scholarly journals Finitely generated groups L with L ≈ L × M, M ≠ 1, M finitely presented

1986 ◽  
Vol 99 (1) ◽  
pp. 232-238 ◽  
Author(s):  
R Hirshon
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Finitely Presented

scholarly journals Groups with a quotient that contains the original group as a direct factor

1992 ◽  
Vol 45 (3) ◽  
pp. 513-520 ◽  
Author(s):  
Ron Hirshon ◽  
David Meier
Keyword(s):  
Free Group ◽  
Normal Subgroups ◽  
Direct Factor ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Finitely Generated Groups ◽  
Original Group ◽  
Hopfian Group ◽  
Finitely Presented

We prove that given a finitely generated group G with a homomorphism of G onto G × H, H non-trivial, or a finitely generated group G with a homomorphism of G onto G × G, we can always find normal subgroups N ≠ G such that G/N ≅ G/N × H or G/N ≅ G/N × G/N respectively. We also show that given a finitely presented non-Hopfian group U and a homomorphism φ of U onto U, which is not an isomorphism, we can always find a finitely presented group H ⊇ U and a finitely generated free group F such that φ induces a homomorphism of U * F onto (U * F) × H. Together with the results above this allows the construction of many examples of finitely generated groups G with G ≅ G × H where H is finitely presented. A finitely presented group G with a homomorphism of G onto G × G was first constructed by Baumslag and Miller. We use a slight generalisation of their method to obtain more examples of such groups.

A QUASI-ISOMETRY INVARIANT LOOP SHORTENING PROPERTY FOR GROUPS

2008 ◽  
Vol 18 (08) ◽  
pp. 1243-1257 ◽  
Author(s):  
STEPHEN G. BRICK ◽  
JON M. CORSON ◽  
DOHYOUNG RYANG
Keyword(s):  
Isoperimetric Inequality ◽  
Metric Spaces ◽  
Cayley Graphs ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Quadratic Isoperimetric Inequality ◽  
Finitely Presented

We first introduce a loop shortening property for metric spaces, generalizing the property considered by M. Elder on Cayley graphs of finitely generated groups. Then using this metric property, we define a very broad loop shortening property for finitely generated groups. Our definition includes Elder's groups, and unlike his definition, our property is obviously a quasi-isometry invariant of the group. Furthermore, all finitely generated groups satisfying this general loop shortening property are also finitely presented and satisfy a quadratic isoperimetric inequality. Every CAT(0) cubical group is shown to have this general loop shortening property.

scholarly journals FINITELY ANNIHILATED GROUPS

2014 ◽  
Vol 90 (3) ◽  
pp. 404-417 ◽  
Author(s):  
MAURICE CHIODO
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Infinite Family ◽  
Finitely Generated ◽  
Perfect Group ◽  
Finitely Generated Groups ◽  
Finitely Presented

AbstractIn 1976, Wiegold asked if every finitely generated perfect group has weight 1. We introduce a new property of groups, finitely annihilated, and show that this might be a possible approach to resolving Wiegold’s problem. For finitely generated groups, we show that in several classes (finite, solvable, free), being finitely annihilated is equivalent to having noncyclic abelianisation. However, we also construct an infinite family of (finitely presented) finitely annihilated groups with cyclic abelianisation. We apply our work to show that the weight of a nonperfect finite group, or a nonperfect finitely generated solvable group, is the same as the weight of its abelianisation. This recovers the known partial results on the Wiegold problem: a finite (or finitely generated solvable) perfect group has weight 1.

scholarly journals Hopfian and co-Hopfian groups

1997 ◽  
Vol 56 (1) ◽  
pp. 17-24 ◽  
Author(s):  
Satya Deo ◽  
K. Varadarajan
Keyword(s):  
Fundamental Group ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Generated Groups ◽  
Aspherical Manifold ◽  
Finitely Presented

The main results proved in this note are the following:(i) Any finitely generated group can be expressed as a quotient of a finitely presented, centreless group which is simultaneously Hopfian and co-Hopfian.(ii) There is no functorial imbedding of groups (respectively finitely generated groups) into Hopfian groups.(iii) We prove a result which implies in particular that if the double orientable cover N of a closed non-orientable aspherical manifold M has a co-Hopfian fundamental group then π1(M) itself is co-Hopfian.

scholarly journals Computability of Følner sets

2017 ◽  
Vol 27 (07) ◽  
pp. 819-830 ◽  
Author(s):  
Matteo Cavaleri
Keyword(s):  
Word Problem ◽  
Finitely Generated ◽  
Amenable Groups ◽  
Finitely Generated Groups ◽  
Distortion Functions ◽  
Solvable Word Problem ◽  
Følner Sets ◽  
Finitely Presented ◽  
Unsolvable Word Problem ◽  
Solvable Word

We define the notion of computability of Følner sets for finitely generated amenable groups. We prove, by an explicit description, that the Kharlampovich groups, finitely presented solvable groups with unsolvable Word Problem, have computable Følner sets. We also prove computability of Følner sets for extensions — with subrecursive distortion functions — of amenable groups with solvable Word Problem by finitely generated groups with computable Følner sets. Moreover, we obtain some known and some new upper bounds for the Følner function for these particular extensions.

scholarly journals TRANSVERSALS AS GENERATING SETS IN FINITELY GENERATED GROUPS

2015 ◽  
Vol 93 (1) ◽  
pp. 47-60
Author(s):  
JACK BUTTON ◽  
MAURICE CHIODO ◽  
MARIANO ZERON-MEDINA LARIS
Keyword(s):  
Finite Index ◽  
Finitely Generated ◽  
Primitive Elements ◽  
Generating Set ◽  
Generating Sets ◽  
Finitely Generated Groups ◽  
Formula Group ◽  
Finite Index Subgroups ◽  
Finitely Presented

We explore transversals of finite index subgroups of finitely generated groups. We show that when $H$ is a subgroup of a rank-$n$ group $G$ and $H$ has index at least $n$ in $G$, we can construct a left transversal for $H$ which contains a generating set of size $n$ for $G$; this construction is algorithmic when $G$ is finitely presented. We also show that, in the case where $G$ has rank $n\leq 3$, there is a simultaneous left–right transversal for $H$ which contains a generating set of size $n$ for $G$. We finish by showing that if $H$ is a subgroup of a rank-$n$ group $G$ with index less than $3\cdot 2^{n-1}$, and $H$ contains no primitive elements of $G$, then $H$ is normal in $G$ and $G/H\cong C_{2}^{n}$.

Collectives of Automata in Finitely Generated Groups

2020 ◽  
Vol 108 (5-6) ◽  
pp. 671-678
Author(s):  
D. V. Gusev ◽  
I. A. Ivanov-Pogodaev ◽  
A. Ya. Kanel-Belov
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Groups
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