Donald solitar in 1962 to provide examples of finitely presented hopfian groups. Nonamenable finitely presented torsion bycyclic groups. Pierre fima, amenable, transitive and faithful actions of groups acting on trees. We construct first examples of infinite finitely generated residually finite torsion groups with positive rank gradient. Constructions of torsionfree countable, amenable, weakly mixing. 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. Pdf nonamenable finitely presented torsionbycyclic groups. Electronic research announcements of the american mathematical society volume 7, pages 6371 july 3, 2001 s 107967620956 nonamenable finitely presented torsionbycyclic groups a. Stable finiteness of group rings in arbitrary characteristic core. The work of lewis bowen on the entropy theory of non. These groups are amenable torsion groups and are not finitely generated.
Inner amenability for groups and central sequences in. Finitely generated elementary amenable groups are never of intermediate growth 48, so that problems 1. I saw the pale student of unhallowed arts kneeling beside the thing he had put together. Finitely presented simple groups and products of trees. The torsion subgroup of a group is the subgroup of all those elements g g, which have finite order, i. Pdf algorithmic and asymptotic properties of groups researchgate. The torsion subgroup of a, denoted ta, is the set ta fa2aj9n2n such that na 0g. 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. Example of an amenable finitely generated and presented group. For every finite or compact subset f of g there is an integrable nonnegative. Our methods use sylvester rank functions and the translation ring of an amenable group. 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. In general, subgroups of finitely generated groups are not finitely generated.
The question is easy for finitely generated amenable. By recent work of hull and osin groups with hyperbolically embedded subgroups. Our group is an extension of a group of finite exponent n. Alexander varieties and largeness of finitely presented groups thomas koberda abstract. Consider the natural action of the group psl 2 r on the projective line p 1 p 1 r. Quotients this group property is quotientclosed, viz. However, in 2002 sapir and olshanskii found finitely presented counterexamples. Stable finiteness of group rings in arbitrary characteristic. Nonamenable finitely presented torsionby cyclic groups. This provides the first torsion free finitely presented counterexample, and admits a presentation with 3 generators and 9 relations. Nonamenable nitely presented torsionbycyclic groups 3 1. 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. The examples are so simple that many additional properties can be established. 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.
Sending a to a primitive root of unity gives an isomorphism between the two. We show that there exist non unitarizable groups without nonabelian free subgroups. Both torsion and torsion free examples are constructed. As a corollary, all the groups constructed by golod and shafarevich groups are nonamenable. G contains a subgroup isomorphic to a free burnside group of exponent n with 2 generators. A survey of problems, conjectures, and theorems about quasiisometric classification and rigidity for finitely generated solvable groups. The original definition, in terms of a finitely additive invariant measure or mean on subsets of g, was introduced. In this paper, we prove that the class of lacunary hyperbolicgroups is very large. These groups are finitely generated, but not finitely presented. G is an ascending hnn extension of a nitely generated in nite group of exponent n. G is an extension of a nonlocally nite group of exponent n by an in nite cyclic group.
Example of an amenable finitely generated and presented. Groups of piecewise projective homeomorphisms pnas. It could, but that result is contained in lemma 10. Mary shelley, introduction to the 1831 edition of frankenstein. A group is torsionfree if there is no such element apart from the neutral element e e itself, i. See burnside problem on torsion groups for finiteness conditions of torsion groups. So analogs of banachtarski paradox can be found in.
Sapir, nonamenable finitely presented torsionbycyclic groups, publ. Inner amenability for groups and central sequences in factors. A variational principle of topological pressure on subsets for amenable group actions. John donnelly, a cancellative amenable ascending union of nonamenable semigroups. It follows from the extension property above that a group is amenable if it has a finite index amenable subgroup. Structure theorem for abelian torsion groups that are not. It possesses a presentation with finitely many generators, and finitely many relations. Example of an amenable finitely generated and presented group with a nonfinitely generated subgroup.
Nonamenable finitely presented torsionbycyclic groups. By the fundamental theorem of finitely generated abelian groups, it follows that abelian groups are amenable. There is a general idea, commonly attributed to rips, which shows that such groups should exist. The following examples may be useful for illustrative or instructional purposes. Nonamenable finitely presented torsionbycyclic groups abstract msc key words authors. 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.
Ams transactions of the american mathematical society. You start running into settheoretic problems, where certain axioms e. Any torsion abelian group splits into a direct sum of primary groups with respect to distinct prime numbers. 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. We construct a nitely presented nonamenable group without. We construct a finitely presented nonamenable group without free noncyclic subgroups thus providing a finitely presented counterexample. A typical realization of this group is as the complex n th roots of unity. We show that the class of lacunary hyperbolic groups contains elementary amenable groups, groups with all proper subgroups cyclic, and torsion groups. Narens, meaningfulness and the erlanger program of felix klein.
Amenable groups without finitely presented amenable covers. Aluffi 09, pages 8384 this is a special case of the structure theorem for finitely generated modules over a principal ideal domain. Citeseerx citation query on residualing homomorphisms. A group is said to be finitely presented or finitely presentable if it satisfies the following equivalent conditions. We construct a finitely presented nonamenable group without free noncyclic subgroups thus providing a finitely.
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. We denote by g the group of all homeomorphisms of p 1 that are piecewise in psl 2 r. Sapir, nonamenable finitely presented torsionbycyclic groups. Mar 19, 20 finitely presented examples were constructed another 20 y later by ol. In 055z, would it be convenient to have the extra generality of allowing to be replaced by any finite module. L 2 betti numbers and nonunitarizable groups without. We endow p 1 with its rtopology, making it a topological circle. 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. Problems on the geometry of finitely generated solvable groups. 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. 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. Im looking for an example of a finitely presented and finitely generated amenable group, that has a subgroup which is not finitely generated. If ais a nitely generated torsion free abelian group.
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. We show that the class of lacunary hyperbolicgroups contains nonvirtually cyclic elementary amenable groups, groups with all proper subgroups cyclic tarski monsters, and torsion. There are several more recent counterexamples 12 14. Recall that the torsion subgroup of abelian group g is the subgroup of g consisting of all elements of g of. Groups of piecewise projective homeomorphisms ergodic and. An abelian group ais said to be torsion free if ta f0g. We show that there exist nonunitarizable groups without nonabelian free subgroups. Nonpositive curvature and complexity for finitely presented.
Our group is an extension of a group of finite exponent n 1 by a cyclic group, so it. We give some applications of this result to the study of. 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. 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. Sapir, title nonamenable finitely presented torsionbycyclic groups. Given any subring a non amenable finitely presented torsion by cyclic groups. Fully explicit quasiconvexification of the meansquare deviation of the gradient of the state in optimal design abstract msc key words. 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. Finitely presented groups whose asymptotic cones are rtrees by m. We construct lattices in aut tn x aut tm which are finitely presented, torsion free, simple groups. Citeseerx citation query on residualing homomorphisms and g. 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.
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. In particular, finite abelian groups split into a direct sum of primary cyclic groups. We also show that the groupmeasure space constructions associated to free, strongly ergodic p. Aluffi 09, pages 8384 this is a special case of the structure theorem for finitely generated modules over a principal ideal domain examples. In 20, yash lodha and justin tatch moore isolated a finitely presented non amenable subgroup of monods group. 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. 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. L 2 betti numbers and nonunitarizable groups without free. 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. Algebraic and combinatorial methods in concrete classes of.
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