scholarly journals A Tits alternative for topological full groups

2019 ◽  
Vol 41 (2) ◽  
pp. 622-640
Author(s):  
NÓRA GABRIELLA SZŐKE
Keyword(s):  
Free Group ◽  
Free Action ◽  
The Other ◽  
Full Group ◽  
Finitely Generated ◽  
Cantor Space ◽  
Cyclic Groups ◽  
Finitely Generated Groups ◽  
Tits Alternative ◽  
The One

We prove a Tits alternative for topological full groups of minimal actions of finitely generated groups. On the one hand, we show that topological full groups of minimal actions of virtually cyclic groups are amenable. By doing so, we generalize the result of Juschenko and Monod for $\mathbf{Z}$-actions. On the other hand, when a finitely generated group $G$ is not virtually cyclic, then we construct a minimal free action of $G$ on a Cantor space such that the topological full group contains a non-abelian free group.

scholarly journals New exotic 4-manifolds via Luttinger surgery on Lefschetz fibrations

2015 ◽  
Vol 26 (01) ◽  
pp. 1550010 ◽  
Author(s):  
Anar Akhmedov ◽  
Kadriye Nur Saglam
Keyword(s):  
Fundamental Group ◽  
Free Product ◽  
Free Group ◽  
Building Blocks ◽  
Finitely Generated ◽  
Lefschetz Fibrations ◽  
Cyclic Groups ◽  
Finitely Generated Groups ◽  
Higher Genus ◽  
Symplectic Cohomology

In [Small exotic 4-manifolds, Algebr. Geom. Topol.8 (2008) 1781–1794], the first author constructed the first known example of exotic minimal symplectic[Formula: see text] and minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to [Formula: see text]. The construction in [Small exotic 4-manifolds, Algebr. Geom. Topol.8 (2008) 1781–1794] uses Yukio Matsumoto's genus two Lefschetz fibrations on [Formula: see text] over 𝕊2 along with the fake symplectic 𝕊2 × 𝕊2 construction given in [Construction of symplectic cohomology 𝕊2 × 𝕊2, Proc. Gökova Geom. Topol. Conf.14 (2007) 36–48]. The main goal in this paper is to generalize the construction in [Small exotic 4-manifolds, Algebr. Geom. Topol.8 (2008) 1781–1794] using the higher genus versions of Matsumoto's fibration constructed by Mustafa Korkmaz and Yusuf Gurtas on [Formula: see text] for any k ≥ 2 and n = 1, and k ≥ 1 and n ≥ 2, respectively. Using our symplectic building blocks, we also construct new symplectic 4-manifolds with the free group of rank s ≥ 1, the free product of the finite cyclic groups, and various other finitely generated groups as the fundamental group.

FREE PRODUCT, PROFINITE TOPOLOGY AND FINITELY GENERATED SUBGROUPS

2001 ◽  
Vol 11 (02) ◽  
pp. 171-184 ◽  
Author(s):  
THIERRY COULBOIS
Keyword(s):  
Free Product ◽  
Free Group ◽  
Finite Groups ◽  
The Other ◽  
Finitely Generated ◽  
Product Operation ◽  
Other Hand ◽  
The One ◽  
Profinite Topology

We consider the following property for a group G:(RZn)ifH1,…,Hnare finitely generated subgroups of G then the setH1 H2⋯ Hn= {h1 ⋯ hn| h1∈ H1, …,hn∈ Hn}is closed with respect to the profinite topology of G. It is obvious that finite groups and finitely generated commutative groups have the property ( RZ n). L. Ribes and P. Zalesskiĭ proved that any free group has ( RZ n). We show that the property ( RZ n) is stable under the free product operation. We use techniques developed by B. Herwig and D. Lascar on the one hand, R. Gitik on the other hand.

scholarly journals Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sam Shepherd ◽  
Daniel J. Woodhouse
Keyword(s):  
Free Group ◽  
Large Family ◽  
Finitely Generated ◽  
Free Vertex ◽  
Cyclic Subgroups ◽  
Graphs Of Groups ◽  
Jsj Decomposition ◽  
Finitely Generated Groups ◽  
Hnn Extensions ◽  
Abelian Subgroups

Abstract We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let G be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that are not quadratically hanging. Our main result is that any group quasi-isometric to G is abstractly commensurable to G. In particular, our result applies to certain “generic” HNN extensions of a free group over cyclic subgroups.

Simple groups and irreducible lattices in wreath products

2020 ◽  
pp. 1-12 ◽  
Author(s):  
ADRIEN LE BOUDEC
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Free Group ◽  
Wreath Products ◽  
Simple Groups ◽  
Finitely Generated ◽  
Locally Compact ◽  
Regular Tree ◽  
Finitely Generated Groups ◽  
A Finite Group

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$ , where $C$ is a finite group and $F$ a non-abelian free group.

scholarly journals Groups with finitely many conjugacy classes of subgroups with large subnormal defect

1995 ◽  
Vol 37 (1) ◽  
pp. 69-71 ◽  
Author(s):  
Howard Smith
Keyword(s):  
Positive Integer ◽  
Free Group ◽  
Conjugacy Classes ◽  
The Other ◽  
Chief Factor ◽  
Simple Groups ◽  
Finitely Generated ◽  
Finite Image ◽  
Chief Factors ◽  
Locally Graded Groups

Given a group G and a positive integer k, let vk(G) denote the number of conjugacy classes of subgroups of G which are not subnormal of defect at most k. Groups G such that vkG) < ∝ for some k are considered in Section 2 of [1], and Theorem 2.4 of that paper states that an infinite group G for which vk(G) < ∝ (for some k) is nilpotent provided only that all chief factors of G are locally (soluble or finite). Now it is easy to see that a group G whose chief factors are of this type is locally graded, that is, every nontrivial, finitely generated subgroup F of G has a nontrivial finite image (since there is a chief factor H/K of G such that F is contained in H but not in K). On the other hand, every (locally) free group is locally graded and so there is in general no restriction on the chief factors of such groups. The class of locally graded groups is a suitable class to consider if one wishes to do no more than exclude the occurrence of finitely generated, infinite simple groups and, in particular, Tarski p-groups. As pointed out in [1], Ivanov and Ol'shanskiĭ have constructed (finitely generated) infinite simple groups all of whose proper nontrivial subgroups are conjugate; clearly a group G with this property satisfies v1(G) = l. The purpose of this note is to provide the following generalization of the above-mentioned theorem from [1].

scholarly journals ON GROUPS WHOSE GEODESIC GROWTH IS POLYNOMIAL

2012 ◽  
Vol 22 (05) ◽  
pp. 1250048 ◽  
Author(s):  
MARTIN R. BRIDSON ◽  
JOSÉ BURILLO ◽  
MURRAY ELDER ◽  
ZORAN ŠUNIĆ
Keyword(s):  
Nilpotent Group ◽  
Finite Index ◽  
The Other ◽  
Normal Closure ◽  
Finitely Generated ◽  
Generating Set ◽  
Finitely Generated Group ◽  
Cyclic Groups ◽  
Growth Functions ◽  
Generating Sets

This paper records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely generated group G has an element whose normal closure is abelian and of finite index, then G has a finite generating set with respect to which the geodesic growth is polynomial (this includes all virtually cyclic groups).

Hierarchy for groups acting on hyperbolic ℤn-spaces

2021 ◽  
pp. 1-28
Author(s):  
Andrei-Paul Grecianu ◽  
Alexei Myasnikov ◽  
Denis Serbin
Keyword(s):  
Abelian Group ◽  
Systematic Study ◽  
Metric Spaces ◽  
Free Groups ◽  
Lexicographic Order ◽  
Finitely Generated ◽  
Princeton University ◽  
Finitely Generated Groups ◽  
The One ◽  
The Right

In [A.-P. Grecianu, A. Kvaschuk, A. G. Myasnikov and D. Serbin, Groups acting on hyperbolic [Formula: see text]-metric spaces, Int. J. Algebra Comput. 25(6) (2015) 977–1042], the authors initiated a systematic study of hyperbolic [Formula: see text]-metric spaces, where [Formula: see text] is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case [Formula: see text] taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic [Formula: see text]-metric spaces. Under certain constraints, the structure of such groups is described in terms of a hierarchy (see [D. T. Wise, The Structure of Groups with a Quasiconvex Hierarchy[Formula: see text][Formula: see text]AMS-[Formula: see text], Annals of Mathematics Studies (Princeton University Press, 2021)]) similar to the one established for [Formula: see text]-free groups in [O. Kharlampovich, A. G. Myasnikov, V. N. Remeslennikov and D. Serbin, Groups with free regular length functions in [Formula: see text], Trans. Amer. Math. Soc. 364 (2012) 2847–2882].

ON ALGORITHMIC SOLVABILITY OF THE PROBLEM OF GENERALIZED CONJUGACY OF SUBGROUPS IN COXETER GROUPS WITH A TREE STRUCTURE

2021 ◽  
Vol 1 (2) ◽  
pp. 7-14
Author(s):  
I. V. Dobrynina ◽  
◽  
E. L. Turenova ◽  
Keyword(s):  
Coxeter Groups ◽  
Conjugacy Problem ◽  
Tree Structure ◽  
The Other ◽  
Combinatorial Group Theory ◽  
Finitely Generated ◽  
Algorithmic Problems ◽  
Algorithmic Solvability ◽  
The One ◽  
Combinatorial Group

The main algorithmic problems of combinatorial group theory posed by M. Den and G. Titze at the beginning of the twentieth century are the problems of word, word conjugacy and of group isomorphism. However, these problems, as follows from the results of P.S. Novikov and S.I. Adyan, turned out to be unsolvable in the class of finitely defined groups. Therefore, algorithmic problems began to be considered in specific classes of groups. The word conjugacy problem allows for two generalizations. On the one hand, we consider the problem of conjugacy of subgroups, that is, the problem of constructing an algorithm that allows for any two finitely generated subgroups to determine whether they are conjugate or not. On the other hand, the problem of generalized conjugacy of words is posed, that is, the problem of constructing an algorithm that allows for any two finite sets of words to determine whether they are conjugated or not. Combining both of these generalizations into one, we obtain the problem of generalized conjugacy of subgroups. Coxeter groups were introduced in the 30s of the last century, and the problems of equality and conjugacy of words are algorithmically solvable in them. To solve other algorithmic problems, various subclasses are distinguished. This is partly due to the unsolvability in Coxeter groups of another important problem – the problem of occurrence, that is, the problem of the existence of an algorithm that allows for any word and any finitely generated subgroup of a certain group to determine whether this word belongs to this subgroup or not. The paper proves the algorithmic solvability of the problem of generalized conjugacy of subgroups in Coxeter groups with a tree structure.

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 Nonconsensual neurocorrectives, bypassing, and free action

2021 ◽  
Author(s):  
Gabriel De Marco
Keyword(s):  
Negative Impact ◽  
Free Action ◽  
Mental States ◽  
The State ◽  
The Other ◽  
Criminal Offenders ◽  
The One

AbstractAs neuroscience progresses, we will not only gain a better understanding of how our brains work, but also a better understanding of how to modify them, and as a result, our mental states. An important question we are faced with is whether the state could be justified in implementing such methods on criminal offenders, without their consent, for the purposes of rehabilitation and reduction of recidivism; a practice that is already legal in some jurisdictions. By focusing on a prominent type of view of free action, which I call bypassing views, this paper evaluates how such interventions may negatively impact the freedom of their subjects. The paper concludes that there will be a tension between the goals of rehabilitation and reduction of recidivism, on the one hand, and the negative impact such interventions may have on free action, on the other. Other things equal, the better that a particular intervention is at achieving the former, the more likely it is to result in the latter.

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