Arithmetic Lattices in Unipotent Algebraic Groups

J. Group Theory 23 (2020), 299–312 DOI 10.1515/jgth-2019-0112 © de Gruyter 2020

Arithmetic lattices in unipotent algebraic groups

Khalid Bou-Rabee and Daniel Studenmund Communicated by John S. Wilson

Abstract. Fixing an arithmetic lattice in an algebraic group G, the commensurability growth function assigns to each n the cardinality of the set of subgroups with Œ Œ n. This growth function gives a new setting where methods of F. Grunewald,W \ D.W Segal\ andD G. C. Smith’s “Subgroups of ﬁnite index in nilpotent groups” apply to study arithmetic lattices in an algebraic group. In particular, we show that, for any unipotent algebraic Z-group with arithmetic lattice , the Dirichlet function associated to the commensurability growth function satisﬁes an Euler decomposition. Moreover, the s local parts are rational functions in p , where the degrees of the numerator and denominator are independent of p. This gives regularity results for the set of arithmetic lattices in G.

1 Introduction

Let G be an algebraic group deﬁned over Z (an algebraic Z-group). Two subgroups 1 and 2 of G.R/ are commensurable if their commensurability index

c.1; 2/ Œ1 1 2Œ2 1 2 ´ W \ W \ is finite. An arithmetic lattice of G is a subgroup of G.R/ that is commensurable with G.Z/. The first purpose of this article is to show that, when G is unipotent, the set of arithmetic lattices in G has a great deal of regularity. The second purpose is to bring attention to a new notion of quantifying commensurability. The main tool of this article is the commensurability growth function N N ! [ ¹1º assigning to each n N the cardinality 2 cn.G.Z/; G.R// G.R/ c.G.Z/; / n : ´ j¹ Ä W D ºj We study cn.G.Z/; G.R// for groups G in the class U of unipotent algebraic Z- groups, starting with the fact that cn.G.Z/; G.R// is finite for all n in Lemma 3.1. K. Bou-Rabee supported in part by NSF grant DMS-1405609. D. Studenmund supported in part by NSF grant DMS-1547292. 300 K. Bou-Rabee and D. Studenmund

Note that the correspondence between G and G.Z/ is the Mal’cev correspondence [12] between unipotent Z-groups and torsion-free finitely generated nilpotent groups (see also [5]). Our proofs follow the outline of F. Grunewald, D. Segal and G. C. Smith [6], who studied the subgroup growth function of torsion-free finitely generated nilpotent groups , defined by an./ Œ n , by decomposing the ´ ¹ Ä W W D º associated zeta function into a product of local functions (see also [4]). We over- come an additional technical hurdle in order to prove rationality of our local zeta functions. While Grunewald, Segal and Smith use a result of J. Denef [2] to prove rationality, we require rationality of an integral over a noncompact set, and there- fore draw from work of A. MacIntyre [11] developed after [6]. See § 1.2 for more details. In view of the relationship cn.; / an./, the function cn extends an to D pairs of groups. While we have the simple relationship

cn.G.Z/; G.R// an.G.Z//; it is clear from the results in this paper that this inequality is strict when G is unipotent. The difference is evident even in the one-dimensional case.

1.1 The setting

Our results about cn are stated in the language of zeta functions. Let G be an algebraic group and F a family of subgroups of G.R/. We associate to such a family the Dirichlet series

X s X1 s .s/ c.G.Z/; / cn.F /n ; F D D F n 1 2 D where cn.F / F c.G.Z/; / n . ´ j¹ 2 W D ºj Following [6], there are two questions one can ask about the sequence cn.F /: (a) how fast does it grow, and (b) how regularly does it behave? Pursuing (a) is the study of commensurability growth, which we do not directly address here. We will address (b) by decomposing F into local parts and proving a local regularity theorem. The families we consider are the following: L.G/ all arithmetic lattices inside G ; ´ ¹ º Ln.G/ L.G/ c.; G.Z// n ; ´ ¹ 2 W D º L .G/ L.G/ c.; G.Z// is a power of p : .p/ ´ ¹ 2 W º Deﬁne the global and local commensurability zeta functions

G and G;p : ´ L.G/ ´ L.p/.G/ Arithmetic lattices in unipotent algebraic groups 301

1.2 The main results We start with an elementary example, proved in § 2.

Proposition 1.1. Let G Ga, the additive algebraic group, so that G.Z/ Z D D and G.R/ R. Then, formally, D 2.s/ G.s/ : D .2s/ The zeta functions introduced in [6] decompose into Euler products. Our ﬁrst main result, proved in § 3.1, is an analogous decomposition of the commensurability zeta function G for G U . 2 Proposition 1.2. If G U, then 2 Y G.s/ G;p.s/ D p as formal products over all primes of Dirichlet series. The main content of this article, our next result, is proved in § 3.2.

s Theorem 1.3. Let G U; then the function G;p.s/ is rational in p , where the degrees of the numerators2 and denominators are bounded independently of the prime p. Proposition 1.2 and Theorem 1.3 provide regularity results for the set of arithmetic lattices, L.G/.

Corollary 1.4. Let G U; then 2 Q e Q ei (1) cn.L.G// cp i .L.G//, where n p is the prime factorization of n; D i D i (2) there exist positive integers l and k such that, for each prime p, the sequence .cpi .L.G///i>l satisﬁes a linear recurrence relation over Z of length at most k. Our proof of Theorem 1.3 uses [11, Theorem 22]. We require a stronger theorem than that used in [6] since we express the local commensurability zeta function as a p-adic integral over an unbounded set (see Proposition 3.12). To ﬁnd this explicit formula, we constructed a parametrization of lattices that gives a closed form for the commensurability index function (see Lemma 3.10). We also discovered an explicit correspondence k k closed K G.Qp/ c.G.Zp/; K/ p G.Q/ c.G.Z/; / p ¹ Ä W D º $ ¹ Ä W D º that may be of independent interest (see Lemma 3.5). 302 K. Bou-Rabee and D. Studenmund

We were moved to pursue this subject after reading the work of N. Avni, S. Lim and E. Nevo [1] on the related concept of commensurator growth. Commensurator growth in [1], subgroup growth in [6] and commensurability growth in this paper are all studied through their associated zeta functions. Associating zeta functions to growth functions in groups is an active area of research, centering around subgroup growth and representation growth. For background reading on these subjects, we recommend the references [8, 10, 15].

Notation c.A; B/ ŒA A BŒB A B. D W \ W \

U is the class of unipotent algebraic Z-groups.

T is the class of torsion-free ﬁnitely generated nilpotent groups. S is the cardinality of the set S. j j Œx; y x 1y 1xy x 1xy. D D Gn gn g G for n Z. D h W 2 i 2

Z.G/ is the center of G.

Tn.R/ is the set of lower-triangular n n matrices over a commutative ring R.

.s/ is the Riemann zeta function.

2 The one-dimensional case

We begin with the integers. In this case, we can directly relate the commensurability zeta function with the classical zeta function.

Proposition 2.1. Let G Ga, the additive algebraic group, so that G.Z/ Z D D and G.R/ R. Then, formally, D 2.s/ G.s/ : D .2s/ Proof. The subgroups of R commensurable with Z are all of the form rZ, where r Q . Writing r n=d in reduced form, we have c.Z; rZ/ nd. Hence, 2 D D

cn.L.G// a=b a; b N; .a; b/ 1; ab n : D j¹ W 2 D D ºj

From this, we get, for distinct primes p1; p2; p3; : : : ; pn,

j1 j2 jn j1 j2 jn cp p pn .L.G// cp .L.G//cp .L.G// cpn .L.G//: 1 2 D 1 2 Arithmetic lattices in unipotent algebraic groups 303

!n And for any prime p, we have c k .L.G// 2. Hence, cn.L.G// 2 , where p D D !n is the number of distinct primes dividing n. Following [7, p. 255], we compute

1 Â 1 Ã X !n s Y X 2 G.s/ 2 n 1 D D C pks n 1 p prime k 1 D D Y Â 2p s Ã Y 1 p 2s 2.s/ 1 : D 1 p s C D .1 p s/2 D .2s/ p prime p prime

3 Unipotent algebraic groups

To any G U, we ﬁx an embedding of G.R/ into a group Tn.R/ such that 2 G.Q/ Tn.Q/ G.Q/ D \ given by Kolchin’s theorem [9]. With this in hand, we ﬁrst show that, for any unipotent algebraic Z-group G, the commensurability growth function takes values in N.

Lemma 3.1. Let G be in U. Then Ln.G/ is finite for any n N. 2 n Proof. Given n N, for any Ln.G/ and for every h , we have h G.Z/. 2 2 2 2 By [13, Exercise 7, p. 114], there exists a finitely generated subgroup G.Q/ Ä that contains every nth root of an element of G. Then every element in Ln.G/ is a subgroup of of index at most nŒ G.Z/. Since is finitely generated, it has W finitely many subgroups of any given index, so Ln.G/ is finite. As an immediate consequence of the proof of Lemma 3.1, we get the well- known fact that arithmetic lattices in G are contained in G.Q/.

Lemma 3.2. If G is in U, then each element of L.G/ is a subgroup of G.Q/. Slight modiﬁcations of a technical idea in the proof of Lemma 3.1 will be used repeatedly throughout the rest of the paper, so we encapsulate them here. Through- out, for any G.R/ and n N, we write 1=n for the group generated by Ä 2 elements g G.R/ such that gn . 2 2 Lemma 3.3. Let G be in U and G.Z/ have nilpotence class c and Hirsch D length d. (1) For any k N, Œ1=k is ﬁnite. 2 k kc.c 1/=2W (2) We have .p /p C . Ä k (3) For any prime p and any k N, we have Œp pdkc.c 1/=2. 2 W Ä C 304 K. Bou-Rabee and D. Studenmund

Proof. Item (1) follows from [13, Exercise 7, p. 114]. Item (2) follows from [13, Proposition 3, p. 113]. For the final statement, by [13, Exercise 7, p. 114], the group k p is a finitely generated nilpotent group. As a finite extension of , it has Hirsch length at most d, and hence

k k kc.c 1/=2 Œp .p /p C pdkc.c 1/=2: W Ä C k Then, by item (2), it follows that we have bounded Œp as desired. W

3.1 The Euler decomposition Proposition 3.4. If G is in U, then Y G.s/ G;p.s/: D p prime

1=n Proof. Any group in Ln.G/ is a subgroup of G.Z/ . Hence, applying item (1) of Lemma 3.3, we see that Ln.G/ is finitely generated and contains each D h i element of Ln.G/ as a subgroup of finite index. Hence, ƒn0 A Ln.G/A is D\ 2 a subgroup of finite index in . Let ƒn be the normal core of ƒn0 in . Consider the group =ƒn. It is a finite nilpotent group, and hence de- ´ composes into a product of its Sylow p-subgroups: Y =ƒn Sp: D p

For any image A in of an element Ln.G/, we get the decomposition 2 Y A Ap; D p where Ap are the Sylow p-subgroups of A and Ap Sp for every p. A similar Ä statement is true for the image Q of G.Z/ in ; we have Y Q Qp; where Qp Sp: D Ä p prime

We compute Y Q A .Ap Qp/: \ D \ p prime Hence, Y c.G.Z/; / c.Q; A/ c.Ap;Qp/: D D p prime Arithmetic lattices in unipotent algebraic groups 305

Further, c.Ap;Qp/ is the greatest power of p that divides n. Since any such element of Ln.G/ arises from such a decomposed A, we have Y cn.L/ c k .L/; D p pk n k and hence the Euler decomposition for the commensurability zeta function above holds.

3.2 p-adic formulation and the proof of Theorem 1.3

Let G U. Fix a Mal’cev basis .x1; : : : ; xn/ for G.Z/ so that 2

1 < x1 < x1; x2 < < x1; : : : ; xn G.Z/ h i h i h i D is a central series with inﬁnite cyclic successive quotients. Elements in G.Qp/ may be identiﬁed with the set of all “p-adic words” of the form

x.a/ xa1 xan ; where a Qn: D 1 n 2 p Note that x parametrizes G.Z/ when restricted to a Zn. Let , , and .i/ for 2 i 2 be polynomials deﬁned over Q as in [6, p. 197] for the T group G.Z/ so that, for a Zn, we have 2 k k x.a/ x.a1; a2; a3; : : : ; an/ x..a; k//; D D .i/ x.a1/x.a2/ x.ai / x. .a1; a2;:::; ai //; D Œx.a1/; x.a2/ x..a1; a2//: D We denote the closure of a subset S of G.Qp/ to be S or S . For any L.G/, 2 we have that is the pro-p completion of [16, Theorem 4.3.5].

Lemma 3.5. For each k, the closure map gives a one-to-one correspondence between elements of Lpk .G/ and closed subgroups H of G.Qp/ with

k c.H; G.Zp// p : D p k Proof. Let G.Z/ . Note that contains every subgroup in Lpk .G/. More- D N over, by item (2) of Lemma 3.3, we have that p G.Z/ for N kc.c 1/=2, Ä D C where c is the nilpotence class of G. It follows that any Lpk .G/ contains N k N k 2 p C . Note that p C is a normal subgroup of and the quotient group pN k = C is a ﬁnite p-group. It follows that, for each L k .G/, there is some 2 p ` N such that Œ p`. 2 W D 306 K. Bou-Rabee and D. Studenmund

Let K be the closure of in G.Qp/. Then K is a finitely generated pro-p group. k Any closed subgroup H G.Qp/ satisfying c.H; G.Zp// p is a finite-index p k Ä D subgroup of G.Zp/ K. The closure map gives an index-preserving bijection D between subgroups of with index a power of p and finite-index subgroups of K. This completes the proof.

r1 ri Deﬁne Gi x ; : : : ; x r1; : : : ; ri Qp . The following is an extension of D h 1 i W 2 i the notion of good basis from [6] to the G.Qp/ setting.

Deﬁnition 3.6. Let H be a closed subgroup of G.Qp/. An n-tuple .h1; : : : ; hn/ of elements in H is called a good basis if h1; : : : ; hi H Gi for each h i D \ i 1; : : : ; n. D

Lemma 3.7. Let hi Gi Gi 1 for i 1; : : : ; n, and let H h1; : : : ; hn . Then the following statements2 n hold. D D h i

(1)c .H; G.Zp// < . 1 (2) .h1; : : : ; hn/ is a good basis for H if and only if

Œhi ; hj h1; : : : ; hi 1 for 1 i < j n: (3.1) 2 h i Ä Ä (3) Suppose equation (3.1) holds, and let h ; : : : ; h H. Then .h ; : : : ; h / is 10 n0 2 10 n0 a good basis for H if and only if there exists ri Zp, wi h1; : : : ; hi 1 ri 2 2 h i such that h wi h for 1 i n. i0 D i Ä Ä Proof. Since hi Gi Gi 1 for i 1; : : : ; n, we have 2 n D ri1 ri2 rin hi x x x ; where rij Qp: D 1 2 n 2 k For any such nonzero rij , we have rij up , where k Z and u Z . If all D 2 2 p such k are nonpositive, then we appeal to [6, Lemma 2.1]. Otherwise, let k be the maximal integer that appears in this way. Then

k k k k p p p p H x ; x ; : : : ; x G.Zp/ : Ä h 1 2 n i Ä p k p k By continuity of taking powers, G.Zp/ is the closure of G.Z/ in G.Qp/. Thus, by applying item (2) of Lemma 3.3, we have that there exists m N such m 2 that H G.Zp/. It follows that G.Zp/ H must have the same dimension Ä \ as G, and so item (1) follows. Items (2) and (3) follow from small modiﬁcations of the proof of [6, Lemma 2.1].

Deﬁnition 3.8. For L.G/, deﬁne U./ to be the collection of pairs .A; B/, 2 where A Tn.Qp/, B Tn.Zp/, such that if ai denotes the ith row of A and bi 2 2 Arithmetic lattices in unipotent algebraic groups 307 denotes the ith row of B,

(1) .x.a1/; : : : ; x.an// is a good basis for ,

(2) .x.b1/; : : : ; x.bn// is a good basis for G.Zp/. \ Combining suitable U./ into one set, we define [1 Â [ Ã Up U./ : ´ k 1 L k .G/ D 2 p The next proposition shows that Up is Lp-defined in the sense of [11] (that is, it can be defined using first-order logic, p-norms and field operations). Note that x p y p can be stated in LP (see the example in [11, p. 71]). j j j j Lemma 3.9. Let .A; B/ Tn.Qp/ Tn.Zp/. Then .A; B/ Up if and only if each of the following holds.2 2

(1) det.A/ 0 and the rows a1; : : : ; an of the matrix A satisfy ¤ 1 i 1 for 1 i < j n; there exist Y ;:::;Y Zp such that Ä Ä ij ij 2 .i 1/ 1 i 1 .ai ; aj / ..a1;Yij /; : : : ; .ai 1;Yij //: D (2) det.B/ 0 and the rows b1; : : : ; bn of the matrix B satisfy ¤ 1 i 1 for 1 i < j n; there exist Y ;:::;Y Zp such that Ä Ä ij ij 2 .i 1/ 1 i 1 .bi ; bj / ..b1;Yij /; : : : ; .bi 1;Yij //: D n Moreover, let y Z G.Qp/ be the function defined by W p ! d1 d2 dn y.d/ x.a1/ x.a2/ x.an/ : ´ n Then there exists vectors c1; : : : ; ci Z such that, for all i 1; : : : ; n, 2 p D y.ci / x.bi /: D (3) det.B/ p is maximal over all B satisfying (2). j j Proof. Let .A; B/ U./. Then conditions (1), (2) are satisfied by Lemma 3.7. 2 To see condition (3), note that det.B/ p ŒG.Z/ G.Z/ , while any matrix j j D W \ B satisfying condition (2) determines a subgroup of G.Zp/ and hence satis- 0 \ fies det.B / p ŒG.Z/ G.Z/ . j 0 j Ä W \ Let .A; B/ Up. Let H be the closed subgroup of G.Qp/ generated by x.ai /, 2 where ai are the rows or A, and let be the subgroup in Lp` .G/ corresponding to H given by Lemma 3.5. The conditions above ensure .A; B/ U./. 2 Our parametrization of arithmetic lattices gives a nice formula for the commensurability index. 308 K. Bou-Rabee and D. Studenmund

Lemma 3.10. Let L.G/. Then, for any .A; B/ U./, we have 2 2 2 1 1 c.; G.Zp// det.B/ det.A/ : D j jp Proof. A direct application of the results of [6, p. 198] shows 1 ŒG.Zp/ G.Zp/ det.B/ : W \ D j jp Let C be the matrix whose ith row ci satisfies y.ci / x.bi /, where y is the D n function defined in Lemma 3.9. Note that C Tn.Zp/. Because y.Z /, it 2 D p follows as above from [6, p. 198] that 1 Œ G.Zp/ det.C / : W \ D j jp Using the fact that the rows of B are coordinate vectors of a good basis, compu- tation shows that bi .qi1; : : : ; qi.i 1/; aii cii ; 0; : : : ; 0/, where qij Qp for all D 2 j < i and aii and cii are the respective diagonal elements of A and C . It follows that det.B/ 1 det.AC / 1: j jp D j jp The desired result follows. For n N, let be the Haar measure on Qn normalized so that .Zn/ 1. 2 p p D Qn Lemma 3.11. For a matrix M Tn.Qp/, let f .M / i 1 m11 mii . Let 2 ´ D L.G/. Then U./ is an open subset of Tn.Qp/ Tn.Zp/ and 2 1 2n .U.// .1 p / f .A/f .B/ p D j j for any .A; B/ U./. 2 Proof. For L.G/ and a ring R, let MR./ be the set of matrices M in Tn.R/ 2 with rows m1; : : : ; mn such that x.m1/; : : : ; x.mn/ are a good basis for . Notice that U./ MQ ./ MZ .G.Z/ /. Hence, that U./ is an open subset D p p \ of Tn.Qp/ Tn.Zp/ is a straightforward application of the first part of the proof of [6, Lemma 2.5, p. 198]. Moreover, the second part of the proof of [6, Lemma 2.5, p. 198] and Fubini’s theorem give Z Z 1 d 1 d U./ D MQp ./ MZp .G.Z/ / \ Z Z 1 d D MQp ./ MZp .G.Z/ / \ Z 1 n .1 p / f .B/ p d D MQp / j j 1 2n .1 p / f .A/f .B/ p: D j j Arithmetic lattices in unipotent algebraic groups 309

Proposition 3.12. Let f be as in Lemma 3.11. We have Z 1 1 2 1 s G;p.s/ 1 2n f .A/f .B/ p det.B/ det.A/ p d: D .1 p / .A;B/ Up j j j j 2 Proof. By Lemma 3.11, for any .A; B/ U./, the integrand evaluates to 2 1 s .U.// c.; G.Z// :

By decomposing Up into disjoint open sets,

[1 Â [ Ã Up U./ ; D k 1 L k .G/ D 2 p using Lemma 3.11, we compute 1 Z f .A/f .B/ 1 det.B/2 det.A/ 1 s d .1 p 1/2n j jp j jp Up Â ÂZ ÃÃ X1 X 1 s .U.// c.; G.Z// d D U./ k 1 L k .G/ D 2 p Â ÂZ ÃÃ X1 X 1 ks .U.// p d D U./ k 1 L k .G/ D 2 p Â Ã X1 X ks X1 cpk .L.G// p G;p.s/: D D pks D k 1 L k .G/ k 1 D 2 p D We are now ready to prove our main result.

Proof of Theorem 1.3. We ﬁrst show that the local commensurability zeta functions converge for sufﬁciently large s. Given Lpk .G/, for any , we pk p k 2 2 have G.Zp/ so k .G.Z// . Then, by item (3) of Lemma 3.3, 2 Ä ´ Dk there exists D, depending only on G, such that k G.Z/ p . Hence, any j .1W D/k j Ä L k .G/ is a subgroup of of index at most p , giving 2 p k C

cpk .L.G// sp.1 D/k .k/ sp.1 D/k .N /; (3.2) Ä C Ä C where sm.G/ is the subgroup growth function (the number of subgroups of index at most m) and N is the free nilpotent group of class and rank equal to that of G.Z/. ˛k It follows from [6] that sp.1 D/k .N / p for some ﬁxed ˛ that does not depend C Ä on p. Hence, inequality (3.2) gives that G;p.s/ is ﬁnite for s > M , where M does not depend on p. 310 K. Bou-Rabee and D. Studenmund

The integrand in Proposition 3.12 and Up are both Lp-deﬁnable in the sense of [11] (see also [2]) by Lemma 3.9. Thus, by [11, Theorem 22], we have that G;p.s/ is a rational function with numerator and denominator degrees bounded by a constant depending only on G.

4 Parting remarks

At present, we are unable to precisely compute closed forms of the zeta functions for unipotent groups of dimension greater than one. Even the two-dimensional case is too difficult for us. However, we can compute an explicit formula for cn in the two-dimensional case. Here we work globally over Q rather than locally as we did in § 3.2. 2 Consider the case G.Z/ Z . Let Vn be the set of pairs of nonnegative matri- D ces ! !! a a c c .A; C / 11 12 ; 11 12 ; D 0 a22 0 c22 where aij Q, cij Z, 0 a12 < a22 and c12 < c22, satisfying the following 2 2 Ä two properties: (1) det.C 2/ det.A/ n, D (2) if D GL2.Q/ satisfies DA GL2.Z/, then det.D/ det.C /. 2 2 j j Then cn.L.G// Vn . The bijection is defined as follows: Suppose G.Q/ D j j Ä is a lattice with c.Z2; / n. The rows of A give an ordered generating set S for D Q2. The rows of C give the coordinates of a generating set of Z2 with Ä \ respect to S. We can simplify this expression. Define D Q N to be the denominator of W ! a rational number in reduced form. Eliminating the cij ’s gives the formula

3 cn.L.G// .a11; a12; a22/ Q 0 a12 < a22; D j¹ 2 W a11a22.lcm.D.a12=a22/; 2 D.a11//D.a22// n : D ºj We do not have such a simple expression for the Heisenberg group. Computing nice expressions for these examples would, at the very least, give an indication of whether the results in [3, 14] hold for commensurability zeta functions.

Acknowledgments. We are grateful to Benson Farb, Tasho Kaletha, Michael Lar- sen, and Andrew Putman for their conversations and support. In particular, Benson Farb provided helpful comments on an early draft. We are very grateful to an anonymous referee for useful comments and corrections on an earlier draft. Arithmetic lattices in unipotent algebraic groups 311

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Received August 8, 2019; revised October 1, 2019.

Author information Khalid Bou-Rabee, Department of Mathematics, City College of New York, New York, USA. E-mail: [email protected] Daniel Studenmund, Department of Mathematics, University of Notre Dame, Notre Dame, USA. E-mail: [email protected]