In this article we will describe a nitely presented subgroup of the group of piecewise projective homeomorphisms of the real projective line. Non amenable finitely presented torsion bycyclic groups. On torsion free groups with infinitely many ends by john r. The deficiency of a finitely presented group h is the largest integer that can occur as the. In 20, yash lodha and justin tatch moore isolated a finitely presented non amenable subgroup of monods group. Our group is an extension of a group of finite exponent n 1 by a cyclic group, so it satisfies the identity x,yn 1.
Cyclic splittings of finitely presented groups 57 for cl with respect to the second splitting. A cohomological characterization of locally virtually. Lustigs nonefficient torsionfree group admits a minimal presentation. Encoding and detecting properties in finitely presented groups. Geometric group theory, finitely presented groups, nonpositive curvature. We show that for testing kaplanskys direct finiteness conjecture, it su. Nonamenable finitely presented torsionbycyclic groups. Sep 25, 2002 we show that every discrete group ring dg of a freeby amenable group g over a division ring d of arbitrary characteristic is stably finite, in the sense that onesided inverses in all matrix rings over dg are twosided. Sapir, title non amenable finitely presented torsionbycyclic groups. In contrast to the other amenable groups, virtually abelian groups are indisputably onpositively curved. Investigation of locally finite varieties of groups, doctor of sci.
These are the first examples of this kind that are not subgroups of hyperbolic groups. If you have understood both these terms, you should have no difficulty in doing this problem. Section 2 of our paper is devoted the proof of the following theorem. Pdf chapter 3 paradoxical decompositions of groups and. Zlil sela 88 solved the isomorphism problem for torsionfree hyperbolic gro. It is shown that certain torsion groups studied by grigorchuk and by gupta and sidki are conjugacy separable. Amenable groups without finitely presented amenable covers. The groups involved are finitely generated free groups.
Brady br constructed a remarkable example of a torsion free hyperbolic group which contains a finitely presented subgroup which is not itself word hyperbolic. Embed it into a finitely presented group g by using a version of the higman embedding theorem. We construct a finitely presented non amenable group without free non cyclic subgroups thus providing a finitely. Let g be a finitely presented abelianbycyclic group with exponen. The group of piecewise projective homeomorphisms of the line provides straightforward torsionfree counterexamples to the socalled. Nonamenable nitely presented torsionbycyclic groups. A free bycyclic group is the extension of a finitely generated free group. How to prove that a finite group of order n is cyclic if and. We construct a finitely presented non amenable group without free non cyclic subgroups. The direct product of two amenable groups is amenable, while the direct product of an infinite family of amenable groups need not be. These examples grew out of an ongoing study of the structure and classification of lattices in the group of automorphisms of a product of trees see 3 and 4. Non amenable finitely presented torsion bycyclic groups abstract msc key words authors.
An example of a finitely presented amenable group not belonging to the class eg of elementary amenable groups is constructed. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a. A corollary of this theorem is that all finitely generated submonoids of the positive monoid of f are not amenable, and therefore if f is amenable these sets have measure zero. Abstract references similar articles additional information. Consider an n nmatrix m with integral entries and det m0. It has been aimed primarily at mathematics students but those studying related disciplines such as computer science or physics should also find it useful. Nonpositivecurvature and complexity for nitely presented. One also asks what type of amenable subgroups can arise, cf. The nontrivial elementary factors of a are the torsion invariants of the group g, and n r is the torsion free rank. Geometry of amenable groups 123 the equivalence of ii above with the usual notion of quasiisometry in the case of finitely generated groups see definition 2. Smiths result, interpreted in group theoretic terms, leads to a method for decomposing a finitely presented abelian group into a direct product of cyclic subgroups. Finitely generated torsion free nilpotent group 2s satisf of clasy as remak krull schmidt condition. Dissertation doctoral thesis etheses universitat wien. We construct a finitely presented non amenable group without free non cyclic subgroups thus providing a finitely presented counterexample.
In contrast to the other amenable groups, virtually abelian groups are indis. You start running into settheoretic problems, where certain axioms e. Embedding free burnside groups in finitely presented. We construct an embedding of a free burnside group bm,n of odd exponent n 248 and rank m 1 in a finitely presented group with some special properties. Non amenable finitely presented torsionbycyclic groups. Bieri, no examples of in nite nitely presented groups that are extensions of a periodic group by a cyclic group but are not almost cyclic have been known before. This provides the first torsion free finitely presented counterexample, and admits a presentation with 3 generators and 9 relations. Feb 11, 2016 we construct uncountably many quasiisometry classes of finitely generated, torsion free groups in which every maximal cyclic subgroup is hyperbolically embedded.
The group of integers is amenable a sequence of intervals of length tending to infinity is a folner sequence. Defined like nilpotent bycyclic groups, finitely generated abelian bycyclic groups are the hnn extensions of finitelygenerated abelian groups. An indicable group is one having an infinite cyclic factor group. Non amenable nitely presented torsionbycyclic groups 3 1. The common opinion i believe is that such groups do exist, but the best result in this direction so far is the olshanskiisapir group, which is finitely presented and infinite torsion bycyclic. The question is easy for finitely generated amenable. We construct a finitely presented nonamenable group without free noncyclic subgroups thus providing a finitely. Lodha later showed that this group satisfies the property. The existence of a shiftinvariant, finitely additive probability measure on the group z also. Nonpositive curvature and complexity for finitely presented groups. We show that if a, b are locally indicable and u is cyclic, then g is locally indicable see theorem 9. Our group is an extension of a group of finite exponent n. Thompsons group f thompsons group f has been studied for several decades. An example of a nitely presented amenable group 77 finally, assertion iii is also of interest because, as i have been told by r.
Our methods use sylvester rank functions and the translation ring of an amenable group. Nonamenable finitely presented torsionbycyclic groups 2002. G is an extension of a nonlocally nite group of exponent n by an in nite cyclic group. Higman 4 showed that the group ring of a locally indicable group has no zero divisors if the coefficient ring has none. Harmonic analysis, cohomology, and the largescale geometry. Finitely presented simple groups and products of trees. Groups of piecewise projective homeomorphisms pnas. The goal of this article is to study results and examples concerning finitely presented covers of finitely generated amenable groups. Is a finitely generated torsion group finite in general. Example of an amenable finitely generated and presented group.
If g is freely indecomposable, both cl and c2 are simultaneously elliptic, or simultaneously hyperbolic. Structure theorem for abelian torsion groups that are not. It is proved that every elementary amenable group of type fp. Constructions of torsionfree countable, amenable, weakly mixing. An example of a finitely presented amenable group not. This follows immediately from the definitions of the two key terms. Stable finiteness of group rings in arbitrary characteristic. Embedding free burnside groups in finitely presented groups. We construct an embedding of a free burnside group bm,n of odd exponent n 2 48 and rank m 1 in a finitely presented group with some special properties. Pdf nonamenable finitely presented torsionbycyclic groups. Finitely presented examples were constructed another 20 y later by ol. On the asymptotic geometry of abelianbycyclic groups. Clearly q is torsion being a quotient of b m, n and is finitely presented. The canonical jsj decomposition we construct generalizes the one introduced in sel for gromov hyperbolic groups.
By this means, a solution of the day problem in the class of finitely presented groups is given. It was posed c1900, and golod and shafarevich proved in 1964 that there existed finitely generated infinite p groups for some primes p, although the groups they constructed had unbounded order for all n. Invertible element or ulie groups, that are universal for existence of onesided invertible elements in a group ring kg, where k is a. Non amenable finitely presented torsion bycyclic groups, electron.
Finite part of operator k theory for groups finitely. Presentations of groups finitely generated groups viewed as metric spaces quasiisometry limit groups free actions homomorphisms of groups the pair hs jriis called a presentation of g, it determines g uniquely up to isomorphism. For amenable groups finitely generated or not, property wm is equivalent to having no nontrivial finite quotients or abelian quo tients 3 in. Then take the quotient group q of g by the relations. Nonpositivecurvature and complexity for nitely presented groups. A nonamenable finitely presented group of piecewise projective homeomorphisms yash lodha and justin tatch moore this paper is dedicated to the memory of william thurston 19462012. We collect examples of groups \g\ with the following properties. Cyclic splittings of finitely presented groups and the. Publications alexander olshanskiy vanderbilt university. In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. Several classes of nonpositively curved groups serve as natural envelopes uniting free and freeabelian groups.
On the asymptotic geometry of abelian bycyclic groups 147 by a result of bieri and strebel bs1, the class of finitely presented, torsion free, abelian bycyclic groups may be described in another way. Pdf thompsons group f and uniformly finite homology. We borrow the jacoshalenjohannson notion of characteristic submanifolds from 3dimensional topology to study cyclic splittings of finitely presented groups. How to prove that a finite group of order n is cyclic if. The rank and generating set for iterated wreath products of cyclic groups.
Nonamenable finitely presented torsionbycyclic groups math. If c is a group which satisfies the maximal condition for normal subgroups, then c may be cancelled from a group a in direct products if and only if the infinite cyclic group can be cancelled from a. Infinite groups with cyclic subgroups, doklady akad. Im looking for an example of a finitely presented and finitely generated amenable group, that has a subgroup which is not finitely generated. Grigorchuk introduced and studied some remarkable new examples of finitely generated torsion groups.
We show that if is a countable amenable group acting on a dendrite, then has either a fixed point or a 2periodic point in. Pdf thompsons group f and uniformly finite homology dan. A torsion free, finitely generated group, having a free subgroup of finite index, is itself free. In the first case, r contains all finitely generated torsion. The presentation hs jriis nite if both sets s and r are nite.
Fully explicit quasiconvexification of the meansquare deviation of the gradient of the state in optimal design abstract msc key words. Examples of amenable groups include finite groups, abelian groups, nilpotent groups. An infinite simple noetherian torsion free group, izvestia akad. It is a classical question whether there exist finitely generated groups where every element is torsion. If g is a torsion free, finitely presented group, with infinitely many ends, then g is a nontrivial free product. In mathematics, an amenable group is a locally compact topological group g carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. Word hyperbolic groups are finitely presented, and examples are provided by finite groups, finitely generated free groups, fundamental groups of compact surfaces except for the torus and the klein bottle, and groups which act properly discontinuously and cocompactly on hyperbolic space of any dimension.
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