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«QUASI-ISOMETRIES AND RIGIDITY OF SOLVABLE GROUPS ALEX ESKIN, DAVID FISHER AND KEVIN WHYTE Abstract. In this note, we announce the first results on ...»

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QUASI-ISOMETRIES AND RIGIDITY OF SOLVABLE GROUPS

ALEX ESKIN, DAVID FISHER AND KEVIN WHYTE

Abstract. In this note, we announce the first results on quasi-isometric rigidity

of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-

isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice

in Sol. We prove analogous results for groups quasi-isometric to R Rn where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasi-isometries of the solvable Lie group. Our classification of self quasi- isometries for R Rn proves a conjecture made by Farb and Mosher in [FM3].

Our techniques for studying quasi-isometries extend to some other classes of groups and spaces. In particular, we characterize groups quasi-isometric to any lamplighter group, answering a question of de la Harpe [dlH]. Also, we prove that certain Diestel-Leader graphs are not quasi-isometric to any finitely gener- ated group, verifying a conjecture of Diestel and Leader from [DL] and answering a question of Woess from [SW, Wo1]. We also prove that certain non-unimodular, non- hyperbolic solvable Lie groups are not quasi-isometric to finitely generated groups.

The results in this paper are contributions to Gromov’s program for classifying finitely generated groups up to quasi-isometry [Gr2]. We introduce a new technique for studying quasi-isometries, which we refer to as coarse differentiation.

Dedicated to Gregory Margulis on the occasion of his 60th birthday.

1. Introduction and statements of rigidity results For any group Γ generated by a subset S one has the associated Cayley graph, CΓ (S). This is the graph with vertex set Γ and edges connecting any pair of elements which differ by right multiplication by a generator. There is a natural Γ action on CΓ (S) by left translation. By giving every edge length one, the Cayley graph can be made into a (geodesic) metric space. The distance on Γ viewed as the vertices of the

Cayley graph is the word metric, defined via the norm:

γ = inf{length of a word in the generators S representing γ in Γ.} Different sets of generators give rise to different metrics and Cayley graphs for a group but one wants these to be equivalent. The natural notion of equivalence in this

category is quasi-isometry:

First author partially supported by NSF grant DMS-0244542. Second author partially supported by NSF grants DMS-0226121 and DMS-0541917. Third author partially supported by NSF grant DMS-0349290 and a Sloan Foundation Fellowship.

1

2 ALEX ESKIN, DAVID FISHER AND KEVIN WHYTE

Definition 1.1. Let (X,

–  –  –

If Γ is a finitely generated group, Γ is canonically quasi-isometric to any finite index subgroup Γ in Γ and to any quotient Γ = Γ/F for any finite normal subgroup F. The equivalence relation generated by these (trivial) quasi-isometries is called weak commensurability. A group is said to virtually have a property if some weakly commensurable group does.

In his ICM address in 1983, Gromov proposed a broad program for studying finitely generated groups as geometric objects, [Gr2]. Though there are many aspects to this program (see [Gr3] for a discussion), the principal question is the classification of finitely generated groups up to quasi-isometry. By construction, any finitely generated group Γ is quasi-isometric to any space on which Γ acts properly discontinuously and cocompactly by isometries. For example, the fundamental group of a compact manifold is quasi-isometric to the universal cover of the manifold (this is called the Milnor-Svarc lemma). In particular, any two cocompact lattices in the same Lie group G are quasi-isometric. One important aspect of Gromov’s program is that it allows one to generalize many invariants, techniques, and questions from the study of lattices to all finitely generated groups.

Given the motivations coming from the study of lattices, one of the first questions in the field is whether a group quasi-isometric to a lattice is itself a lattice, at least virtually. This question has been studied extensively. For lattices in semisimple groups this has been proven, see particularly [P1, S1, FS, S2, KL, EF, E] and also the survey [F] for further references. For lattices in other Lie groups the situation is less clear. It follows from Gromov’s polynomial growth theorem [Gr1] that any group quasi-isometric to a nilpotent group is virtually nilpotent, and hence essentially a lattice in some nilpotent Lie group. However, the quasi-isometry classification of lattices in nilpotent Lie groups remains an open problem.

In the case of solvable groups, even less is known. The main motivating question

is the following:

Conjecture 1.2. Let G be a solvable Lie group, and let Γ be a lattice in G. Any finitely generated group Γ quasi-isometric to Γ is virtually a lattice in a (possibly different) solvable Lie group G.

–  –  –

(2) Examples where G and G need to be different are known. See [FM3] and Theorem 1.4 below.





(3) Conjecture 1.2 can be rephrased to make no reference to connected Lie groups.

In particular, by a theorem of Mostow, any polycyclic group is virtually a lattice in a solvable Lie group, and conversely any lattice in a solvable Lie group is virtually polycyclic [Mo2]. The conjecture is equivalent to saying that any finitely generated group quasi-isometric to a polycyclic group is virtually polycyclic. This means that being polycyclic is a geometric property.

(4) Erschler has shown that a group quasi-isometric to a solvable group is not necessarily virtually solvable [D]. Thus, the class of virtually solvable groups is not closed under the equivalence relation of quasi-isometry. In other words, solvability is not a geometric property.

(5) Some classes of solvable groups which are not polycyclic are known to be quasi-isometrically rigid. See particularly the work of Farb and Mosher on the solvable Baumslag-Solitar groups [FM1, FM2] as well as later work of FarbMosher, Mosher-Sageev-Whyte and Wortman [FM3, MSW, W]. The methods used in all of these works depend essentially on topological arguments based on the explicit structure of singularities of the spaces studied and cannot apply to polycyclic groups.

(6) Shalom has obtained some evidence for the conjecture by cohomological methods [Sh]. For example, Shalom shows that any group quasi-isometric to a polycyclic group has a finite index subgroup with infinite abelianization. Some of his results have been further refined by Sauer [Sa].

Our main results establish Conjecture 1.2 in many cases. We believe our techniques provide a method to attack the conjecture. This is work in progress, some of it joint with Irine Peng.

From an algebraic point of view, solvable groups are generally easier to study than semisimple ones, as the algebraic structure is more easily manipulated. In the present context it is extremely difficult to see that any algebraic structure is preserved and so we are forced to work geometrically. For nilpotent groups the only geometric fact needed is polynomial volume growth. For semisimple groups, the key fact for all approaches is nonpositive curvature. The geometry of solvable groups is quite difficult to manage, since it involves a mixture of positive and negative curvature as well as exponential volume growth.

The simplest non-trivial example for Conjecture 1.2 is the 3-dimensional solvable Lie group Sol. This example has received a great deal of attention. The group Sol ∼ R R2 with R acting on R2 via the diagonal matrix with entries ez/2 and e−z/2.

=

As matrices, Sol can be written as :

    ez/2 x 0  0  (x, y, z) ∈ R3 Sol =  0 1 0 y e−z/2  

4 ALEX ESKIN, DAVID FISHER AND KEVIN WHYTE

The metric e−z dx2 + ez dy 2 + dz 2 is a left invariant metric on Sol. Any group of the form Z T Z2 for T ∈ SL(2, Z) with |tr(T )| 2 is a cocompact lattice in Sol.

The following theorem proves a conjecture of Farb and Mosher and is one of our

main results:

Theorem 1.3.

Let Γ be a finitely generated group quasi-isometric to Sol. Then Γ is virtually a lattice in Sol.

More generally, we can describe the quasi-isometry types of lattices in many solvable groups. Here we restrict our attention to groups of the form R Rn where the action of R on Rn is given by powers of an n by n matrix M. The following theorem proves another conjecture of Farb and Mosher.

Theorem 1.4.

Suppose M is a positive definite symmetric matrix with no eigenvalues equal to one, and G = R M Rn. If Γ is a finitely generated group quasi-isometric to G, then Γ is virtually a lattice in R M α Rn for some α ∈ R.

Remarks:

(1) This theorem is deduced from Theorem 2.2 below and a theorem from the Ph.d. thesis of T. Dymarz.

(2) This result is best possible. All the Lie groups R M α Rn for α = 0 in R are quasi-isometric.

The following is a basic question:

Question 1.5. Given a Lie group G, is there a finitely generated group quasi-isometric to G?

It is clear that the answer is yes whenever G has a cocompact lattice. However, many solvable locally compact groups, and in particular, many solvable Lie groups do not have any lattices. The simplest examples are groups which are not unimodular. However, it is possible for Question 1.5 to have an affirmative answer even if G is not unimodular. For instance, the non-unimodular group solvable group ab a 0, b ∈ R acts simply transitively by isometries on the hyperbolic 0 a−1 plane, and thus is quasi-isometric to the fundamental group of any closed surface of

genus at least 2. Thus the answer to Question 1.5 can be subtle. Our methods give:

Theorem 1.6.

Let G = R R2 be a solvable Lie group where the R action on R2 is defined by z·(x, y) = (eaz x, e−bz y) for a, b 0, a = b. Then there is no finitely generated group Γ quasi-isometric to G.

If a 0 and b 0, then G admits a left invariant metric of negative curvature. The fact that there is no finitely generated group quasi-isometric to G in this case is a result of Kleiner [K], see also [P2]. Both our methods and Kleiner’s prove similar results for groups of the form R Rn. Nilpotent Lie groups not quasi-isometric to any finitely generated group where constructed in [ET].

QUASI-ISOMETRIES OF SOLVABLE GROUPS 5

In addition our methods yield quasi-isometric rigidity results for a variety of solvable groups which are not polycyclic, in particular the so-called lamplighter groups. These are the wreath products Z F where F is a finite group. The name lamplighter comes from the description Z F = F Z Z where the Z action is by a shift. The subgroup F Z is thought of as the states of a line of lamps, each of which has |F | states. The ”lamplighter” moves along this line of lamps (the Z action) and can change the state of the lamp at her current position. The Cayley graphs for the generating sets F ∪ {±1} depend only on |F |, not the structure of F. Furthermore, Z F1 and Z F2 are quasiisometric whenever there is a d so that |F1 | = ds and |F2 | = dt for some s, t in Z.

The problem of classifying these groups up to quasi-isometry, and in particular, the question of whether the 2 and 3 state lamplighter groups are quasi-isometric, were well known open problems in the field, see [dlH].

Theorem 1.7.

The lamplighter groups Z F and Z F are quasi-isometric if and only if there exist positive integers d, s, r such that |F | = ds and |F | = dr.

For a rigidity theorem for lamplighter groups, see Theorem 1.8 below.

To state Theorem 1.8 as well as an analogue of Theorem 1.6 for groups which are not Lie groups, we need to describe a class of graphs. These are the Diestel-Leader graphs, DL(m, n), which can be defined as follows: let T1 and T2 be regular trees of valence m + 1 and n + 1. Choose orientations on the edges of T1 and T2 so each vertex has n (resp. m) edges pointing away from it. This is equivalent to choosing ends on these trees. We can view these orientations at defining height functions f1 and f2 on the trees (the Busemann functions for the chosen ends). If one places the point at infinity determining f1 at the top of the page and the point at infinity determining

f2 at the bottom of the page, then the trees can be drawn as:

–  –  –

Figure 1. The trees for DL(3, 2).

Figure borrowed from [PPS].

The graph DL(m, n) is the subset of the product T1 × T2 defined by f1 + f2 = 0.

The analogy with the geometry of Sol is made clear in section 3. For n = m the Diestel-Leader graphs arise as Cayley graphs of lamplighter groups Z F for |F | = n.

This observation was apparently first made by R.Moeller and P.Neumann [MN] and is described explicitly, from two slightly different points of view, in [Wo2] and [W].

We prove the following:

6 ALEX ESKIN, DAVID FISHER AND KEVIN WHYTE

Theorem 1.8. Let Γ be a finitely generated group quasi-isometric to the lamplighter group Z F. Then there exists positive integers d, s, r such that ds = |F |r and an isometric, proper, cocompact action of a finite index subgroup of Γ on the DiestelLeader graph DL(d, d).

Remark: The theorem can be reinterpreted as saying that any group quasi-isometric to DL(|F |, |F |) is virtually a cocompact lattice in the isometry group of DL(d, d) where d is as above.

In [SW, Wo1], Soardi and Woess ask whether every homogeneous graph is quasiisometric to a finitely generated group. The graph DL(m, n) is easily seen to be homogeneous (i.e. it has a transitive isometry group). For m = n its isometry group is not unimodular, and hence has no lattices. Thus there are no obvious groups

quasi-isometric to DL(m, n) in this case. In fact, we have:

Theorem 1.9.



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