Non amenable finitely presented torsion by cyclic groups pdf

Aluffi 09, pages 8384 this is a special case of the structure theorem for finitely generated modules over a principal ideal domain. Mary shelley, introduction to the 1831 edition of frankenstein. We denote by g the group of all homeomorphisms of p 1 that are piecewise in psl 2 r. The work of lewis bowen on the entropy theory of non.

Aluffi 09, pages 8384 this is a special case of the structure theorem for finitely generated modules over a principal ideal domain examples. As a corollary, all the groups constructed by golod and shafarevich groups are nonamenable. You start running into settheoretic problems, where certain axioms e. Stable finiteness of group rings in arbitrary characteristic core. Nonamenable finitely presented torsionbycyclic groups. Any torsion abelian group splits into a direct sum of primary groups with respect to distinct prime numbers. A survey of problems, conjectures, and theorems about quasiisometric classification and rigidity for finitely generated solvable groups. It could, but that result is contained in lemma 10. See burnside problem on torsion groups for finiteness conditions of torsion groups. These groups are amenable torsion groups and are not finitely generated.

By the fundamental theorem of finitely generated abelian groups, it follows that abelian groups are amenable. We show that every discrete group ring dg of a free by 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. An abelian group ais said to be torsion free if ta f0g. Problems on the geometry of finitely generated solvable groups. Inner amenability for groups and central sequences in. L 2 betti numbers and nonunitarizable groups without free. We construct first examples of infinite finitely generated residually finite torsion groups with positive rank gradient. A variational principle of topological pressure on subsets for amenable group actions. Finitely presented simple groups and products of trees. Sending a to a primitive root of unity gives an isomorphism between the two.

Fully explicit quasiconvexification of the meansquare deviation of the gradient of the state in optimal design abstract msc key words. Alexander varieties and largeness of finitely presented groups thomas koberda abstract. Inner amenability for groups and central sequences in factors. Such splittings are, in general, not unique, but any two splittings of a finitely generated abelian group into direct sums of nonsplit cyclic groups are isomorphic, so that the number of infinite cyclic summands and the collection of the orders of the. Sapir, nonamenable finitely presented torsionbycyclic groups, publ. Some applications to problems about cost and l2betti numbers are discussed. Pdf algorithmic and asymptotic properties of groups researchgate. We construct a finitely presented nonamenable group without free noncyclic subgroups thus providing a finitely. Quotients this group property is quotientclosed, viz. Example of an amenable finitely generated and presented. Consider the natural action of the group psl 2 r on the projective line p 1 p 1 r. Finitely presented freebycyclic groups have received a great deal of attention in recent years in part because they form a rich context in which to draw out distinctions between the different.

Groups of piecewise projective homeomorphisms pnas. Nonpositive curvature and complexity for finitely presented. Given any subring a non amenable finitely presented torsion by cyclic groups. Citeseerx citation query on residualing homomorphisms and g.

We show that there exist non unitarizable groups without nonabelian free subgroups. We construct a finitely presented nonamenable group without free noncyclic subgroups thus providing a finitely presented counterexample. Mar 19, 20 finitely presented examples were constructed another 20 y later by ol. Amenable groups without finitely presented amenable covers. I saw the pale student of unhallowed arts kneeling beside the thing he had put together. It possesses a presentation with finitely many generators, and finitely many relations it is finitely generated and, for any finite generating set, it has a presentation with that generating set and finitely many relations it is finitely generated and, for any. G is an ascending hnn extension of a nitely generated in nite group of exponent n. If ais a nitely generated torsion free abelian group. A cyclic group z n is a group all of whose elements are powers of a particular element a where a n a 0 e, the identity. Pierre fima, amenable, transitive and faithful actions of groups acting on trees. Our group is an extension of a group of finite exponent n.

We show that there exist nonunitarizable groups without nonabelian free subgroups. L 2 betti numbers and nonunitarizable groups without. By recent work of hull and osin groups with hyperbolically embedded subgroups. The question is easy for finitely generated amenable. It follows from the extension property above that a group is amenable if it has a finite index amenable subgroup. Example of an amenable finitely generated and presented group with a nonfinitely generated subgroup. These groups are finitely generated, but not finitely presented. In this paper, we prove that the class of lacunary hyperbolicgroups is very large.

A group is said to be finitely presented or finitely presentable if it satisfies the following equivalent conditions. It follows from the well known theorems on the algorithmic unsolvability of the word problem and related problems that there are no deterministic methods to answer most questions about the structure of finitely presented groups. The following examples may be useful for illustrative or instructional purposes. The torsion subgroup of a group is the subgroup of all those elements g g, which have finite order, i. In general, subgroups of finitely generated groups are not finitely generated. Sapir, title nonamenable finitely presented torsionbycyclic groups. Narens, meaningfulness and the erlanger program of felix klein. Stable finiteness of group rings in arbitrary characteristic.

Finitely presented free by cyclic groups have received a great deal of attention in recent years in part because they form a rich context in which to draw out distinctions between the different. For every finite or compact subset f of g there is an integrable nonnegative. Finitely generated elementary amenable groups are never of intermediate growth 48, so that problems 1. Ams transactions of the american mathematical society. There is a general idea, commonly attributed to rips, which shows that such groups should exist. The original definition, in terms of a finitely additive invariant measure or mean on subsets of g, was introduced. Example of an amenable finitely generated and presented group. Groups of piecewise projective homeomorphisms ergodic and. Nonamenable finitely presented torsionby cyclic groups. The examples are so simple that many additional properties can be established. Nonamenable nitely presented torsionbycyclic groups. There are several more recent counterexamples 12 14. Sapir, nonamenable finitely presented torsionbycyclic groups.

Nonamenable finitely presented torsion bycyclic groups. On proofs in finitely presented groups 4 4 pruned enumeration starting with a successful coset enumeration where the total number of cosets used, t, exceeds the subgroup index, i, it is often possible to prune the sequence of t. It is also interesting to note that both of these examples are based on the constructions of nonamenable torsion groups they are torsionbycyclic and in particular. Both torsion and torsion free examples are constructed. Finitely presented groups whose asymptotic cones are rtrees by m. The details are over my head i am not a group theorist, hardly even a mathematician, but i have it on good hearsay that at one time the existence of a finitely generated infinite simple group was known, but the existence of a finitely presented infinite simple group was still an unsolved problem. Nonamenable nitely presented torsionbycyclic groups 3 1. However, in 2002 sapir and olshanskii found finitely presented counterexamples. G is an extension of a nonlocally nite group of exponent n by an in nite cyclic group. Citeseerx citation query on residualing homomorphisms. Our group is an extension of a group of finite exponent n 1 by a cyclic group, so it. Recall that the torsion subgroup of abelian group g is the subgroup of g consisting of all elements of g of. Electronic research announcements of the american mathematical society volume 7, pages 6371 july 3, 2001 s 107967620956 nonamenable finitely presented torsionbycyclic groups a. Youve concluded that the surjection is finitely generated, so is finitely presented by definition, and there is no need to invoke 4, because the module playing the role of in 4 is, not an arbitrary finitely presented module.

Structure theorem for abelian torsion groups that are not. Our methods use sylvester rank functions and the translation ring of an amenable group. John donnelly, a cancellative amenable ascending union of nonamenable semigroups. We construct a nitely presented nonamenable group without. We show that the class of lacunary hyperbolic groups contains elementary amenable groups, groups with all proper subgroups cyclic, and torsion groups. We give some applications of this result to the study of. Constructions of torsionfree countable, amenable, weakly mixing. We show that the class of lacunary hyperbolicgroups contains nonvirtually cyclic elementary amenable groups, groups with all proper subgroups cyclic tarski monsters, and torsion. Donald solitar in 1962 to provide examples of finitely presented hopfian groups. Pdf nonamenable finitely presented torsionbycyclic groups. The torsion subgroup of a, denoted ta, is the set ta fa2aj9n2n such that na 0g. We also show that the groupmeasure space constructions associated to free, strongly ergodic p. So analogs of banachtarski paradox can be found in. 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.

He introduced the concept of an amenable group he called such groups measurable as a group g which has a left invariant finitely additive measure, g 1, noticed that if a group is amenable, then any set it acts upon freely also has an invariant measure, and proved that a group is not amenable provided it contains a free nonabelian subgroup. G contains a subgroup isomorphic to a free burnside group of exponent n with 2 generators. This provides the first torsion free finitely presented counterexample, and admits a presentation with 3 generators and 9 relations. A finitely generated abelian group is free if and only if it is torsion free, that is, it contains no element of finite order other than the identity. A typical realization of this group is as the complex n th roots of unity. We construct lattices in aut tn x aut tm which are finitely presented, torsion free, simple groups. Nonamenable finitely presented torsionbycyclic groups abstract msc key words authors. In particular, finite abelian groups split into a direct sum of primary cyclic groups. A group is torsionfree if there is no such element apart from the neutral element e e itself, i.

We show that the fundamental group of xis large if and only if there is a nite cover y of xand a sequence of nite abelian covers fy ngof y which satisfy b 1y n n. 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. 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 by cyclic. In 20, yash lodha and justin tatch moore isolated a finitely presented non amenable subgroup of monods group. A finitely generated abelian group is free if and only if it is torsionfree, that is, it contains no element of finite order other than the identity. Im looking for an example of a finitely presented and finitely generated amenable group, that has a subgroup which is not finitely generated. It possesses a presentation with finitely many generators, and finitely many relations. We show that every discrete group ring dg of a freebyamenable 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. In 055z, would it be convenient to have the extra generality of allowing to be replaced by any finite module. Algebraic and combinatorial methods in concrete classes of. We endow p 1 with its rtopology, making it a topological circle. 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 torsionbycyclic.