Quasi-isometries of rank one S-arithmetic lattices

Quasi-isometries of rank one S-arithmetic lattices evin Wortman December 7, 2007 Abstract We complete the quasi-isometric classification of irreducible lattices in semisimple Lie groups over nondiscrete locally compact fields of characteristic zero by showing that any quasi-isometry of a rank one S-arithmetic lattice in a semisimple Lie group over nondiscrete locally compact fields of characteristic zero is a finite distance in the sup-norm from a commensurator. 1 Introduction Throughout we let be an algebraic number field, V the set of all inequivalent valuations on, and V V the subset of archimedean valuations. We will use S to denote a finite subset of V that contains V, and we write the corresponding ring of S-integers in as O S. In this paper, G will always be a connected non-commutative absolutely simple algebraic -group. 1.1 Commensurators For any valuation v V, we let v be the completion of with respect to v. For any set of valuations S V, we define G S = v S G v Supported in part by an N.S.F. Postdoctoral Fellowship. 1

and we identify GO S as a discrete subgroup of G S using the diagonal embedding. We let AutG S be the group of topological group automorphisms of G S. An automorphism ψ AutG S commensurates GO S if ψgo S GO S is a finite index subgroup of both ψgo S and GO S. We define the commensurator group of GO S to be the subgroup of AutG S consisting of automorphisms that commensurate GO S. This group is denoted as Comm AutGS GO S. Notice that it differs from the standard definition of the commensurator group of GO S in that we have not restricted ourselves to inner automorphisms. 1.2 Quasi-isometries For constants L 1 and C 0, an L, C quasi-isometric embedding of a metric space X into a metric space Y is a function φ : X Y such that for any x 1, x 2 X: 1 L d x 1, x 2 C d φx1, φx 2 Ld x 1, x 2 + C We call φ an L, C quasi-isometry if φ is an L, C quasi-isometric embedding and there is a number D 0 such that every point in Y is within distance D of some point in the image of X. 1.3 Quasi-isometry groups For a metric space X, we define the relation on the set of functions X X by φ ψ if sup d φx, ψx < x X In this paper we will call two functions equivalent if they are related by. For a finitely generated group with a word metric Γ, we form the set of all quasi-isometries of Γ, and denote the quotient space modulo by QIΓ. We call QIΓ the quasi-isometry group of Γ as it has a natural group structure arising from function composition. 1.4 Main result In this paper we show: 2

Theorem 1.4.1. Suppose is an algebraic number field, G is a connected non-commutative absolutely simple algebraic -group, S properly contains V, and rank G = 1. Then there is an isomorphism QIGO S = Comm AutGS GO S Two special cases of Theorem 1.4.1 had been previously known: Taback proved it when GO S is commensurable to PGL 2 Z[1/p] where p is a prime number [Ta] we ll use Taback s theorem in our proof, and it was proved when rank v G 2 for all v S by the author in [Wo 1]. Examples of S-arithmetic groups for which Theorem 1.4.1 had been previously unknown include when GO S = PGL 2 Z[1/m] where m is composite. Theorem 1.4.1 in this case alone was an object of study; see Taback- Whyte [Ta-Wh] for their program of study. Theorem 2.4.1 below presents a short proof of this case. 1.5 Quasi-isometry groups of non-cocompact irreducible S-arithmetic lattices Combining Theorem 1.4.1 with the results from Schwartz, Farb-Schwartz, Eskin, Farb, Taback, and Wortman [Sch 1], [Fa-Sch], [Sch 2], [Es], [Fa], [Ta], and [Wo 1] we have Theorem 1.5.1. Suppose is an algebraic number field, and G is a connected, non-commutative, absolutely simple, -isotropic, algebraic -group. If either Q, S V, or G is not Q-isomorphic to PGL 2, then there is an isomorphism QIGO S = Comm AutGS GO S Note that Theorem 1.5.1 identifies the quasi-isometry group of any noncocompact irreducible S-arithmetic lattice in a semisimple Lie group over nondiscrete locally compact fields of characteristic 0 that is not virtually free. Indeed, the condition that G is -isotropic is equivalent to GO S being non-cocompact, and it is well known that GO S is virtually free only when = Q, S = V, and G is Q-isomorphic to PGL 2. In this case, GO S is commensurable with PGL 2 Z. 3

1.6 Cocompact case By leiner-leeb [-L], the classification of quasi-isometries of cocompact S- arithmetic lattices reduces to the classification of quasi-isometries of real and p-adic simple Lie groups. This classification is known; see e.g. [Fa] for an account of the cases not covered in [-L]. 1.7 Function fields Our proof of Theorem 1.4.1 also applies when is a global function field when we add the hypothesis that there exists v, w S such that rank v G = 1 and rank w G 2. For more on what is known about the quasi-isometry groups of arithmetic groups over function fields and for a conjectural picture of what is unknown see [Wo 2]. 1.8 Outline of the proof and the paper We begin Section 2 with a sort of large-scale reduction theory. We examine a metric neighborhood, N, of an orbit of an S-arithmetic group, Γ, inside the natural product of Euclidean buildings and symmetric spaces, X. In Section 2.2 we show that the fibers of N under projections to building factors of X are geometric models for S-arithmetic subgroups of Γ. In Section 2.3, we apply results from [Wo 1] to extend a quasi-isometry φ : Γ Γ to the space X that necessarily preserves factors. Our general approach to proving Theorem 1.4.1 is to restrict φ to factors of X and use the results from Section 2.2 to decompose φ into quasi-isometries of S-arithmetic subgroups of Γ. Once each of these sub-quasi-isometries is understood, they are pieced together to show that φ is a commensurator. An easy example of this technique is given in Section 2.4 by PGL 2 Z[1/m]. We then treat the general case of Theorem 1.4.1 in Section 2.5 Our proof in Section 2.5 relies on the structure of horoballs for S-arithmetic groups associated to the product of a symmetric space and a single tree. We prove the results that we need for these horoballs that they are connected, pairwise disjoint, and reflect a kind of symmetry in factors in Section 3, so that Section 3 is somewhat of an appendix. The proof is organized in this way because, in the author s opinion, it just makes it easier to digest the material. There would be no harm though in reading Section 3 after Section 2.4 4

and before Section 2.5 for those who prefer a more linear presentation. 1.9 Acknowledgements I was fortunate to have several conversations with evin Whyte on the contents of this paper, and am thankful for those. In particular, he brought to my attention that SL 2 Z is the only non-cocompact arithmetic lattice in SL 2 R up to commesurability. I am also grateful to the following mathematicians who contributed to this paper: Tara Brendle, ariane Calta, Indira Chatterji, Alex Eskin, Benson Farb, Dan Margalit, Dave Witte Morris, and Jennifer Taback. 2 Proof of Theorem 1.4.1 Let G,, and S be as in Theorem 1.4.1, and let φ : GO S GO S be a quasi-isometry. 2.1 Geometric models For each valuation v of, we let X v be the symmetric space or Euclidean building corresponding to G v. If S is a finite set of valuations of, we let X S = v S X v Recall that there is a natural inclusion of topological groups AutG S IsomX S. Let O be the ring of integers in, and fix a connected subspace Ω V X V that GO acts cocompactly on. Let D X V be a fundamental domain for this action. For each nonarchimedean valuation w S V, we denote the ring of integers in w by O w. The group GO w is bounded in G w, so GO w fixes a point x w X w. We choose a bounded set D w X w containing x w with GO S D w = X w and such that gx w D w for g GO S implies that gx w = x w. For any set of valuations S satisfying V S S, we define the space Ω S = GO S D D w w S V 5

Note that Ω S is a subspace of X S. We endow Ω S with the path metric. Since GO S acts cocompactly on Ω S, we have the following observation: Lemma 2.1.1. For V S S, the space Ω S is quasi-isometric to the group GO S. 2.2 Fibers of projections to buildings are S-arithmetic In the large-scale, the fibers of the projection of Ω S onto building factors of X S are also S-arithmetic groups or more precisely, S -arithmetic groups. This is the statement of Lemma 2.2.2 below, but we will start with a proof of a special case. Lemma 2.2.1. The Hausdorff distance between Ω S X S {x w } w S S and is finite. Ω S w S S {x w } Proof. There are three main steps in this proof. First, if y Ω S, then y = gd for some g GO S and some d D w S V Since GO S GO w for all w S S, it follows from our choice of the points x w that {y} {x w } = g {d} {x w } Ω S w S S w S S D w Therefore, Ω S {x w } Ω S X S {x w } w S S w S S 1 6

Second, we suppose {y} {x w } Ω S w S S for some y X S. Then there exists a g GO S such that gy D w S V and gx w D w for all w S S. Notice that our choice of D w implies gx w = x w for all w S S. Thus, g is contained in the compact group H w = { h G w hx w = x w } for all w S S. Consequently, g is contained in the discrete group GO S G S D w w S S H w We name this discrete group Γ S. Note that we have shown {y} {x w } Γ S D D w w S S w S V Therefore, Ω S X S {x w } Γ S D w S S w S V D w 2 Third, we recall that GO S = GO S G S GO w w S S and use the definition of Γ S coupled with the fact that GO w H w to see that Γ S contains GO S. Since, Γ S and GO S are lattices in G S w S S H w, the containment GO S Γ S is of finite index. Therefore, the Hausdorff distance between Γ S D D w {x w } w S S w S V 7

and Ω S {x w } = GO S D w S S w S V D w {x w } w S S is finite. Combined with 1 and 2 above, the lemma follows. The more general form of Lemma 2.2.1 that we will use in our proof of Theorem 1.4.1 is the following lemma. We will use the notation of x S S for the point x w w S S X S S. Lemma 2.2.2. Suppose V S S. If y X S S and y GO S x S S, then the Hausdorff distance between Ω S X S {y} and is finite. Ω S {y} Remark. Our assumption in Lemma 2.2.2 that y GO S x S S is not a serious restriction over the assumption that y X S S. Indeed, GO S is dense in G S S, so the orbit GO S x S S is a finite Hausdorff distance from the space X S S. Proof. Let g GO S be such that y = gx S S. Then { h GO S hx S S = y } = g{ h GO S hx S S = x S S } = g GO S G S = gγ S w S S H w where H w and Γ S are as in the proof of the previous lemma. Now by our choice of the points x w X w for w S V at the beginning of this section, we have Ω S X S {y} = GO S D D w X S {y} = gγ S D w S V w S V D w {y} 8

Notice that the final space from the above chain of equalities is a finite Hausdorff distance from ggo S D D w {y} w S V since Γ S is commensurable with GO S. Because g commensurates GO S, the above space is also a finite Hausdorff distance from Ω S {y}. 2.3 Extending quasi-isometries of Ω S to X S Applying Lemma 2.1.1, we can regard our quasi-isometry φ : GO S GO S as a quasi-isometry of Ω S. Our goal is to show that φ is equivalent to an element of Comm AutGS GO S, and we begin by extending φ to a quasi-isometry of X S. Lemma 2.3.1. There is a permutation of S, which we name τ, and there are quasi-isometries φ v : X v X τv such that the restriction of the quasi-isometry φ S = v S φ v : X S X S to Ω S is equivalent to φ. If X v is a higher rank space for any v S, then φ v may be taken to be an isometry. Proof. By Proposition 6.9 of [Wo 1], the quasi-isometry φ : Ω S Ω S extends to a quasi-isometry of X. That is there is some quasi-isometry φ : X X such that φ φ ΩS where φ ΩS is the restriction of φ to Ω S. The map φ preserves factors in the boundary of X and an argument of Eskin s Proposition 10.1 of [Es] can be directly applied to show that φ is equivalent to a product of quasi-isometries of the factors of X, up to permutation of factors. Note that the statement of Proposition 10.1 from [Es] claims that X v and X τv are isometric for v V. This is because quasi-isometric symmetric spaces are isometric up to scale. 9

2.4 Example of proof to come Before continuing with the general proof, we ll pause for a moment to demonstrate the utility of Lemmas 2.2.2 and 2.3.1 by proving the following special case of Theorem 1.4.1. Theorem 2.4.1. If m N and m 1, then QI PGL 2 Z[1/m] = PGL2 Q Proof. Let = Q, G = PGL 2, and S = {v } {v p } p m where v is the archimedean valuation and v p is the p-adic valuation. Thus, GO S = PGL 2 Z[1/m], the space X v is the hyperbolic plane, and X vp is a p + 1- valent regular tree. If φ : PGL 2 Z[1/m] PGL 2 Z[1/m] is a quasi-isometry, then by Lemma 2.3.1 we can replace φ by a quasi-isometry φ S which is the product of quasi-isometries φ : X v X v and φ p : X vp X vτp for some permutation τ of the primes dividing m. If m is prime, then this theorem reduces to Taback s theorem [Ta]. Now suppose l is a prime dividing m. We let S = { v, v l } and S = { v, v τl }. It follows from the density of PGL 2 Z[1/m] in PGL 2 Q p p m ; p τl that any point in X S S is a uniformly bounded distance from the orbit PGL 2 Z[1/m]x S S. Therefore, we may assume that there is some y PGL 2 Z[1/m]x S S such that φ S XS {x S S } X S {y}. Since φω S Ω S, we may assume that φ S ΩS X S {x S S } Ω S X S {y}. It follows from Lemma 2.2.2 that φ S restricts to a quasiisometry between Ω S {x S S } and Ω S {y}. Note that by the product structure of φ S, we can assume that φ φ l restricts to a quasi-isometry between Ω S and Ω S or by Lemma 2.1.1, a quasi-isometry between PGL 2 Z[1/l] and PGL 2 Z[1/τl]. Taback showed that for any such quasi-isometry we must have l = τl and that φ φ l is equivalent to a commensurator g g l PGL 2 R PGL 2 Q l where g 10

and g l must necessarily be included in, and represent the same element of, PGL 2 Q [Ta]. As the above paragraph is independent of the prime l m, the element g PGL 2 Q determines φ S and thus φ. Having concluded our proof of the above special case, we return to the general proof. Our goal is to show that φ S is equivalent to an element of Comm AutGS GO S. At this point, the proof breaks into two cases. 2.5 Case 1: G V is not locally isomorphic to PGL 2 R Notice that Comm AutGS GO S acts by isometries on X S. So a good first step toward our goal is to show that φ S is equivalent to an isometry. First, we will show that φ V is equivalent to an isometry. Lemma 2.5.1. The quasi-isometry φ V : X V X V is equivalent to an isometry of the symmetric space X V. Indeed, it is equivalent to an element of Comm AutGS GO. Proof. Notice that φ V is simply the restriction of φ S to X V {x S V }. Since GO S is dense in G S V, the Hausdorff distance between GO S x S V and X S V is finite. Thus, by replacing φ S with an equivalent quasi-isometry, we may assume that X V {x S V } is mapped by φ S into a space X V {y} for some y X S V with y GO S x S V. Since the Hausdorff distance between φ S Ω S and Ω S is finite, we have by Lemmas 2.2.2 and 2.1.1 that φ V induces a quasi-isometry of GO. The lemma follows from the existing quasi-isometric classification of arithmetic lattices using our assumption in this Case that G V is not locally isomorphic to PGL 2 R; see [Fa]. At this point, it is not difficult to see that Theorem 1.4.1 holds in the case when every nonarchimedean factor of G S is higher rank: Lemma 2.5.2. If rank v G 2 for all v S V to an element of Comm AutGS GO S. then φ S is equivalent 11

Proof. By Lemma 2.3.1, φ v is equivalent to an isometry for all v S V. Combined with Lemma 2.5.1, we know that φ S is equivalent to an isometry. That φ S is equivalent to an element of Comm AutGS GO S follows from Proposition 7.2 of [Wo 1]. Indeed, any isometry of X S that preserves Ω S up to finite Hausdorff distance corresponds in a natural way to an automorphism of G S that preserves GO S up to finite Hausdorff distance, and any such automorphism of G S is shown in Proposition 7.2 of [Wo 1] to be a commensurator. For the remainder of Case 1, we are left to assume that there is at least one w S V such that rank w G = 1. Before beginning the proof of the next and final lemma for Case 1, it will be best to recall some standard facts about boundaries. Tree boundaries. If w is a nonarchimedean valuation of, and rank w G = 1, then X w is a tree. For any minimal w -parabolic subgroup of G, say M, we let ε M be the end of X w such that M w ε M = ε M. Notice that the space of all ends of the form ε P where P is a minimal -parabolic subgroup of G forms a dense subset of the space of ends of X w. Tits boundaries. For any minimal -parabolic subgroup of G, say P, we let δ P be the simplex in the Tits boundary of X V corresponding to the group v V P v. If δ is a simplex in the Tits boundary of X V, and ε is an end of the tree X w, then we denote the join of δ and ε by δ ε. It is a simplex in the Tits boundary of X T where T = V {w}. Lemma 2.5.3. Let w S V be such that rank w G = 1. Then φ w : X w X τw is equivalent to an isometry that is induced by an isomorphism of topological groups G w G τw. Proof. Below, we will denote the set of valuations V {τw} by T τ. We begin by choosing a minimal -parabolic subgroup of G, say P, and a geodesic ray ρ : R 0 X T that limits to the interior of the simplex δ P ε P. By Lemma 2.5.1, the image of φ T ρ under the projection X T X V limits to a point in the interior of δ Q for some minimal -parabolic subgroup Q. Similarly, φ w is a quasi-isometry of a tree, so it maps each geodesic 12

ray into a bounded neighborhood of a geodesic ray that is unique up to finite Hausdorff distance. Thus, the image of φ T ρ under the projection X T X τw limits to ε Q for some minimal τw -parabolic subgroup Q. Together, these results imply that φ T ρ is a finite Hausdorff distance from a geodesic ray that limits to a point in the interior of δ Q ε Q. By Lemma 3.4.1, there is a subspace H P of X T corresponding to P called a T -horoball such that t d ρt, X T H P is an unbounded function. Thus, t d φ T ρt, X T τ φ T H P is also unbounded. Using Lemma 3.2.1 and the fact that φ T Ω T is a finite Hausdorff distance from Ω T τ, we may replace φ T with an equivalent quasi-isometry to deduce that φ T H P is contained in the union of all T τ -horoballs in X T τ. It will be clear from the definition given in Section 3 that each T -horoball is connected. In addition, Lemma 3.2.2 states that the collection of T τ - horoballs is pairwise disjoint, so it follows that φ T H P is a finite distance from a single T τ -horoball H M X T τ where M is a minimal -parabolic subgroup of G. Therefore, t d φ T ρt, X T τ H M is unbounded. Because the above holds for all ρ limiting to δ P ε P, and because φ T ρ limits to δ Q ε Q, we have by Lemma 3.4.2, that Q = M = Q. That is, φ V completely determines the map that φ w induces between the ends of the trees X w and X τw that correspond to -parabolic subgroups of G. Recall that by Lemma 2.5.1, φ V is equivalent to a commensurator of GO G V. Using Lemma 7.3 of [Wo 1], φ V regarded as an automorphism of G V restricts to G as a composition δ σ : G G 13

where σ is an automorphism of, σ : G σ G is the map obtained by applying σ to the entries of elements in G, and δ : σ G G is a -isomorphism of -groups. Thus, if X w and X τw are the space of ends of the trees X w and X τw respectively, and if φ w : X w X τw is the boundary map induced by φ w, then we have shown that φ w ε P = ε δ σ P for any P G that is a minimal -parabolic subgroup of G. Our next goal is to show that the valuation w σ 1 is equivalent to τw. If this is the case, then δ σ extends from a group automorphism of G to a topological group isomorphism α w : G w G τw If α w : X w X τw is the map induced by α w, then α w equals φ w on the subset of ends in X w corresponding to -parabolic subgroups of G since α w extends δ σ. Therefore, α w = φ w on all of X w by the density of the -rational ends in X w. Thus, α w determines φ w up to equivalence. This would prove our lemma. So to finish the proof of this lemma, we will show that w σ 1 is equivalent to τw. For any maximal -split torus S G, we let γs w X w resp. γ τw S X τw be the geodesic that S w resp. S τw acts on by translations. Fix S and T, two maximal -split tori in G such that γs w γw T is nonempty and bounded. We choose a point a γs w γw T. Since SO T is dense in S w, there exists a group element g n SO T for each n N such that d g n γt w, a > n 14

Note that g n γt w = γw. Thus g ntgn 1 d φ w γ w g ntg 1 n, φ wa is an unbounded sequence. As g n Tgn 1 is -split, φ w γ w is a uniformly bounded Hausdorff distance from g ntgn 1 γ τw = δ σ g n γ τw δ σ T δ σ g ntgn 1 because a geodesic in X τw is determined by its two ends. We finally have that d δ σ g n γ τw δ σ T, φ wa is an unbounded sequence. It is this statement that we shall contradict by assuming that w σ 1 is inequivalent to τw. Note that g n GO v for all v V T since g n SO T. Thus, σ 0 g n σ GO v σ 1 for all v V T. If it were the case that w σ 1 is inequivalent to τw, then it follows that σ 0 g n σ GO τw. Hence, δ σ 0 g n defines a bounded sequence in G τw. Therefore, d δ σ g n γ τw δ σ T, φ wa is a bounded sequence, our contradiction. The proof of Theorem 1.4.1 in Case 1 is complete with the observation that applications of Lemma 2.5.3 to tree factors, allows us to apply the Proposition 7.2 of [Wo 1] as we did in Lemma 2.5.2. 2.6 Case 2: G V is locally isomorphic to PGL 2 R It follows that V contains a single valuation v, and that v = R. Thus = Q, and V is the set containing only the standard real metric on Q. Our assumption that G is absolutely simple implies that G V is actually isomorphic to PGL 2 R. Thus, G is a Q-form of PGL 2. As we are assuming that G is Q-isotropic, it follows from the classification of Q-forms of PGL 2 that G and PGL 2 are Q-isomorphic see e.g. page 55 of [Ti]. 15

From our assumptions in the statement of Theorem 1.4.1, S V. As the only valuations, up to scale, on Q are the real valuation and the p-adic valuations, GO S is commensurable with PGL 2 Z[1/m] for some m N with m 1. Thus, Case 2 of Theorem 1.4.1 follows from Theorem 2.4.1. Our proof of Theorem 1.4.1 is complete modulo the material from Section 3. 3 Horoball patterns in a product of a tree and a symmetric space In this section we will study the components of X S Ω S when X S is a product of a symmetric space and a tree. Setting notation. We let w be a nonarchimedean valuation on such that rank w G = 1. Then we set T equal to V {w}. 3.1 Horoballs in rank one symmetric spaces. Let P be a minimal -parabolic subgroup of G. As in the previous section, we let δ P be the simplex in the Tits boundary of X V corresponding to the group v V P v. Note that G being -isotropic and rank w G = 1 together implies that rank G = 1. Borel proved that the latter equality implies that X V Ω V can be taken to be a disjoint collection of horoballs 17.10 [Bo]. To any horoball of X V Ω V, say H, there corresponds a unique δ P as above such that any geodesic ray ρ : R 0 X V that limits to δ P defines an unbounded function t d ρt, X V H 3.2 T -horoballs in X T Let y X w and suppose y GO T x w. Recall that by Lemma 2.2.2, the space Ω T X V {y} is a finite Hausdorff distance from Ω V {y}. For any minimal -parabolic subgroup of G, say P, we let H y X V {y} be the horoball of Ω T X V {y} that corresponds to δ P. 16

For arbitrary x X w, we define H x = H y where y GO T x w minimizes the distance between x and GO T x w. We let H P = H x {x} x X w Each of the spaces H P is called a T -horoball. Let P be the set of all minimal -parabolic subgroups of G. The following lemma follows directly from our definitions. It will be used in the proof of Lemma 2.5.3. Lemma 3.2.1. The Hausdorff distance between X T Ω T and is finite. P P H P We record another observation to be used in the proof of Lemma 2.5.3. Lemma 3.2.2. If P Q are minimal -parabolic subgroups of G, then H P H Q =. Proof. The horoballs comprising X V Ω V {xw } are pairwise disjoint, and are a finite Hausdorff distance from the horoballs of Ω T X V {x w } by Lemma 2.2.1. Hence, if y = gx w for some g GO T, then the horoballs determined by Ω T X V {y} = g [Ω T X V {x w } ] are disjoint. 3.3 Deformations of horoballs over geodesics in X w We let π : X T X V be the projection map. Note that if x X w and P is a minimal -parabolic subgroup of G, then πh x is a horoball in X V that is based at δ P. Recall that for any minimal w -parabolic subgroup of G, say Q, we denote the point in the boundary of the tree X w that corresponds to Q w by ε Q. 17

Lemma 3.3.1. Suppose P is a minimal -parabolic subgroup of G and that Q is a minimal w -parabolic subgroup of G. If γ : R X w is a geodesic with γ = ε P and γ = ε Q then i s t implies π H γs γt π H ii t R π H γt = XV iii t R H γt = iv There exists constants L P, C > 0 such that if hs, t = d π H γs γt, π H then hs, t L P s t C Proof. As the ends of X w corresponding to -parabolic subgroups are a dense subset of the full space of ends, it suffices to prove this lemma when Q is defined over. In this case, the image of γ corresponds to a -split torus S G that is contained in P. Let α be a root of G with respect to S such that α is positive in P. Since the diagonal embedding of SO T in the group v V S v has a dense image, there is some b SO T such that αsb v < 1 for all v V. Thus, Sbπ H γ0 is a horoball in XV that strictly contains π H γ0. Generally, we have Sb m π H γ0 Sb n π H γ0 for all m, n Z with m < n. By the product formula we have αsb w > 1. Thus, there is a positive number λ > 0 such that γnλ = Sb n γ0 for any n Z. It follows for m < n that We let so that π H γmλ = Sb m π H γ0 Sb n π H γ0 γnλ = π H L P = 1 λ d π H γ0 γλ, π H hmλ, nλ = L P λ m n = L P mλ nλ 18

Then we take and, say, C = { max d π H γs γt }, π H 0 s t λ C = 2C + L P d γ0, γλ 3.4 Basepoints in the Tits boundary for T -horoballs The Tits boundary for X is the spherical join of the Tits boundary for the symmetric space X V and the Tits boundary for the tree X w. The purpose of the following two lemmas and of this entire section is to show that each T -horoball H P is geometrically associated with the join of δ P and ε P, denoted δ P ε P. Lemma 3.4.1. Let P be a minimal -parabolic subgroup of G. Any geodesic ray ρ : R 0 X T that limits to the simplex δ P ε P in the Tits boundary of X T defines an unbounded function when composed with the distance from the complement of H P in X T : t d ρt, X T H P Proof. Any such geodesic ray ρ is a product of a geodesic ray b : R 0 X V that limits to δ P and a geodesic ray c : R 0 X w that limits to ε P. Let Y = π H c0 cr 0. Since t d bt, X V π H c0 is unbounded, t d ρt, X T Y is unbounded. The lemma follows from Lemma 3.3.1i which guarantees that Y H P. Lemma 3.4.2. Suppose Q and M are minimal -parabolic subgroups of G, and that Q is a minimal w -parabolic subgroup of G with M Q or 19

M Q. Then there is a geodesic ray ρ : R 0 X with ρ δ Q ε Q such that the function t d ρt, X T H M is bounded. Proof. Choose a geodesic ray b : R 0 X V that limits to δ Q and a geodesic ray c : R 0 X w that limits to ε Q. Let r be the ratio of the speed of b to the speed of c. If M Q, then after ignoring at most a bounded interval of c, we can extend c to a bi-infinite geodesic with c = ε M. With L M as in Lemma 3.3.1, ρt = bl M t, crt defines a geodesic ray satisfying the lemma. In the remaining case, M Q and M = Q. The distance from bt to π HM, b0 is a convex function in t. Since M Q, this function has a positive derivative, u > 0, for some large value of t. Then ρt = bl M t, curt defines a geodesic ray satisfying the lemma. References [Bo] [Es] Borel, A., Introduction aux groupes arithmétiques. Hermann, Paris 1969. Eskin, A., Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces. J. Amer. Math. Soc., 11 1998, 321-361. [Fa] Farb, B., The quasi-isometry classification of lattices in semisimple Lie groups. Math. Res. Letta., 4 1997, 705-717. [Fa-Sch] [-L] Farb, B., and Schwartz, R., The large-scale geometry of Hilbert modular groups. J. Diff. Geom., 44 1996, 435-478. leiner, B., and Leeb, B., Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. Inst. Hautes Études Sci. Publ. Math., 86 1997, 115-197. 20

[Sch 1] [Sch 2] [Ta] [Ta-Wh] [Ti] Schwartz, R., The quasi-isometry classification of rank one lattices. Inst. Hautes Études Sci. Publ. Math., 82 1995, 133-168. Schwartz, R., Quasi-isometric rigidity and Diophantine approximation. Acta Math., 177 1996, 75-112. Taback, J., Quasi-isometric rigidity for P SL 2 Z[1/p]. Duke Math. J., 101 2000, 335-357. Taback, J., and Whyte,., The large-scale geometry of some metabelian groups. Michigan Math. J. 52 2004, 205-218. Tits, J., Classification of algebraic semisimple groups. 1966 Algebraic Groups and Discontinuous Subgroups Proc. Sympos. Pure Math., Boulder, Colo., 1965 p. 33-62 Amer. Math. Soc., Providence, R.I., 1966. [Wo 1] Wortman,., Quasi-isometric rigidity of higher rank S- arithmetic lattices. Geom. Topol. 11 2007, 995-1048. [Wo 2] Wortman,., Quasi-isometric rigidity of higher rank arithmetic lattices over function fields. In appendix to: Einsiedler, M., and Mohammadi, A., A Joining classification and a special case of Raghunathan s conjecture in positive charactersitic. Preprint. evin Wortman Department of Mathematics University of Utah 155 South 1400 East, Room 233 Salt Lake City, UT 84112-0090 wortman@math.utah.edu 21