In this paper we prove quasiisometric rigidity results concerning lattices in sol and lamplighter groups. Along the way, one discovers other features that nilpotent groups share. We give these spaces the path metric of the graph, where every edge is considered to have length 1. This implies the class of discrete groups commensurable to lamplighter groups is not closed under quasiisometries and, combined with work of eskin. We include a discussion of other applications of coarse differentiation to problems in geometric group theory and a comparison of coarse differentiation to other related techniques in nearby. Navas to appear in iumj 2016days fixed point theorem, group cohomology and quasiisometric rigidity.
Quasi isometries and rigidity of solvable groups alex eskin, david fisher and kevin whyte abstract. To understand quasi isometries of solvable lie groups, eskinfisherwhyte developed a technique. Basic construction all of our spaces are the vertex sets of connected graphs of bounded valence. This paper is the first in a sequence of papers proving results announced in our 2007 article quasiisometries and rigidity of solvable groups. In this paper we prove quasi isometric rigidity results concerning lattices in sol and lamplighter groups. The elements of the isometry group are sometimes called motions of the space every isometry group of a metric space is a. A survey of problems, conjectures, and theorems about quasi isometric classification and rigidity for finitely generated solvable groups. Coarse differentiation and quasiisometries of a class of. R n where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. In particular, we prove that any group quasi isometric to the three dimenionsional solvable lie group sol is virtually a lattice in sol. For us, the space xis either a solvable lie group solm,n or dlm,n. Quasiisometries are height respecting a typical step in the study of quasiisometric rigidity of groups is the identi cation of all quasiisometries of some space x quasiisometric to the group.
Quasiconvex groups of isometries of negatively curved. The conference will take place in ventotene, italy from the 9th until the 14th of september 2019. The grigorchuk group is an example of a group which has uncountably many groups quasiisometric to it, but has solvable word problem. For us, the space xis either a solvable lie group solm. In addition to research talks there will be an instructional component in the form of three minicourses. We say that a class c of group is qi rigid if every group g0which is qi to. A rigidity property of some negatively curved solvable lie groups nageswari shanmugalingam, xiangdong xie november 15, 2009 abstract we show that for some negatively curved solvable lie groups, all self quasi isometries are almost isometries. A survey of problems, conjectures, and theorems about quasiisometric classification and rigidity for finitely generated solvable groups. A rigidity property of some negatively curved solvable lie groups nageswari shanmugalingam, xiangdong xie november 15, 2009 abstract we show that for some negatively curved solvable lie groups, all self quasiisometries are almost isometries. We will present the ingredients needed to complete the proof of quasiisometric rigidity for these more general solvable lie groups. Doctoral students and young researchers are particularly encouraged to apply. This paper is concerned with the structure of quasiisometries between.
Counting problems ventotene international workshops. In this paper, we continue with the results in \citepg and compute the group of quasiisometries for a subclass of split solvable unimodular lie groups. Quasiisometric rigidity of solvable groups department of. Quasiisometries kevin whyte berkeley fall 2007 lecture 1 theorem 1. This implies the class of discrete groups commensurable to lamplighter groups is not closed under quasi isometries and, combined with work of eskin. In pseudoeuclidean space the metric is replaced with an isotropic quadratic form. In accordance with the statement of inclusiveness, this event will be open to everybody, regardless of race, sex. A rigidity property of some negatively curved solvable lie. On the asymptotic geometry of abelianbycyclic groups. R n proves a conjecture made by farb and mosher in fm3. Pdf quasiisometries and rigidity of solvable groups. In this paper we provide the final steps in the proof of quasiisometric rigidity of a class of nonnilpotent polycyclic groups. In this note, we announce the rst results on quasi isometric rigidity of nonnilpotent polycyclic groups. The main results are that quasiisometries preserve the product structure, and that in the irreducible higher rank case, quasiisometries are at finite distance from homotheties.
Our techniques for studying quasiisometries extend to some other classes of groups and spaces. Large scale geometry of certain solvable groups springerlink. Suppose that g is a group quasiisometric to a nilpotent group. In fact the proofs in this answer, or a paper by anna erschler. Quasiisometries and rigidity of solvable groups core. To this end, we prove a rigidity theorem on the boundaries of certain negatively curved homogeneous spaces and combine it with work of eskinfisherwhyte and peng on the structure of quasiisometries of certain solvable lie groups.
The basic example of such an action is when k is compact, g. Quasi isometries also play a crucial role in mostows proof of his rigidity theorem. In mathematics, a quasiisometry is a function between two metric spaces that respects largescale geometry of these spaces and ignores their smallscale details. In particular, we prove that any group quasiisometric to the three dimenionsional solvable lie group sol is virtually a lattice in sol. Let h be a properly discontinuous group of isometries of a negatively curved gromov hyperbolic metric space x. An immediate consequence of the bassgromov equivalence is the quasi isometry rigidity for virtually nilpotent groups. A discrete isometry group is an isometry group such that for every point of the space the set of images of the point under the isometries is a discrete set.
A rigidity property of some negatively curved solvable lie groups. Quasiconvex groups of isometries of negatively curved spaces. Whyte, pure and applied mathematics quarterly 3 2007 927947. Quasiisometric rigidity of solvable groups proceedings.
In this paper, we continue with the results in \citepg and compute the group of quasi isometries for a subclass of split solvable unimodular lie groups. The main results are that quasi isometries preserve the product structure, and that in the irreducible higher rank case, quasi isometries are at finite distance from homotheties. Navas to appear in iumj 2016days fixed point theorem, group cohomology and quasi isometric rigidity with x. The quasiisometry claim follows because the vertex groups in each case are quasiisometrically rigid relative to the incident edge groups, and the stretch factor for g i is theratioofthewordlengthsofu iandv i,whicharedi. The results discussed here rely on a new technique for studying quasiisometries of finitely generated groups, which we refer to as coarse differentiation. The goal of the present paper is to show that a much broader class of solvable groups. This involves, among other things, a rigidity theorem on the boundaries. Consequently, we show that any finitely generated group quasi isometric to a member of the subclass has to be polycyclic, and is virtually a lattice in an abelianbyabelian solvable lie group. Jan 22, 2010 in this paper we provide the final steps in the proof of quasiisometric rigidity of a class of nonnilpotent polycyclic groups. We study quasiisometries between products of symmetric spaces and euclidean buildings. If g acts geometrically on x and y proper geodesic metric spaces then x and y are quasiisometric.
Quasi isometries are height respecting a typical step in the study of quasi isometric rigidity of groups is the identi cation of all quasi isometries of some space x quasi isometric to the group. We say that a group gis qi rigid if every group g0which is qi to g, is in fact vi to g. Quasiisometric rigidity of solvable groups proceedings of. As an example, here we mention a quasiisometric rigidity result for solvable lie groups. Quasiisometries and rigidity of solvable groups by alex eskin, david fisher and kevin whyte download pdf 280 kb. Since the identity map is a quasi isometry, and the composition of two quasi isometries is a quasi isometry, it follows that the property of being quasi isometric behaves like an equivalence relation on the class of metric spaces. The grigorchuk group is an example of a group which has uncountably many groups quasi isometric to it, but has solvable word problem.
Xie to appear in groups, geometry and dynamics 2016. Rigidity and classification ventotene, 914 september 2019 quasiisometric rigidity for certain classes of solvable groups tullia dymarz university of wisconsin in this minicourse we will survey results and techniques used to prove quasiisometric rigidity for various solvable. We prove analogous results for groups quasiisometric to r. Quasiisometries also play a crucial role in mostows proof of his rigidity theorem. Various geometries isometry groups of metric spaces, both discrete and. Mostow rigidity theorem for hyperbolic manifolds and will outline a proof of the tits alternative. Quasiisometries and rigidity of solvable groups authors. The paper builds in a substantial way on our earlier paper coarse differentiation of quasiisometries i. We study quasi isometries between products of symmetric spaces and euclidean buildings. The property of being quasiisometric behaves like an equivalence relation on the class of metric spaces the concept of quasiisometry is. Quasiisometries and rigidity of solvable groups alex eskin, david fisher and kevin whyte abstract. Alex eskin, david fisher, kevin whyte submitted on 27 nov 2005 v1, last revised 7 jul 2006 this version, v3.
Pdf quasiisometric rigidity for the solvable baumslag. In the affine actions, elliptic and hyperbolic dynamics coexist, forcing a priori complicated dynamics in. Rigidity of some abelianbycyclic solvable group actions. This follows from the polynomial growth theorem of m. Informally, quasiisometric rigidity is the situation when the arrow viqican be reversed. To understand quasiisometries of solvable lie groups, eskinfisherwhyte developed a technique. In this note, we announce the first results on quasiisometric rigidity of nonnilpotent polycyclic groups. Extension of quasiisometries of hyperbolic spaces to the ideal boundary. Our approach to these problems is to first classify all self quasiisometries of the solvable lie. Eskinfisherwhyte and peng use similar techniques to extend this structure theorem to quasiisometries of more general solvable lie groups. Quasiisometries between groups with twoended splittings. Pdf quasiisometric rigidity for the solvable baumslagsolitar. The aim of this course is to try and create a bridge between these two perspectives. Lukyanenko preprint 2016a matrix model for random nilpotent groups with k.
Then any connected and simply connected solvable lie group gquasiisometric to sis isomorphic to s. Lectures on quasiisometric rigidity uc davis mathematics. Alternatively, the groups can sometimes be treated via other categorizations, e. In the setting of geometric view of groups, the following questions become. A typical step in the study of quasiisometric rigidity of groups is the identi. Traditionally, rigidity results have been most successful for spaces of nonpositive curvature. The paper builds in a substantial way on our earlier paper coarse differentiation of quasi isometries i. An immediate consequence of the bassgromov equivalence is the quasiisometry rigidity for virtually nilpotent groups.
This paper is concerned with the structure of quasi isometries between products of symmetric spaces and euclidean buildings. Rigidity and classification ventotene, 914 september 2019 quasi isometric rigidity for certain classes of solvable groups tullia dymarz university of wisconsin in this minicourse we will survey results and techniques used to prove quasi isometric rigidity for various solvable. Read rigidity of quasiisometries of hmn associated with nondiagonalizable derivation of the heisenberg algebra, the quarterly journal of mathematics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Lectures on geometric group theory hunter college, cuny. A geometric action is a group action that is cocompact, isometric, and properly discontinuous. The main application of this is to give alternate definitions of quasi convex, or rational subgroups of negatively curved word hyperbolic groups. We give equivalent conditions on h to be quasi convex. Quasiisometric classification of graph manifold groups behrstock, jason a. We give equivalent conditions on h to be quasiconvex. Consequently, we show that any finitely generated group quasiisometric to a member of the subclass has to be polycyclic, and is virtually a lattice in an abelianbyabelian solvable lie group. Schaffercohen preprint 2016nonrectifiable delone sets in sol and other solvable groups with a. The primary motivating examples are the cayley graphs of. Problems on the geometry of finitely generated solvable groups. Xis an isomorphism, we say x is quasiisometrically rigid.
Rigidity of quasiisometries for symmetric spaces and. Our approach to these problems is to first classify all self quasiisometries of the solvable lie group. In this note, we announce the first results on quasi isometric rigidity of nonnilpotent polycyclic groups. The main application of this is to give alternate definitions of quasiconvex, or rational subgroups of. Not residually finite groups of intermediate growth, commensurability and nongeometricity show that there are uncountably many such groups. As an example, here we mention a quasi isometric rigidity result for solvable lie groups. Rigidity of some abelianbycyclic solvable group actions on. The results discussed here rely on a new technique for studying quasi isometries of finitely generated groups, which we refer to as coarse differentiation. Two metric spaces are quasiisometric if there exists a quasiisometry between them. For us, the space x is always a solvable lie group. Informally, quasi isometric rigidity is the situation when the arrow viqican be reversed.