Virtually solvable groups pdf

We show that any subgroup of a finitely generated virtually abelian group g grows rationally relative to g, that the set of right cosets of any subgroup of g grows rationally, and that the set of conjugacy classes of g grows rationally. Subgroups of residually pgroups are again residually p. The space of leftorderings of a countable virtually solvable group is either nite or a cantor set. An immediate consequence of the bassgromov equivalence is the quasiisometry rigidity for virtually nilpotent groups. The atheoretic farrelljones conjecture for virtually solvable groups mpsauthors kasprowski, daniel max planck institute for mathematics, max planck society. The middlegame is concerned with moderately large sets and is further handled by the crucial.

Next we make a reduction of the fibered isomorphism conjecture from virtually solvable groups to a class of virtually solvable q. Solvable groups a numerical approach thanos gentimis abstract. Erschler shows that finitely generated solvable groups that are not virtually nilpotent are not choquetdeny 8, and that even some groups of intermediate growth are not. The rst main di erence between the two sets of results lies in the groups considered. Failure of the roth theorem for solvable groups of. Theory of solvable groups farrell 2003 proceedings of. The status of the classification of the finite simple groups michael aschbacher 736 notices of the ams volume 51, number 7 c ommon wisdom has it that the theoremclassifying the finite simple groups was proved around 1980. Suppose that g is a group and that n is a normal subgroup of g. The space of leftorderings of a countable virtually solvable group is either finite or a cantor set. The atheoretic farrelljones conjecture for virtually. But, the most important properties of free solvable groups can be deduced from the magnusremeslennikovsokolov embedding see 9, ch. A countable linear nontorsion group which is not virtually solvable has a maximal subgroup of in. If x g is deformation rigid then x g is a quotient of an abelian variety, in particular, x g is projective.

Theory of solvable groups farrell 2003 proceedings. Pdf we give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were. A group is virtually nilpotent when it has a finite index subgroup which is nilpotent. If g is a power of a prime p, then g is a solvable group. Then g is said to be virtually solvable if it possesses a solvable subgroup of finite index. We first prove that the whitehead group of a torsionfree virtually solvable linear group vanishes. Equivalently, polycyclic groups are solvable groups that satisfy a maximal condition on subgroups, namely that each subgroup is nitelygenerated. Pdf characterization of solvable groups and solvable radical. This context is somehow very close to our theorems a and b and this issue deserves further comments.

The cohomology of virtually torsionfree solvable groups of. Ruping, the farrelljones conjecture for arbitrary lattices in virtually connected lie groups, geom. A polycyclic group is a solvable group where every subgroup is finitely generated. In this note, we announce the rst results on quasiisometric rigidity of nonnilpotent polycyclic groups. Chapter 7 nilpotent groups recall the commutator is given by x,yx. We show that every solvable subgroup of outf n has a finite index subgroup that is finitely generated and free abelian. Quasiisometries and rigidity of solvable groups alex eskin, david fisher and kevin whyte abstract.

In this note, we announce the first results on quasiisometric rigidity of nonnilpotent polycyclic groups. Wolf proved that finitely gener ated groups that are finite extensions of a nilpotent group i. Tits alternative, many quotients, permutation groups. Gromov a finitely generated group g has polynomial. The farrelljones conjecture for virtually solvable groups. Let u be the class of solvable groups g of finite rank such that g contains a nilpotent normal subgroup n satisfying the two conditions below. Yair glasner joint with tsachik gelander school of mathematics institute for advanced study. Rather than just focusing on groups that are virtually torsionfree, the present paper considers the following, larger class of solvable groups of finite rank. Solvable group 3 virtually solvable groups a group g is called virtually solvable if it has a solvable subgroup of finite index. We stipulate a condition on a that ensures the existence of. On fundamental groups of complete affinely flat manifolds core. In this chapter we shall consider groups whose composition factors are all abelian. Similarly, the inclusion of nilpotent groups in polycyclic groups is strict. Problems on the geometry of finitely generated solvable groups.

A group g is called virtually solvable if it has a solvable subgroup of finite index. A characterization is given of which closed 3manifolds can be represented as. The class 20f16 is used for the general theory of solvable groups, while the class 20d10 coming under 20d which is for finite groups focusses on finite solvable groups. Soluble groups we have already met the concept of a composition series for a g roup. Dec 23, 2016 we first prove that the whitehead group of a torsion. Hence the class of finitelygenerated solvable groups is not.

Clearly all solvable groups are virtually solvable, since one can just choose the group itself, which has index. Geometry of nilpotent and solvable groups cornelia drut. Finitely generated nilpotent groups are polycyclic. The isomorphism conjecture for solvable groups in waldhausen. Solvable groups mathematics 581, fall 2012 in many ways, abstract algebra began with the work of abel and galois on the solvability of polynomial equations by radicals. What finitely generated amenable groups are known to be lerf. The cohomology of virtually torsionfree solvable groups of finite rank article pdf available in transactions of the american mathematical society 3679. A critical ingredient in the early treatments was the sumproduct theorembourgain et al. If is a countable leftorderable virtually solvable group of in nite rank, then lo is a cantor set. Introduction naturally all the papers and presentations start by exemplifying the importance of their topics, their various applications. The cohomology of virtually torsionfree solvable groups of finite rank. First, here are three easy identities involving simple. Think of this as the class of groups that can be built using onlyabelian groups. Pdf the cohomology of virtually torsionfree solvable.

As in the previous paper, we will use the shorter term solvable group of finite rank to refer to such a group. So in this paper we focus on non virtually solvable linear groups. The smallest nonsolvable group is the simple group a 5, the alternating group of order 60 inside the symmetric group s 5. Clearly all solvable groups are virtually solvable, since one can just choose the group itself, which has index 1. On the space of leftorderings of virtually solvable groups. Every solvable subgroup of zm is virtually abelian. These results hold regardless of the choice of finite weighted generating set for g. Torsion cohomology for solvable groups of finite rank. One says that a group gis virtually p if there exists a nite index subgroup of g which is p. Alternatively you could use the fact that nilpotent groups have polynomial growth.

The status of the classification of the finite simple groups. However, the proof of the classification is not an ordinary proof because of its length and com. The purpose of this paper is to show that virtually the picture is very di erent. As solvable lie groups have only cocompact lattices, the conjectures is equivalent to saying that any group quasiisometric to lattice in a simply connected solvable lie group is virtually a lattice in a simply connected, solvable lie group. In particular, we prove that any group quasiisometric to the three dimenionsional solvable lie group sol is virtually a lattice in sol. Solvable groups realize their greatest importance in the proof of the insolvability of the quintic in chapter v in.

Abstract we prove that any nitely generated torsion free solvable subgroup of the group iet of all interval exchange transformations is virtually abelian. H2 is a group of order 4, and all of these quotient groups are abelian. Solvable subgroups of out f n are virtually abelian. The cohomology of virtually torsionfree solvable groups of finite rank by p. The smallest non solvable group is the simple group a 5, the alternating group of order 60 inside the symmetric group s 5. For a virtually torsionfree solvable group g of finite rank and a zgmodule a that. Subgroups of residually pgroups are again residually p, so virtually. Recall that if c is a class of groups, then a group is virtually c if it has a csubgroup of finite index equivalently it has a normal. More generally, given any property p such as being commutative, solvable, or torsionfree, a group is virtually p if it admits a subgroup of finite index with. Common uses for this would be when p is abelian, nilpotent, solvable or free. Some classes of solvable groups which are not polycyclic are known to be quasiisometrically rigid. Then it can be proved that g is a solvable group if and only if both gn and n are solvable groups. Solvable groups of interval exchange transformations francoi.

Each member of this family is a solvable sarithmetic group that is related to baumslagsolitar groups and has a quasiisometry group that is virtually a product of a solvable real lie group. The theory really takes o with the borel xedpoint theorem concerning the action of a smooth connected split solvable group on a smooth quasiprojective variety. Furthermore, if gis a virtually solvable subgroup ofm,then gcontains an abelian subgroup, a,such that the index ofa in gis boundedby vm,where vmis a positive integer depending only upon m. Assume that g is a virtually torsionfree solvable group of nite rank and a a zgmodule whose underlying abelian group is torsionfree and has nite rank. The cohomology of virtually torsionfree solvable groups. First and foremost, together with agols virtual fibering theorem ag08 they imply the virtually. The key idea galois had was to transform questions about elds and polynomials into questions about nite groups. For example, virtually solvable groups are one of the two alternatives in the tits alternative, while gromovs theorem states that the finitely generated groups with polynomial growth are precisely the finitely generated virtually nilpotent groups. All of the dihedral groups d2n are solvable groups. Along the way, one discovers other features that nilpotent groups share. If xis an orientable, strongly minimal pd 4complex and. Pdf quasiisometries and rigidity of solvable groups. A solvable group is one whose derived series reaches the trivial subgroup at a finite stage.

S3 is a finite solvable group and hence is polycyclic, but is not nilpotent. It can be proved that if g is a solvable group, then every subgroup of g is a solvable group and every quotient group of g is also a solvable group. Virtually abelian kahler and projective groups 195 corollary 1. Hence a solvable group that is not polycyclic is not f. This note is an addendum to the paper on the conjugacy growth functions of groups by v. Finite decomposition rank for virtually nilpotent groups caleb eckhardt, elizabeth gillaspy, and paul mckenney abstract. The solvable groups are thus those groups whose simple successive quotients in a composition series are prime cyclic abelian groups. The cohomology of virtually torsionfree solvable groups of finite rank peter kropholler and karl lorensen abstract. The case of virtually solvable groups, linear or not, was treated in 4 and independently by m. We first prove that the whitehead group of a torsion. For the last two classes one can state a much stronger statement, since hyperbolic groups are virtually solvable even amenable if and only if they are virtually cyclic see 4, besides, tits alternative holds also for finitely generated subgroups of hyperbolic groups which may not be hyperbolic, and any such subgroup too is virtually. So theorem 1 is a step towards proving a standard conjecture.