scholarly journals Admitting a Coarse Embedding is Not Preserved Under Group Extensions

2018 ◽  
Vol 2019 (20) ◽  
pp. 6480-6498 ◽  
Author(s):  
Goulnara Arzhantseva ◽  
Romain Tessera
Keyword(s):  
Hilbert Space ◽  
Finitely Generated ◽  
Group Extensions ◽  
Finitely Generated Group ◽  
Finitely Generated Groups ◽  
Coarse Embedding ◽  
Monster Group

AbstractWe construct a finitely generated group which is an extension of two finitely generated groups coarsely embeddable into Hilbert space but which itself does not coarsely embed into Hilbert space. Our construction also provides a new infinite monster group: the first example of a finitely generated group that does not coarsely embed into Hilbert space and yet does not contain a weakly embedded expander.

scholarly journals BNS INVARIANTS AND ALGEBRAIC FIBRATIONS OF GROUP EXTENSIONS

2021 ◽  
pp. 1-15
Author(s):  
Stefan Friedl ◽  
Stefano Vidussi
Keyword(s):  
Fundamental Group ◽  
Broad Class ◽  
Interesting Case ◽  
Finitely Generated ◽  
Group Extensions ◽  
Finitely Generated Group ◽  
Surface Bundles ◽  
Surface Bundle

Abstract Let G be a finitely generated group that can be written as an extension $$ \begin{align*} 1 \longrightarrow K \stackrel{i}{\longrightarrow} G \stackrel{f}{\longrightarrow} \Gamma \longrightarrow 1 \end{align*} $$ where K is a finitely generated group. By a study of the Bieri–Neumann–Strebel (BNS) invariants we prove that if $b_1(G)> b_1(\Gamma ) > 0$ , then G algebraically fibres; that is, admits an epimorphism to $\Bbb {Z}$ with finitely generated kernel. An interesting case of this occurrence is when G is the fundamental group of a surface bundle over a surface $F \hookrightarrow X \rightarrow B$ with Albanese dimension $a(X) = 2$ . As an application, we show that if X has virtual Albanese dimension $va(X) = 2$ and base and fibre have genus greater that $1$ , G is noncoherent. This answers for a broad class of bundles a question of J. Hillman ([9, Question 11(4)]). Finally, we show that there exist surface bundles over a surface whose BNS invariants have a structure that differs from that of Kodaira fibrations, determined by T. Delzant.

Omitting quantifier-free types in generic structures

1972 ◽  
Vol 37 (3) ◽  
pp. 512-520 ◽  
Author(s):  
Angus Macintyre
Keyword(s):  
Word Problem ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Prove Theorem ◽  
Finitely Generated Groups ◽  
Solvable Word Problem ◽  
Closed Groups ◽  
Central Result ◽  
Generic Structures ◽  
Solvable Word

The central result of this paper was proved in order to settle a problem arising from B. H. Neumann's paper [10].In [10] Neumann proved that if a finitely generated group H is recursively absolutely presentable then H is embeddable in all nontrivial algebraically-closed groups. Harry Simmons [14] clarified this by showing that a finitely generated group H is recursively absolutely presentable if and only if H can be recursively presented with solvable word-problem. Therefore, if a finitely generated group H can be recursively presented with solvable word-problem then H is embeddable in all nontrivial algebraically-closed groups.The problem arises of characterizing those finitely generated groups which are embeddable in all nontrivial algebraically-closed groups. In this paper we prove, by a forcing argument, that if a finitely generated group H is embeddable in all non-trivial algebraically-closed groups then H can be recursively presented with solvable word-problem. Thus Neumann's result is sharp.Our results are obtained by the method of forcing in model-theory, as developed in [1], [12]. Our method of proof has nothing to do with group-theory. We prove general results, Theorems 1 and 2 below, about constructing generic structures without certain isomorphism-types of finitely generated substructures. The formulation of these results requires the notion of Turing degree. As an application of the central result we prove Theorem 3 which gives information about the number of countable K-generic structures.We gratefully acknowledge many helpful conversations with Harry Simmons.

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.

scholarly journals Cup products and group extensions

1991 ◽  
Vol 50 (1) ◽  
pp. 108-115 ◽  
Author(s):  
P. A. Linnell
Keyword(s):  
Commutative Ring ◽  
Finitely Generated ◽  
Group Extensions ◽  
Finitely Generated Group ◽  
Cup Product ◽  
Cup Products

AbstractLet G be a finitely generated group and let R be a commutative ring, regarded as a G-module with G acting trivially. We shall determine when the cup product of two elements of H1(G, R) is zero. Our method will use the interpretation of H2(G, R) as extensions of G by R. This will give an alternative demonstration of results of Hillman and Würfel.

On R0-Closed Classes, and Finitely Generated Groups

1970 ◽  
Vol 22 (1) ◽  
pp. 176-184 ◽  
Author(s):  
Rex Dark ◽  
Akbar H. Rhemtulla
Keyword(s):  
Normal Subgroups ◽  
Finitely Generated ◽  
Maximal Condition ◽  
Soluble Groups ◽  
Finitely Generated Group ◽  
Finitely Generated Groups ◽  
Central Factors ◽  
Subnormal Subgroups

1.1. If a group satisfies the maximal condition for normal subgroups, then all its central factors are necessarily finitely generated. In [2], Hall asked whether there exist finitely generated soluble groups which do not satisfy the maximal condition for normal subgroups but all of whose central factors are finitely generated. We shall answer this question in the affirmative. We shall also construct a finitely generated group all of whose subnormal subgroups are perfect (and which therefore has no non-trivial central factors), but which does not satisfy the maximal condition for normal subgroups. Related to these examples is the question of which classes of finitely generated groups satisfy the maximal condition for normal subgroups. A characterization of such classes has been obtained by Hall, and we shall include his result as our first theorem.

EMBEDDABILITY OF GENERALISED WREATH PRODUCTS

2014 ◽  
Vol 91 (2) ◽  
pp. 250-263 ◽  
Author(s):  
CHRIS CAVE ◽  
DENNIS DREESEN
Keyword(s):  
Hilbert Space ◽  
Wreath Product ◽  
Metric Spaces ◽  
Wreath Products ◽  
Finitely Generated ◽  
Product Construction ◽  
Finitely Generated Groups

AbstractGiven two finitely generated groups that coarsely embed into a Hilbert space, it is known that their wreath product also embeds coarsely into a Hilbert space. We introduce a wreath product construction for general metric spaces $X,Y,Z$ and derive a condition, called the (${\it\delta}$-polynomial) path lifting property, such that coarse embeddability of $X,Y$ and $Z$ implies coarse embeddability of $X\wr _{Z}Y$. We also give bounds on the compression of $X\wr _{Z}Y$ in terms of ${\it\delta}$ and the compressions of $X,Y$ and $Z$.

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 Infinitely presented small cancellation groups have the Haagerup property

2015 ◽  
Vol 07 (03) ◽  
pp. 389-406 ◽  
Author(s):  
Goulnara Arzhantseva ◽  
Damian Osajda
Keyword(s):  
Hilbert Space ◽  
Hyperbolic Groups ◽  
Main Step ◽  
Small Cancellation ◽  
Finitely Generated ◽  
Haagerup Property ◽  
Cancellation Condition ◽  
Finitely Generated Groups ◽  
Direct Limits ◽  
Small Cancellation Condition

We prove the Haagerup property (= Gromov's a-T-menability) for finitely generated groups defined by infinite presentations satisfying the C'(1/6)-small cancellation condition. We deduce that these groups are coarsely embeddable into a Hilbert space and that the strong Baum–Connes conjecture holds for them. The result is a first nontrivial advancement in understanding groups with such properties among infinitely presented non-amenable direct limits of hyperbolic groups. The proof uses the structure of a space with walls introduced by Wise. As the main step we show that C'(1/6)-complexes satisfy the linear separation property.

scholarly journals GROUPS QUASI-ISOMETRIC TO H2×R

2001 ◽  
Vol 64 (1) ◽  
pp. 44-60 ◽  
Author(s):  
ELEANOR G. RIEFFEL
Keyword(s):  
Differential Geometry ◽  
Riemannian Manifolds ◽  
Finite Extension ◽  
Finitely Generated ◽  
Cocompact Lattice ◽  
Finitely Generated Group ◽  
Finitely Generated Groups ◽  
Homogeneous Riemannian Manifolds ◽  
Carry Over ◽  
Simply Connected

The most powerful geometric tools are those of differential geometry, but to apply such techniques to finitely generated groups seems hopeless at first glance since the natural metric on a finitely generated group is discrete. However Gromov recognized that a group can metrically resemble a manifold in such a way that geometric results about that manifold carry over to the group [18, 20]. This resemblance is formalized in the concept of a ‘quasi-isometry’. This paper contributes to an ongoing program to understand which groups are quasi-isometric to which simply connected, homogeneous, Riemannian manifolds [15, 18, 20] by proving that any group quasi-isometric to H2×R is a finite extension of a cocompact lattice in Isom(H2×R) or Isom(SL˜(2, R)).

scholarly journals Topological properties of full groups

2009 ◽  
Vol 30 (2) ◽  
pp. 525-545 ◽  
Author(s):  
JOHN KITTRELL ◽  
TODOR TSANKOV
Keyword(s):  
Hilbert Space ◽  
Equivalence Relation ◽  
Separable Hilbert Space ◽  
Equivalence Relations ◽  
Full Group ◽  
Topological Properties ◽  
Finitely Generated ◽  
Dense Subgroup ◽  
Separable Group ◽  
Finitely Generated Group

AbstractWe study full groups of countable, measure-preserving equivalence relations. Our main results include that they are all homeomorphic to the separable Hilbert space and that every homomorphism from an ergodic full group to a separable group is continuous. We also find bounds for the minimal number of topological generators (elements generating a dense subgroup) of full groups allowing us to distinguish between full groups of equivalence relations generated by free, ergodic actions of the free groups Fm and Fn if m and n are sufficiently far apart. We also show that an ergodic equivalence relation is generated by an action of a finitely generated group if an only if its full group is topologically finitely generated.

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