John donnelly, a cancellative amenable ascending union of nonamenable semigroups. We show that the class of lacunary hyperbolicgroups contains nonvirtually cyclic elementary amenable groups, groups with all proper subgroups cyclic tarski monsters, and torsion. I saw the pale student of unhallowed arts kneeling beside the thing he had put together. In this paper, we prove that the class of lacunary hyperbolicgroups is very large. Nonpositive curvature and complexity for finitely presented. Mary shelley, introduction to the 1831 edition of frankenstein. Any torsion abelian group splits into a direct sum of primary groups with respect to distinct prime numbers. 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. We construct lattices in aut tn x aut tm which are finitely presented, torsion free, simple groups. Pierre fima, amenable, transitive and faithful actions of groups acting on trees. 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. 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. We construct first examples of infinite finitely generated residually finite torsion groups with positive rank gradient.
Nonamenable finitely presented torsion bycyclic groups. So analogs of banachtarski paradox can be found in. Ams transactions of the american mathematical society. Our group is an extension of a group of finite exponent n 1 by a cyclic group, so it.
As a corollary, all the groups constructed by golod and shafarevich groups are nonamenable. 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. G contains a subgroup isomorphic to a free burnside group of exponent n with 2 generators. If ais a nitely generated torsion free abelian group. Pdf algorithmic and asymptotic properties of groups researchgate. The following examples may be useful for illustrative or instructional purposes. There is a general idea, commonly attributed to rips, which shows that such groups should exist.
Finitely generated elementary amenable groups are never of intermediate growth 48, so that problems 1. It possesses a presentation with finitely many generators, and finitely many relations. Consider the natural action of the group psl 2 r on the projective line p 1 p 1 r. 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 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. A group is said to be finitely presented or finitely presentable if it satisfies the following equivalent conditions. A variational principle of topological pressure on subsets for amenable group actions. Problems on the geometry of finitely generated solvable groups. Mar 19, 20 finitely presented examples were constructed another 20 y later by ol. In particular, finite abelian groups split into a direct sum of primary cyclic groups. The torsion subgroup of a group is the subgroup of all those elements g g, which have finite order, i. The torsion subgroup of a, denoted ta, is the set ta fa2aj9n2n such that na 0g.
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. However, in 2002 sapir and olshanskii found finitely presented counterexamples. Sapir, nonamenable finitely presented torsionbycyclic groups, publ. Finitely presented groups whose asymptotic cones are rtrees by m. We construct a finitely presented nonamenable group without free noncyclic subgroups thus providing a finitely presented counterexample. By recent work of hull and osin groups with hyperbolically embedded subgroups. We construct a nitely presented nonamenable group without. Example of an amenable finitely generated and presented group. Given any subring a non amenable finitely presented torsion by cyclic groups. 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. Constructions of torsionfree countable, amenable, weakly mixing.
See burnside problem on torsion groups for finiteness conditions of torsion groups. 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. Some applications to problems about cost and l2betti numbers are discussed. L 2 betti numbers and nonunitarizable groups without.
We show that there exist non unitarizable groups without nonabelian free subgroups. A typical realization of this group is as the complex n th roots of unity. The question is easy for finitely generated amenable. Sapir, nonamenable finitely presented torsionbycyclic groups. 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. Inner amenability for groups and central sequences in factors. The examples are so simple that many additional properties can be established. Stable finiteness of group rings in arbitrary characteristic core. Fully explicit quasiconvexification of the meansquare deviation of the gradient of the state in optimal design abstract msc key words. In 20, yash lodha and justin tatch moore isolated a finitely presented non amenable subgroup of monods group. Inner amenability for groups and central sequences in. Sapir, title nonamenable finitely presented torsionbycyclic groups.
Groups of piecewise projective homeomorphisms pnas. Pdf nonamenable finitely presented torsionbycyclic groups. Donald solitar in 1962 to provide examples of finitely presented hopfian groups. Citeseerx citation query on residualing homomorphisms.
By the fundamental theorem of finitely generated abelian groups, it follows that abelian groups are amenable. We show that the class of lacunary hyperbolic groups contains elementary amenable groups, groups with all proper subgroups cyclic, and torsion groups. Sending a to a primitive root of unity gives an isomorphism between the two. In 055z, would it be convenient to have the extra generality of allowing to be replaced by any finite module. Quotients this group property is quotientclosed, viz. You start running into settheoretic problems, where certain axioms e. In general, subgroups of finitely generated groups are not finitely generated.
Alexander varieties and largeness of finitely presented groups thomas koberda abstract. We give some applications of this result to the study of. Nonamenable finitely presented torsionbycyclic groups. L 2 betti numbers and nonunitarizable groups without free. 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. Aluffi 09, pages 8384 this is a special case of the structure theorem for finitely generated modules over a principal ideal domain examples. 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. Nonamenable finitely presented torsionby cyclic groups. Aluffi 09, pages 8384 this is a special case of the structure theorem for finitely generated modules over a principal ideal domain. Im looking for an example of a finitely presented and finitely generated amenable group, that has a subgroup which is not finitely generated. Amenable groups without finitely presented amenable covers.
A survey of problems, conjectures, and theorems about quasiisometric classification and rigidity for finitely generated solvable groups. Groups of piecewise projective homeomorphisms ergodic and. Our methods use sylvester rank functions and the translation ring of an amenable group. We construct a finitely presented nonamenable group without free noncyclic subgroups thus providing a finitely.
We denote by g the group of all homeomorphisms of p 1 that are piecewise in psl 2 r. Example of an amenable finitely generated and presented group with a nonfinitely generated subgroup. These groups are amenable torsion groups and are not finitely generated. Nonamenable nitely presented torsionbycyclic groups. G is an ascending hnn extension of a nitely generated in nite group of exponent n. An abelian group ais said to be torsion free if ta f0g. Both torsion and torsion free examples are constructed. The work of lewis bowen on the entropy theory of non. We show that there exist nonunitarizable groups without nonabelian free subgroups.
Finitely presented simple groups and products of trees. We endow p 1 with its rtopology, making it a topological circle. A group is torsionfree if there is no such element apart from the neutral element e e itself, i. Electronic research announcements of the american mathematical society volume 7, pages 6371 july 3, 2001 s 107967620956 nonamenable finitely presented torsionbycyclic groups a. 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. 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. It follows from the extension property above that a group is amenable if it has a finite index amenable subgroup. 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.
Structure theorem for abelian torsion groups that are not. Algebraic and combinatorial methods in concrete classes of. Nonamenable nitely presented torsionbycyclic groups 3 1. 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. This provides the first torsion free finitely presented counterexample, and admits a presentation with 3 generators and 9 relations. 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. Our group is an extension of a group of finite exponent n. Narens, meaningfulness and the erlanger program of felix klein. There are several more recent counterexamples 12 14. 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. Citeseerx citation query on residualing homomorphisms and g. Stable finiteness of group rings in arbitrary characteristic.
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