scholarly journals Farrell–Jones via Dehn fillings

2018 ◽  
Vol 10 (04) ◽  
pp. 873-895 ◽  
Author(s):  
Yago Antolín ◽  
Rémi Coulon ◽  
Giovanni Gandini
Keyword(s):  
Free Product ◽  
Infinite Order ◽  
Dehn Filling ◽  
Residually Finite ◽  
Finite Subgroups ◽  
Dehn Fillings

Following the approach of Dahmani, Guirardel and Osin, we extend the group theoretical Dehn filling theorem to show that the pre-images of infinite order subgroups have a certain structure of a free product. We then apply this result to establish the Farrell–Jones conjecture for groups hyperbolic relative to a family of residually finite subgroups satisfying the Farrell–Jones conjecture, partially recovering a result of Bartels.

Residual finiteness of free products of combinatorial strict inverse semigroups

1994 ◽  
Vol 124 (1) ◽  
pp. 137-147 ◽  
Author(s):  
Karl Auinger
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Inverse Semigroups ◽  
Residual Finiteness ◽  
Free Products ◽  
Residually Finite

It is shown that the free product of two residually finite combinatorial strict inverse semigroups in general is not residually finite. In contrast, the free product of a residually finite combinatorial strict inverse semigroup and a semilattice is residually finite.

scholarly journals Infinitely-ended hyperbolic groups with homeomorphic Gromov boundaries

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
Alexandre Martin ◽  
Jacek Świątkowski
Keyword(s):  
Free Product ◽  
Hyperbolic Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Subgroups ◽  
Necessary And Sufficient ◽  
Gromov Boundary

AbstractWe show that the Gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its factors. We generalize this result to graphs of hyperbolic groups over finite subgroups. Finally, we give a necessary and sufficient condition for the Gromov boundaries of any two hyperbolic groups to be homeomorphic (in terms of the topology of the boundaries of factors in terminal splittings over finite subgroups).

scholarly journals Some properties of endomorphisms in residually finite groups

1977 ◽  
Vol 24 (1) ◽  
pp. 117-120 ◽  
Author(s):  
Ronald Hirshon
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Free Product ◽  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Residually Finite Groups ◽  
A Finite Group ◽  
Torsion Free

AbstractIf ε is an endomorphism of a finitely generated residually finite group onto a subgroup Fε of finite index in F, then there exists a positive integer k such that ε is an isomorphism of Fεk. If K is the kernel of ε, then K is a finite group so that if F is a non trivial free product or if F is torsion free, then ε is an isomorphism on F. If ε is an endomorphism of a finitely generated resedually finite group onto a subgroup Fε (not necessatily of ginite index in F) and if the kernel of ε obeys the minimal condition for subgroups then there exists a positive integer k such that ε is an isomorphism on Fεk.

scholarly journals Residually finite subgroups of locally nilpotent groups

2008 ◽  
Vol 320 (1) ◽  
pp. 81-85 ◽  
Author(s):  
Martyn R. Dixon ◽  
Martin J. Evans ◽  
Howard Smith
Keyword(s):  
Nilpotent Groups ◽  
Residually Finite ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
Finite Subgroups

scholarly journals EXAMPLES OF REDUCIBLE AND FINITE DEHN FILLINGS

2010 ◽  
Vol 19 (05) ◽  
pp. 677-694 ◽  
Author(s):  
SUNGMO KANG
Keyword(s):  
Hyperbolic Manifolds ◽  
Dehn Filling ◽  
Dehn Fillings

If a hyperbolic 3-manifold M admits a reducible and a finite Dehn filling, the distance between the filling slopes is known to be 1. This has been proved recently by Boyer, Gordon and Zhang. The first example of a manifold with two such fillings was given by Boyer and Zhang. In this paper, we give examples of hyperbolic manifolds admitting a reducible Dehn filling and a finite Dehn filling of every type: cyclic, dihedral, tetrahedral, octahedral and icosahedral.

On Nilpotent Products of Cyclic Groups—Reexamined by the Commutator Calculus

1975 ◽  
Vol 27 (6) ◽  
pp. 1185-1210 ◽  
Author(s):  
Hermann V. Waldinger ◽  
Anthony M. Gaglione
Keyword(s):  
Finite Number ◽  
Nilpotent Group ◽  
Free Product ◽  
Infinite Order ◽  
Lower Central Series ◽  
Central Series ◽  
Cyclic Groups ◽  
Collection Process

Ruth R. Struik investigated the nilpotent group , where G is a free product of a finite number of cyclic groups, not all of which are of infinite order, and Gm is the mth subgroup of the lower central series of G. Making use of the “collection process” first given by Philip Hall in [8], she determined completely for 1 ≦ n ≦ p + 1, where p is the smallest prime with the property that it divides the order of at least one of the free factors of G. However, she was unable to proceed beyond n = p + 1.

scholarly journals Near Frattini subgroups of residually finite generalized free products of groups

2001 ◽  
Vol 26 (2) ◽  
pp. 117-121
Author(s):  
Mohammad K. Azarian
Keyword(s):  
Free Product ◽  
Finite Index ◽  
Frattini Subgroup ◽  
Free Products ◽  
Residually Finite ◽  
Identical Relation ◽  
Products Of Groups ◽  
Generalized Free Product ◽  
Amalgamated Subgroup ◽  
Generalized Free Products

LetG=A★HBbe the generalized free product of the groupsAandBwith the amalgamated subgroupH. Also, letλ(G)andψ(G)represent the lower near Frattini subgroup and the near Frattini subgroup ofG, respectively. IfGis finitely generated and residually finite, then we show thatψ(G)≤H, providedHsatisfies a nontrivial identical relation. Also, we prove that ifGis residually finite, thenλ(G)≤H, provided: (i)Hsatisfies a nontrivial identical relation andA,Bpossess proper subgroupsA1,B1of finite index containingH; (ii) neitherAnorBlies in the variety generated byH; (iii)H<A1≤AandH<B1≤B, whereA1andB1each satisfies a nontrivial identical relation; (iv)His nilpotent.

Residually finite subgroups of some countable McLain groups

2009 ◽  
Vol 93 (6) ◽  
pp. 513-520 ◽  
Author(s):  
Giovanni Cutolo ◽  
Howard Smith
Keyword(s):  
Residually Finite ◽  
Finite Subgroups

Packings, free products and residually finite groups

1963 ◽  
Vol 59 (3) ◽  
pp. 555-558 ◽  
Author(s):  
A. M. Macbeath
Keyword(s):  
Free Product ◽  
Finite Groups ◽  
Cardinal Number ◽  
Weak Sense ◽  
Free Products ◽  
Residually Finite ◽  
Residual Properties ◽  
Residually Finite Groups ◽  
The Family ◽  
Discontinuous Groups

In this note a simple principle is explained for constructing a transformation group which is a free product of given transformation groups. The principle does not seem to have been formulated explicitly, though it has been used in a more or less vague form in the theory of discontinuous groups (see, for instance, L. R. Ford, Automorphic functions, vol. I, pp. 56–59). It is perhaps of interest that the formulation given here is purely set-theoretic, without any topology, and that it can apply to any free product, whatever the cardinal number of the set of factors. The principle is used to establish the closure under the formation of countable free product of the family of groups which can be represented as discontinuous subgroups of a certain group of rational projective transformations. (The word ‘discontinuous’ is used here in a weak sense, defined later.) Finally, these results are applied to give a new proof of the theorem of Gruenberg that a free product of residually finite groups is itself residually finite (K. W. Gruenberg: Residual properties of groups, Proc. London Math. Soc. (3), 7 (1957), 29–62. See Corollary (ii) of Theorem 4.1, p. 44). The present proof is completely different from Gruenberg's and seems to be of interest for its own sake, though it does not appear to lead to Gruenberg's other results in this connexion. I have to thank Mr W. J. Harvey and Mr C. Maclachlan for checking a first draft of this paper and pointing out a few errors.

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