
Normal Subgroup ◽  
Factor Group ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Negatively Curved ◽  
Quasiconvex Subgroups ◽  
Trivial Subgroup

AbstractLet N be a finitely generated normal subgroup of a finitely generated group G. We show that if the trivial subgroup is tame in the factor group G/N, then N is that in G. We also give a short new proof of the fact that quasiconvex subgroups of negatively curved groups are tame. The proof utilizes the concept of the geodesic core of the subgroup and is related to the Dehn algorithm.

Second Fox Subgroups of Arbitrary Groups

1995 ◽  
Vol 38 (2) ◽  
pp. 177-181 ◽  
Author(s):  
Mario Curzio ◽  
C. Kanta Gupta
Keyword(s):  
Normal Subgroup ◽  
Finitely Generated ◽  
Finitely Generated Group

AbstractWe give a complete description of the second Fox subgroup G ∩ (1 + Δ2(G)Δ(H)) relative to a given normal subgroup H of an arbitrary finitely generated group G.

scholarly journals On a theorem of Avez

2019 ◽  
Vol 22 (3) ◽  
pp. 383-395
Author(s):  
Murray Elder ◽  
Cameron Rogers
Keyword(s):  
Normal Subgroup ◽  
Probability Measure ◽  
Finitely Generated ◽  
Amenable Groups ◽  
Finitely Generated Group ◽  
Amenable Radical

Abstract For each symmetric, aperiodic probability measure μ on a finitely generated group G, we define a subset {A_{\mu}} consisting of group elements g for which the limit of the ratio {{\mu^{\ast n}(g)}/{\mu^{\ast n}(e)}} tends to 1. We prove that {A_{\mu}} is a subgroup, is amenable, contains every finite normal subgroup, and {G=A_{\mu}} if and only if G is amenable. For non-amenable groups we show that {A_{\mu}} is not always a normal subgroup and can depend on the measure. We formulate some conjectures relating {A_{\mu}} to the amenable radical.

A Note on the Hopficity of Finitely Generated Free-by-Nilpotent Groups

1977 ◽  
Vol 20 (1) ◽  
pp. 33-34
Author(s):  
James Boler
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Finitely Generated Group

Our purpose is to deduce from a theorem of P. Hall the following observation.Let G be a finitely generated group and F a free normal subgroup of G with G/F nilpotent. Then G is hopfian.Here a group G is hopfian if every epiendomorphism G→G is an automorphism.

A note on a centrality property of finitely generated soluble groups

1974 ◽  
Vol 75 (1) ◽  
pp. 23-24 ◽  
Author(s):  
J. C. Lennox
Keyword(s):  
Normal Subgroup ◽  
Positive Integer ◽  
Finite Index ◽  
Finitely Generated ◽  
Abelian Normal Subgroup ◽  
Soluble Groups ◽  
Finitely Generated Group ◽  
Torsion Free ◽  
Simpler Method

We recall from (3) that a group G is (centrally) eremitic if there exists a positive integer e such that, whenever an element of G has some power in a centralizer, it has its eth power. The eccentricity of an eremitic group G is the least such positive integer e.In ((4), Theorem A) we proved that if A is a torsion free Abelian normal subgroup of a finitely generated group G with G/A nilpotent, then G has a subgroup of finite index with eccentricity 1. In this note we use a much simpler method to prove a stronger result.

scholarly journals Large normal subgroup growth and large characteristic subgroup growth

2020 ◽  
Vol 23 (1) ◽  
pp. 1-15
Author(s):  
Yiftach Barnea ◽  
Jan-Christoph Schlage-Puchta
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Open Subgroup ◽  
Positive Answer ◽  
Characteristic Subgroup ◽  
Finitely Generated ◽  
Growth Type ◽  
Subgroup Growth ◽  
Finitely Generated Group ◽  
Standing Problem

AbstractThe fastest normal subgroup growth type of a finitely generated group is {n^{\log n}}. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let Γ be a group and Δ a subgroup of finite index. Suppose Δ has normal subgroup growth of type {n^{\log n}}. Does Γ have normal subgroup growth of type {n^{\log n}}? We give a positive answer in some cases, generalizing a result of Müller and the second author and a result of Gerdau. For instance, suppose G is a profinite group and H an open subgroup of G. We show that if H is a generalized Golod–Shafarevich group, then G has normal subgroup growth of type {n^{\log n}}. We also use our methods to show that one can find a group with characteristic subgroup growth of type {n^{\log n}}.

scholarly journals On Characteristic Morphisms: In Memoriam Thomas Macfarland Cherry

1969 ◽  
Vol 9 (3-4) ◽  
pp. 478-488 ◽  
Author(s):  
B. H. Neumann
Keyword(s):  
Normal Subgroup ◽  
Factor Group ◽  
In Memoriam

This note is concerned with a translation of some concepts and results about characteristic subgroups of a group into the language of categories. As an example, consider strictly characteristic and hypercharacteristic subgroups of a group: the subgroup H of the group G is called strictly characteristic in G if it admits all ependomorphisms of G; that is all homomorphic mappings of G onto G; and H is called hypercharacteristic2 in G if it is the least normal subgroup with factor group isomorphic to G/H, that is if H is contained in every normal subgroup K of G with G/K ≅ G/H.

Acylindrical hyperbolicity, non-simplicity and SQ\hbox{-}universality of groups splitting over ℤ

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Jack O. Button
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Group

AbstractWe show, using acylindrical hyperbolicity, that a finitely generated group splitting over

ALGEBRO-GEOMETRIC INVARIANTS OF GROUPS (THE DIMENSION SEQUENCE OF A REPRESENTATION VARIETY)

2011 ◽  
Vol 21 (04) ◽  
pp. 595-614 ◽  
Author(s):  
S. LIRIANO ◽  
S. MAJEWICZ
Keyword(s):  
Algebraic Group ◽  
Free Group ◽  
Algebraic Variety ◽  
Commutator Subgroup ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Geometric Invariants ◽  
Torus Knot ◽  
Irreducible Components ◽  
Irreducible Variety

If G is a finitely generated group and A is an algebraic group, then RA(G) = Hom (G, A) is an algebraic variety. Define the "dimension sequence" of G over A as Pd(RA(G)) = (Nd(RA(G)), …, N0(RA(G))), where Ni(RA(G)) is the number of irreducible components of RA(G) of dimension i (0 ≤ i ≤ d) and d = Dim (RA(G)). We use this invariant in the study of groups and deduce various results. For instance, we prove the following: Theorem A.Let w be a nontrivial word in the commutator subgroup ofFn = 〈x1, …, xn〉, and letG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety andV-1 = {ρ | ρ ∈ RSL(2, ℂ)(Fn), ρ(w) = -I} ≠ ∅, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). Theorem B.Let w be a nontrivial word in the free group on{x1, …, xn}with even exponent sum on each generator and exponent sum not equal to zero on at least one generator. SupposeG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). We also show that if G = 〈x1, . ., xn, y; W = yp〉, where p ≥ 1 and W is a word in Fn = 〈x1, …, xn〉, and A = PSL(2, ℂ), then Dim (RA(G)) = Max {3n, Dim (RA(G′)) +2 } ≤ 3n + 1 for G′ = 〈x1, …, xn; W = 1〉. Another one of our results is that if G is a torus knot group with presentation 〈x, y; xp = yt〉 then Pd(RSL(2, ℂ)(G))≠Pd(RPSL(2, ℂ)(G)).

A finitely generated, infinitely related group with trivial multiplicator

1971 ◽  
Vol 5 (1) ◽  
pp. 131-136 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Metabelian Group ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Related Group ◽  
Finitely Related

We exhibit a 3-generator metabelian group which is not finitely related but has a trivial multiplicator.1. The purpose of this note is to establish the exitense of a finitely generated group which is not finitely related, but whose multiplecator is finitely generated. This settles negatively a question whichb has been open for a few years (it was first brought to my attention by Michel Kervaire and Joan Landman Dyer in 1964, but I believe it is somewhat older). The group is given in the follwing theorem.

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