DEHN FUNCTIONS OF HIGHER RANK ARITHMETIC GROUPS OF TYPE A n IN PRODUCTS OF SIMPLE LIE GROUPS

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1 DEHN FUNCTIONS OF HIGHER RANK ARITHMETIC GROUPS OF TYPE A n IN PRODUCTS OF SIMPLE LIE GROUPS MORGAN CESA Abstract. Suppose Γ is an arithmetic group defined over a global field K, that the K-type of Γ is A n with n 2, and that the ambient semisimple group that contains Γ as a lattice has at least two noncocompact factors. We use results from Bestvina-Eskin- Wortman and Cornulier-Tessera to show that Γ has a polynomially bounded Dehn function. 1. Introduction Let K be a global field, and S a finite, nonempty set of inequivalent valuations on K. Denote by O S the ring of S-integers in K, and let K v be the completion of K with respect to v S. Let G be a noncommutative absolutely almost simple K-isotropic K-group, and let G = v S G(K v). Note that S is the number of simple factors of G, and that G(O S ) is a lattice in G under the diagonal embedding. The geometric rank of G is k(g, S) = v S rank K v (G). The Lie group G is endowed with a left invariant metric, which we will denote d G. Lubotzky-Mozes-Raghunathan showed that if k(g, S) 2, then the word metric on G(O S ) is Lipschitz equivalent to the restriction of d G to G(O S ) [LMR00]. The following is a slight generalization of a conjecture due to Gromov [Gro93]: Conjecture 1. If k(g, S) 3, then the Dehn function of G(O S ) is quadratic. Druţu showed that if k(g, S) 3, rank K (G) = 1, and S contains only archimedean valuations, then the Dehn function of G(O S ) is bounded above by the function x x 2+ɛ for any ɛ > 0 [Dru98]. Young showed that if G(O S ) is SL n (Z) and n 5 (i.e. k(g, S) 4) then the Dehn function of G(O S ) is quadratic [You13]. Cohen showed that if G(O S ) is Sp 2n (Z) and n 5 (i.e. k(g, S) 5) then Date: October 17,

2 2 MORGAN CESA the Dehn function of G(O S ) is quadratic [Coh14]. Bestvina-Eskin- Wortman proved that if S 3 (that is, G has at least 3 factors, which implies that k(g, S) 3)), then the Dehn function of G(O S ) is polynomially bounded [BEW13]. In this paper, we will prove: Theorem 2. If the K-type of G is A n, n 2, and S 2, then the Dehn function of G(O S ) is bounded by a polynomial of degree 3 2 n. For example, Theorem 2 implies that the following groups have polynomially bounded Dehn functions: SL 3 (Z[ 2]), or more generally SL n+1 (O K ) where n 2, O K is a ring of algebraic integers in a number field K, and O K is not isomorphic to Z or Z[i]; SL n+1 (Z[1/k]) where n 2 and k N {1}; and SL n+1 (F p [t, t 1 ]) where n 2 and p is prime. Indeed, SL n+1 is of type A n regardless of the relative global field K, and Z[ 2], O K, Z[1/k], and F p [t, t 1 ] are rings of S-integers with S Dehn Functions and Isoperimetric Inequalities. If H = s 1,..., s n r 1,..., r k is a finitely presented group, and w is a word in the generators of H which represents the identity, then there is a finite sequence of relators which reduces w to the trivial word. Let δ H (w) be the minimum number of steps to reduce w to the trivial word. The Dehn function of H is defined as δ H (n) = max δ H(w) length(w) n While the Dehn function depends on the presentation of H, the growth class of the Dehn function is a quasi-isometry invariant of H. The Dehn function of a simply connected CW -complex X is defined analogously. For any loop γ X, let δ X (γ) be the minimal area of disk in X that fills γ. The Dehn function of X is then δ X (n) = max δ X(γ) length(γ) n If X is quasi-isometric to H (for example, if X has a free, cellular, properly discontinuous cocompact H-action), then the growth class of δ X is the same as that of δ H Coarse Manifolds. An r-coarse manifold in a metric space X is the image of a map from the vertices of a triangulated manifold M into X, with the property that any pair of adjacent vertices in M are mapped to within distance r of each other. We will abuse notation slightly and refer to the image of the map as an r-coarse manifold as well. If Σ is a coarse manifold, then Σ is the restriction of the map to

3 DEHN FUNCTIONS OF LATTICES OF TYPE A n IN PRODUCTS 3 M. If M is an n-manifold, we will say Σ is a coarse n-manifold, and we define the length or area of Σ to be the number of vertices in Σ. We say two coarse n-manifolds, Σ and Σ, have the same topological type if the underlying manifolds M and M have the same topological type Bounds. We will write a = O(C) to mean that there is some constant k, which depends only on G and G(O S ), such that a kc Acknowledgements. The author would like to thank her Ph.D. advisor, Kevin Wortman, under whose direction this work was carried out, for suggesting this problem and for his support and encouragement. 2. Preliminaries 2.1. Parabolic Subgroups. Let K, S, and G be as above. There is a minimal K-parabolic subgroup P G, and P contains a maximal K-split torus which we will call A. Let Φ be the root system for (G, A), and observe that P determines a positive subset Φ + Φ. Let denote the set of simple roots in Φ +. (Note that = rank K (G).) For any I, [I] will denote the linear combinations generated by I. Let Φ(I) + = Φ + [I] and [I] + = [I] Φ +. If α Φ, let U (α) be the corresponding root group. For any Ψ Φ + which is closed under addition, let Note that U Ψ = α Ψ U (α) U Ψ (K v ) v S can be topologically identified with a product of vector spaces and therefore can be endowed with a norm,. Let A I be the connected component of the identity in ( α I ker(α)). The centralizer of A I in G, Z G (A I ), can be written as Z G (A I ) = M I A I, where M I is a reductive K-group with K-anisotropic center. Notice that M I A I normalizes U Φ(I) +, and M I commutes with A I. We define the standard parabolic subgroup P I of G to be P I = U Φ(I) +M I A I Note that P = P and that when α, P α is a maximal proper K-parabolic subgroup of G.

4 4 MORGAN CESA We will use unbolding to denote taking the product over S of the local points of a K-group, as in G = v S G(K v ) 2.2. Parabolic regions and reduction theory. The following theorem was proved in different cases by Borel, Behr, and Harder (cf. [Bor91] Proposition 15.6, [Beh69] Satz 5 and Satz 8, and [Har69] Korrolar 2.2.7). A summary of the individual results and how they imply the theorem is given in [BEW13]. Theorem 3 (Borel, Behr, Harder). There is a finite set F G(K) of coset representatives for G(O S )\G(K)/P(K). Any proper K-parabolic subgroup Q of G is conjugate to P I for some I. Let Λ Q = {γf G(O S )F (γf) 1 P I (γf) = Q for some I } By Theorem 3, Λ Q is nonempty. For a A and α Φ, let α(a) = v S α(a) v where v is the norm on K v. For t > 0 and I, let and A + I to be A + I (t) = {a A I α(a) t if α I} = A + I (1). We define the parabolic region corresponding to Q R Q (t) = Λ Q U Φ(I) +M I (O S )A + I (t) The boundary of A + I (t) is A + I (t) = {a A+ I there exists α I with α(a) α(b) for all b A+ I } and the boundary of the parabolic region R Q (t) is R Q (t) = Λ Q U Φ(I) +M I (O S ) A + I (t) For 0 n <, let P(n) be the set of K-parabolic subgroups of G that are conjugate via G(K) to some P I with I = n. The necessary reduction theory is proved in [BEW13]: Theorem 4 (Bestvina-Eskin-Wortman, 2013). There exists a bounded set B 0 G, and given a bounded set B n G and any N n 0 for 0 n <, there exists t n > 1 and a bounded set B n+1 G such that (i) G = Q P(0) R Q B 0 ; (ii) if Q, Q P(n) and Q Q, then the distance between R Q (t n )B n and R Q (t n )B n is at least N n ;

5 DEHN FUNCTIONS OF LATTICES OF TYPE A n IN PRODUCTS 5 (iii) G(O S ) R Q (t n )B n = for all n; (iv) if n 2 then ( Q P(n) R Q B n ) ( Q P(n) R Q (2t n )B n ) is contained in Q P(n+1) R Q B n+1 ; (v) ( Q P( 1) R Q B 1 ) ( Q P( 1) R Q (2t 1 )B 1 ) is contained in G(O S )B ; and (vi) if Q P(n), then there is an (L, C) quasi-isometry R Q (t n )B n U Φ(I) +M I (O S )A + I for some I with I = n. The quasiisometry restricts to an (L, C) quasi-isometry R Q (t n )B n U Φ(I) +M I (O S ) A + I where L 1 and C 0 are independent of Q Filling coarse manifolds in the boundaries of parabolic regions. For I, we let R I = U Φ(I) +M I (O S )A + I. Proposition 5. Suppose I is a set of simple roots, and let R I denote the corresponding parabolic region of G. Given r > 0, there is some r R >0 such that if Σ R I is an r-coarse 2-manifold of area L, then there is an r -coarse 2-manifold Σ R I such that Σ = Σ. Furthermore, if I 2, then area(σ ) = O(L 2 ) and if I = 1 then area(σ ) = O(L 3 ). Proposition 5 is proved in Sections 3 (for nonmaximal parabolics) and 4 (for maximal parabolics). That Proposition 5 implies Theorem 2 is essentially proved in Bestvina- Eskin-Wortman (see [BEW13] Sections 6 and 7). We restate it here in the specific case we require, and add explicit bounds on filling areas. Proof of Theorem 2. Let X be a simply connected CW complex on with a free, cellular, properly discontinuous and cocompact G(O S )- action. Let x X be a basepoint, and define the orbit map φ : G(O S ) G(O S ) x Note that φ is a bijective quasi-isometry between G(O S ) with the left invariant metric d G and the orbit G(O S ) x with the path metric from X. Let l X be a cellular loop. The G(O S )-action on X is cocompact, so every point in l is a uniformly bounded distance from G(O S ). Therefore, there is a constant r 0 > 0 such that after a uniformly bounded perturbation, l G(O S ) is an r 0 -coarse loop and the Hausdorff distance between l and l G(O S ) is bounded. Let L be the length of l G(O S ). There is a constant r 1 > 0 which depends only on r 0 and the quasiisometry constants of φ such that φ 1 (l G(O S )) is an r 1 -coarse loop in G(O S ). Since G is quasi-isometric to a CAT (0) space (a product of Euclidean buildings and symmetric spaces), there is an r 1 -coarse disk D G with D = φ 1 (l G(O S ) x) and area O(L 2 ).

6 6 MORGAN CESA Set D = D 0 and N 0 = 2r 1. Let B 1 and t 0 be as in Theorem 4. If Q P(0), let D 0,Q = D 0 R Q (t 0 )B 0 Note that D 0,Q and D 0,Q are disjoint if Q Q. For each Q P(0), we can perturb D 0,Q by at most r 1 to ensure that D 0,Q R Q (t 0 )B 0. By Proposition 4(vi), R Q (t 0 )B 0 is quasi-isometric to R. By Proposition 5, there is some r 2 depending only on r 1 and the quasi-isometry constants and an r 2 -coarse 2-manifold D 0,Q R Q(t 0 )B 0 such that the 2-manifold obtained by replacing each D 0,Q by D 0,Q, D 1 = D 0 Q P(0) D 0,Q Q P(0) D 0,Q is an r 1 -coarse 2-disk, and area(d 1 ) = O(area(D 0 ) 2 ) = O(L 4 ). By Proposition 4(iv), D 1 R Q B 0 R Q (2t 0 )B 0 R Q B 1 Q P(0) Q P(0) Q P(1) By Proposition 4(iii), G(O S ) R Q (t 0 )B 0 =, and therefore D 0 = D 1. For 1 n 1 repeat the above process with n in place of 0, to obtain an r n+1 -coarse disk D n+1 with D n+1 = D 0 and area(d n+1 ) = O(L k n+1 ), where k n+1 = 2 n+2 if n 2 and k = 3 2. Furthermore, D n+1 R Q B n R Q (2t n )B n Q P(n) Q P(n) which implies that D G(O S )B by Proposition 4(v). Since G(O S )B is finite Hausdorff distance from G(O S ), there is some r > 0 such that there is an r -coarse disk D G(O S ) with D = φ 1 (l G(O S ) x) and area(d ) = O(L k ), where k = 3 2. There is some r > 0 which depends only on r and the quasiisometry constants of φ such that φ(d ) X is an r -coarse disk with boundary l G(O S ) x. First connect pairs of adjacent vertices in φ(d ) by 1-cells to obtain D, then add 2-cells whose 1-skeleton is in D to obtain D. Note that D = l, D is a bounded Hausdorff distance to φ(d ), and the number of cells in D is O(area(D )) = O(L k ) where k = 3 2. Recall that = rank K (G), so the Dehn function of G(O S ) is bounded by a polynomial of degree 3 2 rank K(G).

7 DEHN FUNCTIONS OF LATTICES OF TYPE A n IN PRODUCTS More preliminaries. Lemma 6. Given r > 0 sufficiently large, I, and S S, there is some a A I (O S ) that strictly contracts all root subgroups of v S U Φ(I) +(K v ), and d G (a, 1) r. Proof. Lemma 12 in Bestvina-Eskin-Wortman show that the projection of A I (O S ) to v S A I (K v ) is a finite Hausdorff distance from v S A I (K v ) [BEW13]. (The proof is independent of S.) This implies that there is some a A I (O S ) such that α(a) v < 1 for all α I and v S. Therefore, if u v S U (β) (K v ) for some β Φ(I) +, then a 1 ua < u. We will make use of the following lemma in both the maximal and nonmaximal parabolic cases: Lemma 7. Let r > 0 and I. If u U Φ(I) +, then there is an r- coarse path p u U Φ(I) +A + I (O S) joining u to 1 such that length(p u ) = O(d G (u, 1)). Proof. Let L = d G (u, 1), and notice that u O(e L ). Letting S = {v 1,..., v k }, we can write u = (u 1,..., u k ), where u i U Φ(I) +(K vi ). By the bound on u, we also have u i O(e L ). By Lemma 6, we can choose a i A + I (O S) such that a i strictly contracts U Φ(I) +(K vi ) and d G (a i, 1) r. For some T i = O(L), d G (a T i i u i a T i i, 1) = d G(u i a T i i, at i i ) 1. Let p i = {a k i 0 k T i } {ua k i 0 k T i }. Note that p i is an r-coarse path from 1 to u i of length O(L). Taking ( ) p u = p 1 (u 1,..., u i 1, 1,..., 1) p i 2 i k gives the desired path from 1 to u. 3. Nonmaximal Parabolic Subgroups In this section, we will prove Proposition 5 in the case where I 2. First, we will divide R I into two pieces. Recall that R I = U Φ(I) +M I (O S ) A + I A + I = {a A+ I there exists α I with α(a) α(b) for all b A+ I } For α I, we define A + I,α, Z+ I,α, B I,α, and ˆB I,α as follows: A + I,α = { a A + I α(a) α(b) for all b } A+ I Z + I,α = β (I α) A + I,β

8 8 MORGAN CESA B I,α = U Φ(I) +M I (O S )A + I,α ˆB I,α = U Φ(I) +M I (O S )Z + I,α Note that A + I = α I A + I,α and that R I = B I,α ˆB I,α. We also observe that A + I,α A+ I α, since A I(O S ) A + I,α for any α I, but A I (O S ) A + I α in general. Since A + I is quasi-isometric to a Euclidean space, there is a projection to A + I which is distance nonincreasing. Note that M I(O S ) commutes with A + I, so there is a distance nonincreasing map M I(O S )A + I M I (O S ) A + I. Let π I : R I R I be the composition of the distance nonincreasing maps U Φ(I) +M I (O S )A I M I (O S )A + I and M I(O S )A + I M I (O S ) A + I. Lemma 8. Suppose I is a set of simple roots such that I 2 and let r > 0 and α I be given. If Σ R I is an r- coarse 2-manifold with boundary and Σ R I, then Σ = Σ 1 Σ 2 for r-coarse 2-manifolds with boundary, Σ 1 and Σ 2, such that (i) π I ( Σ 1 ) B I,α and π I ( Σ 2 ) ˆB I,α, (ii) Σ 1 Σ B I,α and Σ 2 Σ ˆB I,α, and (iii) Σ 1 Σ 2 consists of finitely many r-coarse paths p 1,..., p k, with π I (p i ) B I,α and finitely many r-coarse loops γ 1,..., γ n with π I (γ l ) B I,α. Proof. By transversality, π I (Σ) intersects B I,α in an r-coarse 1-manifold which is made up of finitely many r-coarse paths ( p 1,..., p k ) with endpoints in π I ( Σ) and finitely many r-coarse loops ( γ 1,..., γ n ) which do not intersect π I ( Σ). Furthermore, π I (Σ) intersects B I,α (respectively ˆB I,α ) in a 2-manifold with boundary, Σ 1 (respectively Σ 2 ), and (1) Σ i = ( Σ i π I ( Σ)) ( p 1 p k ) ( γ 1 γ n ) For x R I, note that π I (x) B I,α if and only if x B I,α (since π I only changes the unipotent coordinates of points in R I ). Let Σ 1 and Σ 2 be the respective preimages of Σ 1 and Σ 2 under π I restricted to Σ. Note that p i and γ i lift to r-coarse paths and loops p i and γ i in Σ. Conclusion (i) holds because Σ i = π I (Σ i ), and conclusions (ii) and (iii) hold by (1) and the definition of p i and γ l. Lemma 9. Suppose I is a set of simple roots such that I 2 and let r > 0 and α I be given. If Ω is a closed r-coarse 1-manifold in B I,α or ˆB I,α with diameter and distance to B I,α ˆB I,α bounded by L, then there is an r -coarse 2-manifold A R I such that A = Ω uπ I (Ω) for some u U Φ(I) + and area(a) = O(L 2 ).

9 DEHN FUNCTIONS OF LATTICES OF TYPE A n IN PRODUCTS 9 Proof. We will begin with the case where Ω B I,α. For x Ω, we can write x = u x m x a x with u x U Φ(I) +, m x M I (O S ), and a x A + I,α. Since the diameter of Ω is bounded by L, u 1 x u y O(e L ) for any x, y Ω. Choose b int(a + I α ) with d G(b, 1) r. Note that b commutes with U [I α], M I (O S ), and A + I, and that conjugation by b 1 strictly contracts U Φ(I α) +. Also, U Φ(I) + = U Φ(I α) +U [I α] Φ(I) +, so conjugation by b 1 does not expand any root group in U Φ(I) +. Since d G (b, 1) r, d G (gb, g) r for any g G by left-invariance of the metric on G. Right multiplication by b k is distance nonincreasing on Ω when k 0, since for any x, y Ω, d G (xb k, yb k ) = d G (u x m x a x b k, u y m y a y b k ) = d G (u x b k m x a x, u y b k m y a y ) = d G (b k u 1 y u x b k m x a x, m y a y ) d G (u 1 y u x m x a x, m y a y ) = d G (u x m x a x, u y m y a y ) = d G (x, y) Therefore, 0 k n Ωb k is a 2r-coarse 2-manifold for any n N, which has the topological type of Ω [0, 1], boundary Ω Ωb n and whose area is bounded by Ln. There is some T = O(L) such that the U Φ(I α) +- coordinates of Ωb T are nearly constant. More precisely, there is some fixed u U Φ(I α) + and some v x U [I α] Φ(I) + for each x such that d(u x m x a x b T, u v x m x a x b T ) r for every x in Ω. Let Ω 1 = {u v x m x a x b T } x Ω and let A 1 = Ω 1 ( 0 k T Ωb k ) be the 2r-coarse 2-manifold with boundary Ω Ω 1. Note that area(a 1 ) = O(L 2 ). Let Ω 2 = {u v x m x a x } x Ω. Note that Ω 2 is an r-coarse 1-manifold of the same diameter as Ω, since d G (u v x m x a x, u v y m y a y ) = d G (u b T v x m x a x, u b T v y m y a y ) = d G (u v x m x a x b T, u v y m y a y b T ) r Again, there is a 2r-coarse 2-manifold formed by 0 k T Ω 1 b k, with area O(L 2 ) and boundary Ω 1 Ω 2. After left translation, (u ) 1 Ω 2 U [I α] Φ(I) +M I (O S )A + I,α. Since b commutes with U [I α] Φ(I) +M I(O S )A + I,α, the 2r-coarse 2-manifold formed by k Z (u ) 1 Ω 2 b k intersects B I,α in a 2r-coarse closed 1-manifold of length O(L). Call this Ω 3 and let A 3 be the portion of k Z (u ) 1 Ω 2 b k bounded by (u ) 1 Ω 2 and Ω 3.

10 10 MORGAN CESA Since the distance from Ω to B I,α is bounded by L, the area of A 3 is O(L 2 ). Note that if x = v x m x a x (u ) 1 Ω 2, then x = v x m x ā x Ω 3, where ā x A + I,α. The bound on the diameter of Ω 3 implies that vx 1 v y e L for all x Ω 3. Choose c A + I such that d G (c, 1) r, and for every v S, α(c) v > 1 and β(c) v = 1 for every β α. There is some T = O(L) such that Ω 3 c T has nearly constant U [I α] Φ(I) + coordinates. That is, there is some v U [I α] Φ(I) + such that D(v x m x ā x c T, v m x ā x c T ) 2r for all x Ω 3. Let Ω 4 = {v m x ā x c T } x Ω, and let A 4 be the 4r-coarse 2-manifold Ω 4 ( 0 k T Ω 3 c k ). The area of A 4 is O(L 2 ). Since c commutes with M I (O S ) and A + I, Ω 5 = Ω 4 c T is a 2r-coarse 1-manifold of diameter O(L), and there is a 4r-coarse 2-manifold A 5 = 0 k T Ω 4 c k which has boundary Ω 4 Ω 5, and area O(L 2 ). Finally, observe that Ω 5 = {v m x ā x } x Ω has the same M I (O S )-coordinates as Ω, and that b commutes with Ω 5. Therefore, there is a 2r-coarse 1-manifold Ω 6 k Z Ω 5 b k which has the form Ω 6 = {v m x a x } x Ω, and there is a 4rcoarse 2-manifold A 6 bounded by Ω 5 and Ω 6 with area O(L 2 ). Taking A = A 1 A 2 u A 3 u A 4 u A 5 u A 6 and r = 4r completes the proof. Lemma 10. Suppose I is a set of simple roots such that I 2, and let α I and r > 0 be given. If p R I is an r-coarse path with endpoints x, y B I,α such that π I (p) B I,α, then there is an r-coarse path p B I,α joining x to y of length O(length(p)), and π I (p) π I (p ) bound a disk of area O(length(p) 2 ) in R I. Proof. Let length(p) = L. We can write x = u x m x a x and y = u y m y a y for u x, u y U Φ(I) +; m x, m y M I (O S ); and a x, a y A + I,α. Since π I is distance nonincreasing, π I (p) is an r-coarse path of length L from m x a x to m y a y. Left multiplication by u x gives an r-coarse path p 1, with length L, joining x to u x m y a y. Note that u = (m y a y ) 1 (u 1 x u y )(m y a y ) U Φ(I) + because U Φ (I) + is normalized by M I (O S )A + I. Also, d G (u, 1) = d G (u x m y a y, u y m y a y ) d G (u x m y a y, u x m x a x ) + d G (u x m x a x, u y m y a y ) 2L By Lemma 7, there is an r-coarse path in U Φ(I) +A + I (O S) from u to 1, with length O(L). Left multiplication by u x m y a y gives an r-coarse path p 2 m y a y U Φ(I) +A + I (O S) of length O(L) joining u x m y a y to y.

11 DEHN FUNCTIONS OF LATTICES OF TYPE A n IN PRODUCTS 11 Let p = p 1 p 2. Note that p p is a loop in R I, and that π I (p 1 ) = π I (p). Therefore π I (p 2 ) forms a loop in m y A + I (O S). Since A + I (O S) is quasi-isometric to a Euclidean space of dimension ( I )( S 1), it has a quadratic Dehn function, and therefore π I (p 2 ) bounds an r-coarse disk of area O(L 2 ) in m y A + I (O S) R I. We will now prove Proposition 5 in the case when I 2. Proof of Proposition 5 for nonmaximal parabolics. We will prove the lemma in two cases: first the case where Σ intersects both B I,α and ˆB I,α nontrivially for some α; second the case where Σ B I,α for some α. These two cases are sufficient, because R I = α I B I,α, so Σ must intersect B I,α for at least one α. By Lemma 8, Σ can be written as the union of two r-coarse 2- manifolds, Σ 1 and Σ 2, such that Σ 1 Σ B I,α and Σ 2 Σ ˆB I,α, and Σ 1 Σ 2 is a collection of r-coarse loops and r-coarse paths in R I with endpoints in Σ. Suppose p j is an r-coarse path in Σ 1 Σ 2, with endpoints in B I,α. Lemma 8 implies that π I (p j ) B I,α, so we can apply Lemma 10 to obtain an r-coarse path p j in B I,α which has the same endpoints as p j and length O(length(p j )). If γ l is an r-coarse loop in Σ 1 Σ 2, choose x l γ l and write x l = u l g l for u l U Φ(I) + and g l M I (O S )A + I. Let γ l = u lπ I (γ l ) and note that γ l B I,α and π I (γ l ) = π I(γ l ). Note that Σ i is a closed 1-manifold, and Σ i = (Σ i Σ) (p 1 p k ) (γ 1 γ n ) Although Σ i R I, we can replace p j by p j and γ l by γ l to obtain a closed 1-manifold of the same topological type as Σ i which is contained in R I. Let Ω i = (Σ i Σ) (p 1 p k) (γ 1 γ n) By Lemmas 8 and 10, the length of Ω i is O(area(Σ)). Lemma 9 implies the existence of r -coarse 2-manifolds A i such that A i = Ω i u i π I (Ω i ) for some u i U Φ(I) +, and area(a) = O(area(Σ) 2 ). By Lemma 10, there is a family of disks D i,j R I such that Σ i = A i D i u i π I (Σ i ) is an r -coarse 2-manifold of the same topological type as Σ i. Note that k j=1 length(p j) L, which implies that k j=1 area(d i,j) L 2 and therefore area(σ i) = O(area(Σ) 2 ). Taking Σ = Σ 1 Σ 2 completes the first case of the proof. We now assume that Σ B I,α. Let Ω = Σ and let L be the total length of Σ. Every point x Σ can be written as x = u x m x a x for

12 12 MORGAN CESA u x U Φ(I) +, m x M I (O S ), and a x A + I,α. Note that u 1 x u y = O(e L ) for x, y Σ. Choose some b int(a + I α ) which strictly contracts U Φ(I α) +. As in the proof of Lemma 9, right multiplication by b k is distance nonincreasing on Σ when k 0, and there is some T = O(L) such that Ωb T has nearly constant U Φ(I α) +-coordinates. Let u U Φ(I α) + be such that d G (u x m x a x b T, u v x m x a x b T ) r for every x Ω. Let Ω 1 = {u v x m x a x x Ω}. As in the proof of Lemma 9, there is a 2r-coarse 2-manifold A with boundary Ω Ω 1 and area O(L 2 ). There is a distance nonincreasing map f : U Φ(I) +M I (O S )A + I U [I α] Φ(I) +M I (O S )A + I,α. Taking r = 2r and Σ = f(σ) A completes the proof. 4. Maximal Parabolic Subgroups In this section, we will prove Proposition 5 in the case where R I is a maximal parabolic subgroup of G (when I = 1). There is a simple root α such that I = α. There is a distance nonincreasing map π I : U Φ(I) +M I (O S )A I M I (O S ) A I. Note that A I = A which is quasi-isometric to A(O S ), so M I (O S ) A I is quasi-isometric to (M I A)(O S ). Lemma 11. Given r 1 and x R I, with d G (x, 1) bounded by L, there is an r-coarse path in R I joining x to π I (x) which has length O(L). Proof. We can write x = uma for u U Φ(I) +, m M I (O S ) and a A(O S ). Then π I (x) = ma. Note that (M I A)(O S ) normalizes U Φ(I) +. So finding an r-coarse path from x to π I (x) of length O(L) can be reduced to the problem of finding an r-coarse path from (ma) 1 u(ma) U Φ(I) + to 1 of length O(L). Since (ma) 1 u(ma) O(L), Lemma 7 completes the proof. Fix some w S. Let T I be a K-defined K-anisotropic torus in M I such that gt I g 1 = M I A. Since T I is K-anisotropic, Dirichlet s unit s theorem tells us that T I (O S ) is cocompact in T I, so in particular, the projection of T I (O S ) to T I (K w ) is a finite Hausdorff distance from T I (K w ). Let T I be the projection of T I (O S ) to T I (K w ). Lemma 12. Suppose β Φ(I) +, so that U (β) (K w ) U Φ(I) +(K w ). There is some t T I such that gtg 1 strictly contracts U (β) (K w ).

13 DEHN FUNCTIONS OF LATTICES OF TYPE A n IN PRODUCTS 13 Proof. It suffices to show that there is some t M I (K w ) A(K w ) which strictly contracts U (β). We first note that since the K-type of G is A n, = {α 1,..., α n }, and a general root γ Φ has the form k γ = ± where 1 j k n. Because P I s a maximal parabolic, I = α m for some m such that 1 m n. Let 1 = {α 1,..., α m 1 } and 2 = {α m+1,..., α n }. At least one of these sets must be nonempty. We will assume that 2 is non-empty for the sake of simplicity. We can write M I = M 1 M 2, where i=j α i M 1 = U (αi ), U ( αi ) i<m M 2 = U (αi ), U ( αi ) i>m Let A i = A M i, and note that P M i is a minimal parabolic subgroup of M i, A i is a maximal K-split torus in P M i, and i is the set of simple roots with respect to A i. Since β Φ( α m ) +, we know that β = α j + + α m + + α k for fixed choices of j and k such that 1 j m k n. Suppose that k > m, and choose a A + 2 (K w ) such that α i (a) w < 1 for all α i 2. Note that α i (a) w = 1 for α i 1, since a M 2 (K w ). Conjugation by a acts on U (β) (K w ) by scalar multiplication by the constant k C = α i (a) w i=j By our choice of a, we know that C = α m (a) w C where C < 1. If α m (a) w < 1, then C < 1, and a contracts U C (β) (K w ) by a factor of C. If α m (a) w > 1, then a 1 contracts U C (β) (K w ) by a factor of 1. (Note C that either a or a 1 must contract U (γ) (K w ) for any other γ Φ(I) + with k > m.) If C = 1, choose a m i=1 ker(α i ) such that α i (a ) w 1 for all α i 2 and α k (a ) w < 1. Note that k α i (aa ) w = C i=j so aa contracts U (β) (K w ). k i=m+1 α i (a ) w < C

14 14 MORGAN CESA If β = α j + + α m, a different approach is required. Consider the group M 3 = U αm, U αm, U αm+1, U αm+1 and let A 3 = M 3 A. Note that 3 = {α m, α m+1 } is the set of simple roots of M 3, and the K-type of M 3 is A 2. Furthermore, α m determines a maximal parabolic subgroup P M 3, with ker(α m ) = P A 3. Let L = U αm+1 (K w ), U αm+1 (K w ), and choose a L A 3 (K w ) with α m+1 (a) w < 1. We argue that a contracts U (β) (K w ). Since L A 1 (K w ) is trivial, α i (a) w = 1 for all i < m. So the action of a on U (β) (K w ) depends only on α m (a) w. Let φ be the K-automorphism of M 3 which stabilizes A 3 and transposes P with its opposite with respect to A 3. Note that ker(α m ) L is trivial, since φ preserves L but does not preserve P. Therefore, α m (a) w 1, and after possibly replacing a by its inverse, we find that a contracts U (β) (K w ) by a factor of α m (a) w. Lemma 13. The Dehn function of U Φ(I) + TI A I (O S ) is quadratic. Proof. We observe that T I A I (O S ) is a free abelian group. Also, U Φ(I) + is normalized by T I A I (O S ), and since the K-type of G is A n, U Φ(I) + is abelian and U Φ(I) +(K v ) isomorphic to a direct sum of one or more copies of K v. Therefore, U Φ(I) + TI A I (O S ) can be written as U Φ(I) +(K v ) T I A I (O S ) v S By Theorem 3.1 in [CT10], it suffices to show that for any two unipotent coordinate subgroups, U (β1 )(K v ) and U (β2 )(K v ), of U Φ(I) +, there is some element of T A I (O S ) which simultaneously contracts U (β1 )(K v ) and U (β2 )(K v ). If v = v, then U (β1 )(K v ) and U (β2 )(K v ) are contained in the same factor of U Φ(I) +. By Lemma 6, there is some a A I (O S ) which simultaneously contracts U (β1 )(K v ) and U (β2 )(K v ). If v v, then U (β1 )(K v ) and U (β2 )(K v ) are in different factors of U Φ(I) +. In this case, either S 3 or S = 2. If S 3, then we may again apply Lemma 6 to obtain a A I (O S ) which simultaneously contracts U Φ(I) +(K v ) U Φ(I) +(K v ). If S = 2, we may assume that v = w. Let g M I (K w ) {1} be the element which diagonalizes T I. Note that g commutes with A I (O S ) and normalizes U Φ(I) +, so U Φ(I) + TI A I (O S ) is conjugate to

15 DEHN FUNCTIONS OF LATTICES OF TYPE A n IN PRODUCTS 15 U Φ(I) +(g T I g 1 )A I (O S ), and it suffices to prove the lemma for the latter group. By Lemma 12, there is some gtg 1 g T I g 1 which contracts U (β1 )(K w ) and commutes with U (β2 )(K v ). There is some a A I (O S ) which contracts U (β2 )(K v ). If a expands U (β1 )(K w ), then there is a positive power of gtg 1 such that gt k g 1 a simultaneously contracts U (β1 )(K w ) and U (β2 )(K v ). Proof of Proposition 5 for maximal parabolics. Since π I is distance nonincreasing, π I (Σ) is a 2-manifold in R I with area O(L 2 ), so if we can create an annulus between Σ and π I ( Σ) which has area O(L 3 ), then taking Σ to be the union of this annulus with π I (Σ) completes the proof. By Lemma 11, there is a path from each point in Σ to its image in π I ( Σ) which has length O(L). Two adjacent points in Σ, along with their images in π I ( Σ) and these two paths give a loop of length O(L) in U Φ(I) +A I (O S )B where B is a ball in M I (O S ) of radius r around 1. Note that this subset of G is quasi-isometric to U Φ(I) +A I (O S ), and by Lemma 13, these loops have quadratic fillings in R I. Since there are O(L) such loops formed by adjacent pairs of points in Σ, this gives an annulus A with A = Σ π I ( Σ), and area(a) = O(L 3 ), completing the proof. References [Har69] [Beh69] H. Behr. Endliche Erzeugbarkeit arithmetischer Gruppen über Funktionenkörpern. Invent. Math, 7:1 32, [BEW13] M. Bestvina, A. Eskin, and K. Wortman. Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups. J. Eur. Math. Soc., 15: , [Bor91] A. Borel. Linear Algebraic Groups, volume 126 of Graduate Texts in Mathematics. Springer-Verlag, New York, [Coh14] D. B. Cohen. The Dehn function of Sp(2n; z). preprint, [CT10] Y. Cornulier and R. Tessera. Metabelian groups with quadratic Dehn function and Baumslag-Solitar groups. Confluentes Math., 2(4): , [Dru98] C. Druţu. Remplissage dans des réseaux de Q-rang 1 et dans des groupes résolubles. Pacific J. Math, 185: , [Gro93] M. Gromov. Asymptotic invariants of infinite groups, volume 182 of London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, G. Harder. Minkowskische Reduktionstheorie über Funtionenkörpern. Invent. Math., 7:33 54, [LMR00] A. Lubotzky, S. Mozes, and M. S. Raghunathan. The word and Riemannian metrics on lattices of semisimple groups. Inst. Hautes Études Sci. Publ. Math., 91:5 53, [You13] R. Young. The Dehn function of SL(n; Z). Annals of Mathematics, 177: , 2013.