Geometric Retracts of Siegel's Upper Half Space

Geometric retracts of Siegel’s upper half space

Justin Martel

B.Sc., The University of Ottawa, 2011

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

The Faculty of Graduate Studies

(Mathematics)

THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2013 c Justin Martel 2013 Abstract

The purpose of this thesis is to construct a codimension 1 cocompact Sp2gZ-equivariant strong deformation retract Wg of Siegel’s upper half space hg. This yields a partial con- tribution towards the problem of constructing a strong spine for the real linear symplectic group Sp2gR.

ii Table of Contents

Abstract ...... ii

Table of Contents ...... iii

Acknowledgements ...... v

Dedication ...... vi

1 Introduction ...... 1 1.1 What is a spine? ...... 1 1.2 Outline of thesis ...... 2

1.3 The wellrounded retract for SLn ...... 3

2 Siegel’s upper half space hg ...... 5 2.1 Symmetric space structure ...... 6 2.1.1 Cartan decomposition ...... 6

2.1.2 Killing form and the Ug-invariant metric ...... 7 2.1.3 Nonpositive curvature ...... 7 2.1.4 K¨ahlerstructure ...... 8 2.1.5 Iwasawa decomposition ...... 9 2.2 Symplectic lattices ...... 9

3 Some symplectic linear algebra ...... 11 3.1 Intersecting lagrangian subspaces ...... 11 3.2 Heights of ω-orthogonals ...... 12

4 Systoles on hg ...... 14 4.1 Lagrangian systole function ...... 14

iii Table of Contents

4.2 L -wellrounded lattices ...... 14

5 The equivariant retract Wg ...... 18

6 Cocompactness of Wg ...... 21

7 Codimension of Wg ...... 25

8 Bavard’s retract Yg ...... 29

9 Weak and strong spines ...... 31

10 MacPherson and McConnell’s explicit reduction theory ...... 33 10.1 Voronoi’s cell decomposition ...... 33

10.2 Embedding h2 into S4 ...... 36 10.3 Barycentric retracts ...... 36

11 Conclusion ...... 38

Bibliography ...... 39

iv Acknowledgements

I am grateful to the Topology group at UBC, whose participants were always generous with their time and energy. I have been particularly inspired by my interactions with Lior Silberman, Juan Souto, and Alexandra Pettet - all of whom have contributed towards my development as a mathematician. Thank you.

v Dedication

To my parents - Andr´eand Debra.

vi Chapter 1

Introduction

1.1 What is a spine?

We are compelled to the topic of this thesis by the following consideration on the cohomology of arithmetic groups. Given a semisimple algebraic group G (e.g. SLn, Sp2g,...), o the connected component of its R-points G = G(R) yields a semisimple Lie group. For a maximal compact subgroup K, the quotient X = K\G is a finite-dimensional contractible nonpositively curved space. The subgroup of Z-points Γ = G(Z) has a properly discon- tinuous right action on K\G given by (Kx, γ) 7→ Kxγ for x ∈ G, γ ∈ Γ. The discrete arithmetic subgroup Γ contains nontrivial finite subgroups, and hence does not act freely on the contractible space X, i.e. there are points Kx whose Γ-stabilizers are nontrivial. The presence of torsion elements means Γ = G(Z) has infinite cohomological dimension. However it has been known since Minkowski that Γ contains many finite index torsion-free subgroups Γ0, e.g. the congruence subgroups

Γ(N) := ker (SLnZ → SLn(Z/NZ)) for N ≥ 3. These finite index torsion-free subgroups act freely on X, and thus have finite cohomological dimension with the space X serving as explicit model for the classifying space EΓ0. One therefore considers Γ to virtually have finite cohomological dimension. For a given finite index torsion free subgroup Γ0 of Γ, the maximal integer q for which the q 0 0 cohomology module H (Γ , Z) does not vanish gives the cohomological dimension cd Γ of Γ0. One can show (c.f. [5]) that this integer q is constant for every finite-index torsion free subgroup of Γ, and thus we define q to be the virtual cohomological dimension vcd Γ of Γ. The virtual cohomological dimension thus describes the minimal possible dimension of a classifying space for the finite index torsion-free subgroups of Γ = G(Z). A remarkable paper of Borel-Serre [4] computes the virtual cohomological dimension of the semisimple

1 1.2. Outline of thesis algebraic groups as follows:

vcd G(Z) = dimRK\G − rankQ G.

Thus the formula expresses the vcd in terms of the dimension of the symmetric space minus the Q-rank of the group, where we recall that rankQG is the dimension of the maximal

Q-split torus in G. For instance SLn, Sp2g have rankQ = n − 1, 2g. The computation of Borel-Serre motivates us to formulate the following problem.

Problem of the Spine. For a semisimple Lie group G with maximal compact K construct a subset W of K\G for which

•W is a Γ-equivariant strong deformation retract of K\G;

• Γ acts cocompactly on W; and

•W has dimension equal to the virtual cohomological dimension vcd(Γ) of Γ.

The existence of a spine for SL2 has probably been a long recognized fact (c.f. [5]). The question of constructing a spine is probably first due to Souléin his 1973 thesis [20]. Afterwords Ash [2], motivated by group cohomology, expanded Soulé’sretraction and clearly formulated the wellrounded retract (see 1.3).

1.2 Outline of thesis

The present thesis concerns itself with the question of constructing a spine for the linear symplectic group Sp2g and can only offer a partial solution. Our main object of study are symplectic lattices and their associated Teichmüellerspace hg. The basic facts on the symmetric space hg are described in 2. As preliminary to our main results we require some elementary facts concerning the symplectic linear algebra which are presented in 3. The important lagrangian systole function is introduced in 4. Our main result is contained in chapters 5, 6, 7, wherein we construct a cocompact Sp2gZ-equivariant strong deformation retract Wg having codimension 1. We present Bavard’s own codimension retract Yg in 8 which is essentially different from our own, and MacPherson and McConnell’s weak spine for Sp4 in 10. Finally, the general existence question for spines (weak and strong) is described in 9.

2 1.3. The wellrounded retract for SLn

While our result is well-short of constructing a spine for hg (our original motivation), we do believe our retract to have some interesting features. In particular the retract is based on the lagrangian systole function sysL . This is the first instance of a retract being defined by a higher-dimensional systole function, where all previous retracts have been defined against the 1-systole function. Of course the lagrangian systole function is special to our lattices being symplectic. In 4 we establish some basic systolic properties of sysL (e.g. it is bounded from above on hg) which we believe to have some independant interest for the systolic geometry of symplectic tori. These preliminary results naturally suggest some interesting questions on the existence of systolic inequalities within the symplectic geometry which we shall investigate in the future.

1.3 The wellrounded retract for SLn

Before proceeding to our general problem, it is worthwhile to describe the wellrounded retract, i.e. a spine for the special linear group.

In dimension two the symmetric space S2 := SO2R\SL2R is well known to coincide with the hyperbolic upper half space H. We see a point SO2R·A ∈ S2 as describing an isometry 2 2 class of marked 2-dimensional unimodular lattices Λ = SO2R · AZ in R . It is convenient 2 2 to write AZ ⊂ R and work SO2R-equivariantly. An element in the arithmetic group 2 2 2 SL2Z acts on AZ by changing the marking, i.e. (AZ , γ) 7→ AγZ . One should observe 2 2 2 that γ acts as volume-preserving group automorphism on Z (that is, γZ = Z ). The ! ! 1 ±1 0 −1 arithmetic group SL2Z is seen to be generated by the elements and . 0 1 1 0 From this, one readily identiﬁes a fundamental domain for the action of SL2Z on S2 to be given by F := {x + iy : y > 0 |x| ≤ 1/2, x2 + y2 ≥ 1}.

This is the ‘key-hole’ description of the so-called modular domain. The upper half arc of the unit circle in F, namely the set {x + iy ∈ F : x2 + y2 = 1}, is a compact codimension

1 subspace in F whose SL2Z-translates generate a tree X2 for S2. The tree X2 (which also corresponds to the Bass-Serre tree for SL2Z) is a spine for S2. The equivariant retraction of S2 to X2 admits several descriptions. The ﬁrst description of an equivariant strong deformation retract will generalize to the so-called wellrounded retract of Soul´eand Ash ([20], [2]) yielding a spine for the higher

3 1.3. The wellrounded retract for SLn

dimensional symmetric spaces Sn, (n ≥ 2). It begins with interpreting the subset X2 as consisting of those 2-dimensional unimodular lattices Λ whose shortest nonzero vectors span 2 R . It is well to observe that the length of this shortest nonzero vector is SL2Z-invariant. For any Λ ∈ S2 \X2 there will exist, up to a sign, a unique nonzero shortest vector. Call it v. Consider the rank-1 Q-split torus TΛ,ρ deﬁned by expanding by ρ ≥ 1 along Rv, and −1 o contracting by ρ along the orthogonal complement Rv . There is a minimal time τΛ for which expanding along Rv will yield a new shortest nonzero vector. At this minimal time 0 τΛ we have deformed the lattice Λ into a ‘wellrounded lattice’ Λ . For a lattice Λ ∈ X2, we do nothing and leave it stationary, i.e. τΛ = 0. An important technical point is to demonstrate that the function τ : S2 → R is continuous. Then r : [0, 1] × S2 → S2 deﬁned by

r(s, Λ) := TΛ,(1−s)+sτΛ Λ is a well-defined SL2Z-equivariant continuous strong deformation retract of S2 onto X2. An alternative description, which is generalized to arbitrary semisimple algebraic groups by Borel-Serre [4] begins with the following observation. The flow generated by the wellrounded retract on the fundamental domain F can be seen to arise from the geodesic flow on H. More precisely consider the geodesic flow on the fundamental domain F. The wellrounded retract coincides with flowing each point along geodesics away from the cusp of the modular domain.

The 2-dimensional spine X2 extends, as described by Soul´eand Ash to an SLnZ- equivariant strong deformation retract Xn, where Xn consists of those unimodular n- n dimensional lattices whose shortest nonzero vectors (i.e. 1-systoles) span R . Using Mahler’s compactness criterion (c.f. 6) it is immediate that indeed Xn is cocompact, and moreover has codimension equal to rankQ SLn. The deformation of Sn onto Xn is 0 known as the wellrounded retract. The emphasis on distinguishing Xn from the subset Xn consisting of lattices whose 1-systoles span the lattice as a Z-module is essential. Pettet and Souto [16] established that in fact Xn is a minimal cocompact equivariant retract of 0 Sn. In particular the subset Xn is not homotopy equivalent to Sn for n ≥ 5.

4 Chapter 2

Siegel’s upper half space hg

We set some notation and terminology. Throughout this thesis we fix an identification of 2g the standard 2g-dimensional nondegenerate symplectic linear space (R , ω) where ω is the ! t 2g 0 −I 2-form defined by ω(x, y) = xJy for x, y ∈ R and J = . A subspace H in I 0 2g (R , ω) is totally isotropic if ω(x, y) = 0 for every x, y ∈ H. If H is totally isotropic, then JH is a subspace transverse to H and the direct sum (H ⊕JH, ω|H⊕JH ) is a nondegenerate symplectic subspace. The maximal dimension of a totally isotropic subspace is seen to be g.A lagrangian subspace is a g-dimensional (maximal) totally isotropic subspace of 2g (R , ω). The genus g symplectic group Sp2gR consists of those 2g×2g matrices A satisfying t AJA = J. Equivalently, Sp2gR consists of those linear transformations A satisfying 2g ω(Ax, Ay) = ω(x, y) for all x, y ∈ R . One has J ∈ Sp2gR and since every A ∈ Sp2gR satisfies JAJ −1 = tA−1, the symplectic group is stable under transposition. The linear isomorphism J satisfies J 2 = −id and defines a bilinear form h, i given by 2g the formula hx, yi = ω(x, Jy) for x, y ∈ R . Evidently h, i coincides with the standard 2g euclidean inner product on R . Trivially J is a h, i-isometry and ω-symplectomorphism. 2g We call J the standard almost-complex structure on (R , ω), and the pair (ω, J) defines 2g 2g ⊥ o the basic linear Kahler structure on R . For a subspace H ∈ R we denote by H ,H the ω, h, i-orthogonal complements of H respectively. Trivially H⊥ = (JH)o. The radical 2g of a subspace H ⊂ (R , ω) consists of those x ∈ H for which ωx|H ≡ 0. We say vectors 2g x1, . . . , x2g ∈ (R , ω) form a symplectic basis if the matrix x1 . . . x2g ∈ Sp2gR. The standard symplectic basis e1, . . . , eg, f1, . . . , fg constitute the columns of the 2g×2g identity matrix I2g.

5 2.1. Symmetric space structure

2.1 Symmetric space structure

2.1.1 Cartan decomposition

t −1 On the real linear symplectic group Sp2gR, the mapping θ : g 7→ g is called the Cartan involution (evidently θ2 = id). The ﬁxed point set θ(g) = g coincides with the maximal compact subgroup K = Ug in Sp2gR (we have also Ug = Sp2gR ∩ SO2gR). The diﬀerential t dθ acts on the Lie algebra sp2gR via X 7→ − X, splitting the Lie algebra into (+1)- and (−1)-eigenspaces k, p. Thus we recover Cartan’s decomposition

sp2gR = k ⊕ p.

The (+1)-eigenspace coincides with the Lie algebra of the maximal compact Ug. One observes the following relations

[k, k] ⊂ k, [k, p] ⊂ p, [p, p] ⊂ k. (2.1)

In particular p is not a subalgebra. Explicitly we ﬁnd p to consist of those matrices of the form ! AB , B −A where A, B are g × g symmetric real matrices. Cartan’s polar decomposition aﬃrms that the mapping

K × p → Sp2gR (K,X) 7→ K · exp(X) is an analytic diﬀeomorphism. It follows that the homogeneous space hg := Ug\Sp2gR is contractible and that the tangent space at the coset Ke = Ug · e can be identiﬁed as

TKehg ≈ p.

Notice that since k ∩ p = 0 any subalgebra a of sp2g contained in p is necessarily abelian. The Cartan involution θ : K × P → K × P now assumes the form (k, p) 7→ (k, p−1). Let a be a maximal subalgebra of p, necessarily commutative, and set P = exp p,A = exp a. The endomorphisms ad(a) for a ∈ a are simultaneously diagonalizable over R,

6 2.1. Symmetric space structure

and sp2gR splits into the centralizer z(a) of a and proper subspaces vα, where α varies ∗ over a finite set of nonzero functionals in a (the so-called roots of sp2gR relative to a). By computation one knows [vα, vβ] ⊂ vα+β, and therefore each vα consists of elements X for which adX is nilpotent (since there are only finitely many nonzero roots). The ∗ Cartan involution satisfies dθ : vα 7→ v−α. Choosing an ordering on the roots in a we P set n = α>0 vα, and find n to be a nilpotent lie algebra normalized by a. Setting A = expa,N = expn yields closed analytic subgroups of Sp2gR. The Iwasawa decomposition asserts that (k, a, n) 7→ kan is a diffeomorphism K×A×N → Sp2gR. As N is unipotent and A diagonizable over R, the subgroup AN is a connected solvable (triagonalizable) subgroup over R (i.e. the identity component of the R-points of an R-split Borel subgroup).

2.1.2 Killing form and the Ug-invariant metric

Recall that for a semisimple lie algebra g, the Killing form B yields a nondegenerate bilinear form deﬁned by

B(X,Y ) = trace(ad(X) ◦ ad(Y)), where ad : g → End(g) is the usual adjoint representation given by ad(X): Z 7→ [X,Z]. t For the standard representation of sp2gR (2g×2g real matrices X satisfying XJ +J X = 0) the Killing form (up to a constant) takes the form B(X,Y ) = tr(XY ). With respect the Killing form , the decomposition k⊕p is orthogonal. Moreover from the standard lie theory [reference] one ﬁnds

B is negative deﬁnite on k, and positive deﬁnite on p. (2.2)

We deﬁne a right-invariant riemannian metric h·, ·i on hg by deﬁning hX,Y iKe = −1 −1 tr(XY ) for all X,Y ∈ p ≈ TKehg and h·, ·iKg = hg ·, g ·iKe. In other words we extend the Killing form by translation obtaining a Ug-invariant metric on hg for which Sp2gR acts isometrically.

2.1.3 Nonpositive curvature

Via the identiﬁcation TKehg ≈ p, the riemannian curvature tensor satisﬁes R(X,Y )Z =

−[[X,Y ],Z] and the sectional curvature of a 2-plane spanned by orthonormal vectors Y1,Y2 in the tangent space Tphg satisﬁes K(Y1 ∧ Y2) = h[Y2,Y1], [Y2,Y1]i (c.f. [14]). From the

7 2.1. Symmetric space structure

relations (2.1) and (2.2) it follows that K(Y1 ∧ Y2) < 0 unless [Y1,Y2] = 0. Thus we find the basic fact that the sectional curvature of a tangent plane is strictly negative unless any two linearly independant tangent vectors have vanishing bracket, i.e. commute. Therefore the abelian subalgebras a of p correspond to tangents of flat subspaces. Evenmore since commuting families of symmetric endomorphisms are simultaneously diagonalizable we a find a direct 1 − 1 correspondance between

• ﬂat k-dimensional totally geodesic subspaces in hg;

• k-dimensional abelian subalgebras of p;

• k-dimensional R-split algebraic tori in Sp2gR.

The maximal dimension of an R-split torus in Sp2gR, the so-called R-rank rankRSp2gR is known to be g. Indeed, the maximal R-split tori are all conjugate over R to the standard torus −1 −1 diag(λ1, . . . , λg, λ1 , . . . , λg ) with respect to the standard symplectic splitting e1, . . . , eg, f1, . . . , fg.

2.1.4 K¨ahlerstructure

The vector space p splits into an h, i-orthogonal direct sum

! ! A 0 0 B p = { } ⊕ { }, 0 −A B 0 where A, B are symmetric g × g matrices. The mapping j : p → p defined by ! ! ! ! A 0 0 −A 0 B B 0 j : 7→ , 7→ 0 −A −A 0 B 0 0 −B yields an almost-complex structure on p, i.e. j2 = −id. We extend j to an almost-complex −1 structure j : T hg → T hg by defining jg = g · j ◦ g . Evidently j is pointwise an isometry with respect to the metric h, i. One finds that j yields a nondegenerate symplectic form

Ω on hg deﬁned by Ω(·, ·) = hj·, ·i. Thus hg is a (quasi) K¨ahlermanifold with metric h, i, almost-complex structure j, and compatible symplectic form Ω. Observe that this is 2 2 consistent with dimRhg = (2g + g) − g being even.

8 2.2. Symplectic lattices

2.1.5 Iwasawa decomposition

Every element A ∈ Sp2gR has a unique polar decomposition A = UP , where U ∈ Ug and P is a positive definite symmetric symplectic matrix. A positive definite symmetric matrix P ! ! ! I 0 Y 0 IX is symplectic if and only if P admits a decomposition P = XI 0 Y −1 0 I where X,Y are real g×g symmetric matrices and Y is positive definite (cf. chapter 5 in [19] ). The X,Y are uniquely determined by P and constitute the so-called Iwasawa coordinates 2 on Ug\Sp2gR. The vector space hg consisting of all pairs Ω = X + iY (i = −1) with X,Y g × g symmetric and Y positive definite defines the genus g Siegel upper half space. The

Iwasawa coordinates give an explicit identiﬁcation hg ' Ug\Sp2gR. For Ω = X + iY ∈ hg we set <Ω = X (real part), =Ω = Y (imaginary part).

2.2 Symplectic lattices

From our point-of-view the Siegel upper half space hg is a Teichm¨uellerspace for 2g- 2g dimensional symplectic lattices. Speciﬁcally, for A ∈ Sp2gR we say Λ := AZ is a symplec- 2g tic lattice. The symplectic form ω is integral and unimodular on Z . Thus ω is integral and unimodular on every symplectic lattice Λ. The unitary group Ug acts simultaneously by isometries and symplectomorphisms on the left. Therefore a choice of representative for

A ∈ Ug\Sp2gR determines a well defined isometry and symplectomorphism class of marked 2g 2g symplectic lattice Λ := AZ . For any such Λ, a subspace H in (R , ω) is Λ-rational if H ∩ Λ is cocompact in H, or equivalently, admits a Z-basis. Given a d-dimensional Λ- rational subspace H, let x1, . . . , xd be a Z-basis for H ∩ Λ. Define the volume of H in Λ, denoted vol(H, Λ), to be equal to vol(H/H ∩ Λ). In terms of the Z-basis x1, . . . , xd r t 0 0 we have vol(H, Λ) = + det xixj . Since any other Z-basis x1, . . . , xd of H ∩ Λ differs from x1, . . . , xd by an integral unimodular substitution, the above formula for vol(H, Λ) is well-defined. We set the volume of the zero subspace in any lattice to be equal to unity. Whenever there is no confusion concerning the ambient lattice Λ we denote the squared- volume of any Λ-rational subspace H by the shorthand d(H). For vectors x1, . . . , xd, set d(x1, . . . , xd) = d(spanZ{x1, . . . , xd}). The following inequality is used constantly (cf. [10], [13], [18]):

N Hadamard inequality. Fix an arbitrary cocompact lattice Γ in R . Then for any Γ-

9 2.2. Symplectic lattices rational subspaces A, B one has

d(A + B)d(A ∩ B) ≤ d(A)d(B).

From the identity hx, yi = ω(x, Jy) and the fact that ω is integral and unimodular 2g on a symplectic lattice Λ in (R , ω), it follows that symplectic lattices are self-dual (i.e. ∗ 2g isomorphic to their dual lattice Λ which is deﬁned to consist of those vectors v ∈ R ∗ such that hx, yi ∈ Z for all y ∈ Λ) with identiﬁcation given by JΛ = Λ ). The following result of Conway and Sloane (c.f [6], Appendix 2) shows this property in fact characterizes symplectic lattices.

2g Conway-Sloane’s Characterization. A lattice Λ in R is symplectic if and only if there is an orthogonal transformation ι on the space such that ι2 = −1 and ι(Λ) is the dual lattice Λ∗.

As corollary we obtain that a so-called classically integral lattice Λ (i.e. Λ∗ = Λ) is symplectic if and only if it is unimodular and has an automorphism (i.e. lattice isometry) ι satisfying ι2 = −1. The result of Conway-Sloane has the merit of showing that many interesting and important lattices are symplectic (after possibly rescaling their volume). Following the nomenclature of [9] we record some instances. All 2-dimensional unimodular lattices are symplectic (this is obviously so, since SL2 = Sp2). The extremal lattice D4 4 6 on R is symplectic whereas the extremal quadratic form E6 in R is not self-dual and thus not symplectic. The so-called Barnes-Wall lattices are all symplectic, and moreover + (2) D4m(m ≥ 1),A6 ,E8,K12, Λ16 and the Leech lattice Λ24 are all symplectic. Their explicit Gram matrices can be found in ([6], Appendix 2). Remark 1. Symplectic lattices also arise naturally from the theory of Riemann surfaces, where the Abel-Jacobi period mapping describes a holomorphic equivariant embedding from the Teichm¨uellerspace of closed hyperbolic genus g surfaces to hg. The problem of characterizing the image of the period mapping, i.e. which symplectic lattices are the period lattices for a riemann surface, is the so-called Schottky problem.

10 Chapter 3

Some symplectic linear algebra

With the above preliminaries set, we now describe some symplectic linear algebraic con- structions which are basic to our entire paper.

3.1 Intersecting lagrangian subspaces

Our ﬁrst point of study concerns intersections of lagrangian subspaces, where our ﬁrst lemma is crucial.

2g Lemma 2. Let `1, `2,... be lagrangian subspaces in (R , ω). Set W = spanR{`1, `2,...} and K = ∩i=1,2,...`i. Then

(i) K is the radical of the symplectic subspace (W, ω|W ×W )

2g (ii) we have an h, i-orthogonal splitting R = W ⊕ JK

(iii) the ω-orthogonal complement of K is W , i.e. K⊥ = W .

(iv) dimK + dimW = 2g

0 0 Proof. If `, ` are distinct lagrangians with x ∈ `, then ωx is identically zero on ` if and 0 only if also x ∈ ` (by maximality). So for x ∈ W , we have ωx identically zero exactly when x ∈ K. This proves (i). Now JK is trivially h, i-orthogonal to W exactly because K is the radical of W . Moreover ω is nondegenerate on W ⊕ JK. And since W ⊕ JK contains g-dimensional totally isotropic subspaces (i.e. `1, `2,...) it follows that W ⊕ JK has dimension at least 2g. The claim in (ii) follows. To establish (iii), we observe that ⊥ ⊥ ⊥ for i = 1, 2 ... one has K ⊆ ì and hence K ⊇ ì = ì. So K ⊇ W . But both have dimension equal to 2g − dimRK (by (ii)), hence equality. Finally (iv) simply follows from either (iii) or (ii).

Our second lemma will play a role in 4.

11 3.2. Heights of ω-orthogonals

Lemma 3. Fix a symplectic lattice Λ. Let x1, . . . , xk be k ≤ g linearly independant lattice elements satisfying:

⊥ ⊥ xj+1 ∈ xj ∩ ... ∩ x1 , j = 1, . . . , k − 1.

Then

⊥ ⊥ (i) for 1 ≤ j ≤ k − 1 the subspace xj ∩ ... ∩ x1 is (2g − j)-dimensional, and

⊥ ⊥ ⊥ 2g (ii) for 2 ≤ j ≤ k − 1 one has the equality xj + (xj−1 ∩ ... ∩ x1 ) = R .

Proof. For k = 1, (i) is trivial and (ii) is vacuous. Hence we assume k ≥ 2 and proceed by ⊥ induction on k. For the case k = 2, suppose x2, x1 are linearly independant and x2 ∈ x1 . ⊥ ⊥ Deﬁning |H| := dimRH, the dimension formula from linear algebra yields |x2 ∩ x1 | = ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ |x2 |+|x1 |−|x2 +x1 |. Hence (i) and (ii) are equivalent for k = 2, since |x2 | = |x1 | = 2g−1. ⊥ ⊥ 2g ⊥ ⊥ ⊥ ⊥ ⊥ But x1 +x2 6= R if and only if either x2 ⊆ x1 or x2 ⊃ x1 . Since the hypotheses x2 ∈ x1 ⊥ ⊥ ⊥ and x1 ∈ x2 are equivalent, we are free to suppose x2 ⊇ x1 . But this containment ⊥ o ⊥ o is equivalent to having R · Jx2 = (x2 ) ⊆ (x1 ) = R · Jx1, contradicting the linear independance of x2, x1. Supposing the lemma has been proven for some 2 ≤ k < g, we’ll now establish the case ⊥ ⊥ k + 1. The dimension formula shows (i), (ii) are equivalent (since |xk ∩ ... ∩ x1 | = 2g − k ⊥ ⊥ ⊥ 2g ⊥ ⊥ ⊥ by induction). Now xk+1 + (xk ∩ ... ∩ x1 ) 6= R if and only if xk+1 ⊇ (xk ∩ ... ∩ x1 ) ⊥ (since xk+1 is a codimension 1 subspace and 2g − 2 ≥ 2g − k). But this ﬁnal containment ⊥ ⊥ o ⊥ ⊥ o occurs if and only if Jxk+1 ∈ (xk ∩ ... ∩ x1 ) . Now by induction |(xk ∩ ... ∩ x1 ) | = k. ⊥ ⊥ o But Jxk, . . . , Jx1 are k linearly independant vectors ∈ (xk ∩ ... ∩ x1 ) . Hence we have ⊥ ⊥ ⊥ 2g xk+1 + (xk ∩ ... ∩ x1 ) 6= R if and only if Jxk+1, . . . , Jx1 are linearly dependant, which evidently does not occur by the hypotheses of the lemma. This establishes the claim for k + 1, and the lemma.

3.2 Heights of ω-orthogonals

The next lemma may properly belong to the category of symplectic geometry-of-numbers, and gives a convenient estimate for the heights of rational lagrangians.

12 3.2. Heights of ω-orthogonals

2g Lemma 4. Let Λ be a symplectic lattice in (R , ω) and x ∈ Λ\0. Set Nx = miny∈Λ−x⊥ {|ω(x, y)|}. Then the ω-orthogonal complement x⊥ is a Λ-rational subspace with volume d(x⊥) = ⊥ 2 2 −1 ⊥ 2 vol(x , Λ) = ||x|| Nx . If x is primitive, then d(x ) = ||x|| .

Proof. We essentially exploit the trivial identity x⊥ = (Jx)o. Indeed the Λ-rationality of x⊥ = (Jx)o is immediate: the almost-complex structure J yields an explicit isomorphism of the symplectic lattice Λ onto its dual Λ∗ = JΛ (ie. symplectic lattices are self-dual), where the equality trivially follows from the fact that ω is integral and unimodular on Λ. The result then follows from §I.5 in [7]. To calculate the volume of x⊥ in Λ we recall the following basic principle in the geometry-of-numbers (cf. §5 in [18]). Let Λ be a 2g-dimensional lattice in the euclidean 2g d space (R , h, i), and H a d-dimensional Λ-rational subspace with volume vol(H, Λ) = d(H). Set Λ0 to be the image of the orthogonal projection of Λ onto the (2g−d)-dimensional h, i-orthogonal complement Ho. Then Λ0 is a lattice in Ho with covol(Λ0) = covol(Λ)/d(H). The h, i-orthogonal complement of x⊥ = (Jx)o is the 1-dimensional subspace spanned 0 by Jx. To determine the image Λ of the orthogonal projection of Λ onto spanRJx we must describe those possible z = z0 + λJx ∈ Λ, for which z0 belongs to x⊥. But ω(x, z) = 1 λω(x, Jx) must be an integer (since ω is integral on a symplectic lattice Λ). So λ ∈ ω(x,Jx) Z. 0 Note that this explicitly shows Λ is discrete in spanRJx. Furthermore it is apparent that

0 1 1 covol(Λ ) = min ⊥ {|ω(x, z)|} = N . ||x||2 z∈Λ−x ||x||2 x

⊥ 2 −1 Since Λ is unimodular we have indeed d(x ) = ||x|| Nx , as desired. To see Nx = 1 for x primitive: the quantity Nx is invariant under linear symplecto- 2g morphisms, hence we can take Λ = Z . That x is primitive means that with respect to the standard Z-basis e1, . . . , eg, f1, . . . , fg, one has an expression x = Σi(aiei + bifi), for ai, bi ∈ Z with gcd(a1, . . . , bg) = 1. Consequently there exist some ci, di ∈ Z with 1 = Σi(aici + bidi). Set z = Σi(cifi − diei). Then ω(x, z) = 1.

13 Chapter 4

Systoles on hg

4.1 Lagrangian systole function

In this section we introduce a systolic invariant of symplectic lattices. This yields an

Sp2gZ-equivariant function on hg. It is against this function that we deﬁne our equivariant retract.

2g Deﬁnition 5. For A ∈ hg let L A denote the set of AZ -rational lagrangian subspaces. Take 2g sysL A := min`∈L Avol(`, AZ ). (4.1)

We call sysL the lagrangian systole function. g 2g The rational lagrangians L A form a discrete subset of the covolume 1 lattice ∧ Z in g 2g ∧ R . Hence sysL is everywhere ﬁnite and nonzero on hg. Moreover the integral sym- 2g plectic group Sp2gZ acts transitively on L Z . Consequently sysL is a Sp2gZ-equivariant function on hg.

4.2 L -wellrounded lattices

Before making our next deﬁnition, we ﬁx some further

2g Notation 6. For A ∈ hg with Λ = AZ a symplectic lattice, 2g • let LminΛ denote the set of minimal volume Λ-rational lagrangian subspaces in R ;

• take WΛ := spanR{LminΛ};

• take KΛ = ∩`∈LminΛ `; 2g For a 2g-dimensional symplectic lattice Λ, our earlier Lemma 2 tells us that WΛ = R if and only if KΛ = 0. With this equivalence in mind, we now make the central deﬁnition of this chapter:

14 4.2. L -wellrounded lattices

Deﬁnition 7. A symplectic lattice Λ is L -wellrounded if KΛ = 0 (or equivalently WΛ = 2g Λ ⊗ R). We denote by Wg the collection of A ∈ hg for which AZ is L -wellrounded.

Remark 8. We recall that the 1-systole of an n-dimensional unimodular lattice Γ, denoted sys1Γ, is defined as sys1Λ := minx∈Λ−0||x||. The 1-systole thus yields an SLnZ-equivariant function on the symmetric space Sn. Alternatively one can define sys1Γ to be the minimal n volume (length) of a Γ-rational 1-dimensional subspace in R . Generalizing one obtains the higher dimensional systoles syskΓ. The classical Hermite constant γn is defined as the supremum of sys1Γ, where Γ varies over Sn. Minkowski’s first convex body theorem

(c.f. [7]) readily impliies γn < ∞, i.e. sys1 is bounded on the noncompact quotient Sn/SLnZ. Minkowski’s convex body theorem yields an estimate which is far from optimal, and whose improvement is an old and central problem within the geometry-of-numbers.

One can deﬁne the analagous Hermite constant γL ,g := supA∈hg {systL A}. The following proposition shows that γL ,g is always ﬁnite.

Proposition 9. Fix an integer g ≥ 1. There exists a constant C depending only on g for which every 2g-dimensional symplectic lattice Λ has systL Λ ≤ C.

Proof. We describe a procedure using Minkowski’s second theorem on successive minima

(cf. §III.2 in [7]) to a construct a rational lagrangian in A ∈ hg whose volume is bounded above by some positive constant depending only on g. The following estimate provided by Lemma 3 will be required: if x1, . . . , xk(k ≤ g) are linearly independant lattice vectors ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ 2g with x2 ∈ x1 , x3 ∈ x2 ∩ x1 , . . . , xk ∈ xk−1 ∩ ... ∩ x1 , then xk + (xk−1 ∩ · · · ∩ x1 ) = R , ⊥ ⊥ ⊥ i.e. d(xk + (xk−1 ∩ · · · ∩ x1 )) ≥ 1. 2g As Λ := AZ is unimodular, there exists a constant c1 = γ2g depending only on the integer g such that Λ has a nonzero vector x1 of length ||x1|| ≤ c1. The Λ-rational subspace ⊥ ⊥ 2 x1 has volume d(x1 ) = ||x1|| by Lemma 4. 2g−1 ⊥ Consider the (2g − 1)-dimensional unit ball B in x1 , with euclidean volume σ(2g − 1). Now B2g−1 is a compact 0-symmetric convex body and hence by Minkowski’s second theorem on successive minima [[7]] we have

2g−1 2 λ1λ2 ··· λ2g−1σ(2g − 1) ≤ 2 ||x1|| , 0 < λ1 ≤ λ2 ... ≤ λ2g−1 < ∞.

⊥ ⊥ However x1 ∈ x1 , and since x1 is the shortest nonzero lattice point in x1 , we ﬁnd the ﬁrst minimum λ1 = ||x1||. To estimate the second minimum λ2, (i.e. the minimal ⊥ length of a lattice vector x2 ∈ x1 ∩ Λ linearly independant from x1), we observe that

15 4.2. L -wellrounded lattices

2g−2 λ1 λ2 ≤ λ1λ2 ··· λ2g−1 and hence

2−2g 2g−1 2 −1 λ2 ≤ λ1 2 ||x1|| σ(2g − 1)

≤ c2, where indeed c2 is a constant depending only on g, which explicitly can be taken to be 2g−1 −1 4−2g 2 σ(2g − 1) c1 . ⊥ ⊥ ⊥ ⊥ By lemma 3 and Hadamard’s inequality d(x2 ∩ x1 ) ≤ d(x1 )d(x2 ) ≤ c2c1. We now consider, as before, the (2g − 2)-dimensional unit ball B2g−2 of euclidean ⊥ ⊥ volume σ(2g − 2) in the rational subspace x2 ∩ x1 . According to Minkowski’s second theorem the new successive minima λ1, λ2, . . . , λ2g−2 (which now depend on the unit ball 2g−2 ⊥ ⊥ B in x2 ∩ x1 ) satisfy

2g−2 2 2 λ1λ2 ··· λ2g−2σ(2g − 2) ≤ 2 ||x2|| ||x1|| .

⊥ ⊥ As indeed x1, x2 are the 1st, 2nd shortest nonzero lattice vectors in x2 ∩ x1 , we ﬁnd that λ1 = ||x1||, λ2 = ||x2||. Consequently, we can estimate λ3 (the minimal length of a lattice vector x3 linearly independant from x2, x1 such that z, x2, x1 span a totally isotropic subspace) as before:

−1 4−2g 2g−2 2 2 −1 λ3 ≤ λ1 λ2 2 ||x1|| ||x2|| σ(2g − 2)

≤ c3, where c3 is again some constant depending only on g. ⊥ ⊥ ⊥ Continuing this process for x3 ∩x2 ∩x1 , etc we ﬁnd linearly independant lattice vectors ⊥ ⊥ xg, . . . , x1 spanning a rational lagrangian subspace L := xg ∩ ... ∩ x1 with ||xk|| equal to th ⊥ ⊥ the k successive minima of the (2g − k + 1)-unit ball in xk−1 ∩ · · · ∩ x1 . The Hadamard Q 2 inequality gives d(L) ≤ 1≤i≤g ||xi|| ≤ cg, where cg will be a positive contant depending only on g.

Remark 10. Siegel’s upper half space hg admits the following interpretation. For a given symplectic manifold (M, ω) an endomorphism J : TM → TM of the tangent bundle (ﬁbrewise linear) satisfying J 2 = −id is known as an almost-complex structure. We say an almost-complex structure J is ω-compatible if the symmetric form gJ (·, ·) := ω(·,J·) is

16 4.2. L -wellrounded lattices

a riemannian metric. The collection Jc(ω) of ω-compatible almost-complex structures J on (M, ω) is a contractible space. Consider the 2g-dimensional torus T 2g endowed with a unimodular and integral translation- 2g 2g invariant symplectic form ω (i.e. R /Z ). A marking A ∈ hg yields a ω-compatible ! −1 0 −I almost-complex structure J := AJstdA , where Jstd := . The metrics obtained I 0 2g flat are flat metrics on T . Thus we find that hg can be identified with the collection Jc (ω) of flat ω-compatible almost-complex structures on (T 2g, ω).

Our lagrangian systole function sysL thus extends to a function deﬁned on the (inﬁnite- dimensional) space Jc(ω). The question of whether or not supJ∈Jc(ω) sysL < ∞ is unknown. This author sees the present thesis as marking preliminary progress towards answering the previous question, i.e. does sysL satisfy a systolic-inequality on Jc(ω), and hopes to address these issues in the future.

17 Chapter 5

The equivariant retract Wg

In this chapter we make precise the continuity of our deformation of hg onto the set of

L -wellrounded symplectic lattices Wg.

Proposition 11. For g ≥ 1, we have a strong Sp2gZ-equivariant deformation retract hg → Wg.

We begin with a preliminary

Deﬁnition 12. For j = g, g + 1,..., 2g − 1 let

(j) hg := {A ∈ hg|dimR(spanRLminA) ≥ j}.

(j) The subspaces hg are Sp2gZ-equivariant and yield the following ﬁltration of the genus g Siegel upper half space

(g) (g+1) (2g) hg = hg ⊇ hg ⊇ · · · ⊇ hg = Wg.

The inclusions are not generally strict. Nonetheless, we still have our next

(j+1) (j) Proposition 13. For j = g, . . . , 2g − 1, the space hg is a closed subset of hg .

2g 2g Proof. It is here convenient to view hg as a space of certain flat metrics on Z and R : 2g 2g for A ∈ hg we define a metric gA on both Z , R by setting gA(x, y) := ω(Ax, JAy). The metric gA is independant of choice of representative of A ∈ Sp2gR since any U ∈ Ug is a symplectomorphism with tU = U −1. (j+1) Suppose now (Ak) is a sequence in hg with limit A∞ in hg. Take gk := gAk and g∞ := gA∞ . Then (gk) is a sequence of flat metrics for which the set Lmingk of minimal gk- 2g volume Z -rational lagrangians span a real subspace of dimension ≥ j+1. As the collection 2g of Z -rational lagrangians is discrete, by passing to a subsequence we can assume that the set Lmingk is constant as k → ∞. Therefore the real span of Lmingk is some constant

18 Chapter 5. The equivariant retract Wg

subspace, say W . Hence the real span of Lming∞ is also W . Therefore g∞ and A∞ belong (j+1) to hg .

(j) We will construct our retract hg → Wg from a sequence of retractions hg = hg → (j+1) (2g) hg → ... → hg = Wg. The following proposition constructs these auxiliary retractions.

(j+1) Proposition 14. For j = g, . . . , 2g−1, the space hg is a Sp2gZ-equivariant deformation (j) retract of hg .

Proof. To construct a retraction, we first describe some 1-parameter families of linear sym- 2g plectomorphisms. For K a totally isotropic subspace of (R , ω, J), take a h, i-orthocomplement ⊥ K V to K in K . Now for each µ ∈ R consider the map Tµ : K ⊕ V ⊕ JK → K ⊕ V ⊕ JK −1 K −1 K defined as k +v +x 7→ µk +v +µ x (ie. Tµ = µ⊕id⊕µ ). Hence Tµ dilates radially by µ on K, is identity on the orthocomplement V , and contracts by µ−1 on JK. Now we claim K Tµ ∈ Sp2gR for µ ∈ R whenever K is totally isotropic. Indeed K,JK span a J-invariant subspace with ω-orthogonal (equivalently, h, i-orthogonal) V , so ω(V,JK) = ω(K,V ) = 0. K K Thus Tµ can directly be seen to preserve ω. The definition of Tµ is independant of choice of h, i-orthocomplement V in K⊥.

KA (j) Consider the action of Tµ on a marking A ∈ hg , where we expand along the total 2g intersection KA of the minimal volume rational lagrangian subspaces in Λ = AZ . That A KA is, consider the half-ray Tµ A := Tµ A in hg where µ ≥ 1. The linear symplectomorphisms A KA Tµ = Tµ have the eﬀect of expanding the volumes of the rational minimal lagrangians d/2 LminA proportionally by a factor of µ (where d = dimKA = 2g−j−1) while contracting in the direction of JK. Now there shall exist a minimal time τ = τ (j)A ≥ 1 for which A (j+1) A Tτ A ∈ hg , ie. for which LminTµ A spans a linear subspace of dimension ≥ j + 1. (j) (j) (j+1) Evidently τ is Sp2gZ-equivariant and τ A = 1 for A ∈ hg . (j) (j) We claim the function τ is continuous on hg . Indeed suppose (Ak) is a sequence in (j) hg with Ak → A∞. We want to show

(j) (j) τ Ak → τ A∞. (5.1)

(j+1) (j+1) If A∞ ∈ hg , then for sufficiently large k we’ll have Ak ∈ hg and (5) will be clear. (j) (j+1) Suppose now instead that A∞ ∈ hg \ hg . Then WAk ,KAk Gromov-Hausdorff converge 2g 2g 2g to WA∞ ,KA∞ in R . Consequently for sufficiently large k, both the lattices AkZ ,A∞Z Ak A∞ (j) (j) and the deformations Tµ ,Tµ will be comparably close. So the times τ Ak, τ A∞

19 Chapter 5. The equivariant retract Wg for which the corresponding deformations generate a “new” minimal lagrangian will be (j) (j) comparably close. That is, τ Ak → τ A∞. (j) (j) The continuity of the function A 7→ τ A on hg implies that

(j) (j) (j) A r : [0, 1] × hg → hg , (t, A) 7→ T(1−t)+tτ (j)(A)A is a continuous homotopy. It is evidently Sp2gZ-equivariant.

We now complete the proof of Proposition 11:

Proof of Prop 11. The successive homotopies r(g), r(g+1), . . . , r(2g−1) constructed above yield an equivariant strong deformation retract hg → Wg.

20 Chapter 6

Cocompactness of Wg

In this section we prove that our closed Sp2gZ-equivariant deformation retract Wg is cocompact, ie. the quotient Wg/Sp2gZ is compact in hg/Sp2gZ. To verify the cocompactness we use a variation on the well-known Mahler compactness criterion. To present this criterion we let Pg denote the vector space of g × g real symmetric positive deﬁnite matrices and t recall the natural right action of GLgZ on Pg given by (g, P ) 7→ gP g.

Mahler criterion. A closed subset K ⊆ Pg/GLgZ is compact if and only if the following two conditions are satisﬁed:

−1 (i) there exists C > 1 such that C ≤ |detgY | ≤ C for every Y ∈ K (where detg is the

g-dimensional determinant function on Pg);

(g) (g) (ii) there exists > 0 such that syst1 Y ≥ for every Y ∈ K (where syst1 is the g-dimensional 1-systole function on Pg).

The above criterion is proven by examining the cusps in a fundamental domain M of

GLgZ in Pg (cf. [11], [19]). We shall likewise identify a cocompactness criterion for closed subsets of hg by considering a fundamental domain for the integral symplectic group ﬁrst constructed by C.L. Siegel in [19].

Proposition 15 (Siegel). Let F be that subset of hg which in the Iwasawa coordinates consists of those Ω = X + iY for which the following three conditions are satisﬁed:

(i) |xij| ≤ 1/2, for X = <Ω = (xij);

∗ ∗ (ii) |detY | ≤ |detY | for all Y = =(Ω.γ), where γ ∈ Sp2gZ and Y = =Ω;

(iii) Y ∈ M

Since the cube |xij| ≤ 1/2 is compact, Siegel’s fundamental domain F and the Mahler criterion for Pg yields the following

21 Chapter 6. Cocompactness of Wg

Mahler-Siegel criterion. A closed subset K ⊆ hg/Sp2gZ is compact if the following conditions are satisﬁed:

(i) detg is bounded on =K;

(g) (ii) there exists > 0 such that syst1 Y ≥ for all Y ∈ =K. (2g) To apply Prop 6 we shall require systL and syst1 to both be bounded away from zero. This is established by the following

2 Proposition 16. (i) If A ∈ W , then 1 ≤ covol(spanZLminA) ≤ (systL A) ;

(ii) there exists % > 0 such that if A ∈ hg admits a nonzero lattice vector x with ||x|| < %, ⊥ then LminA ⊂ x .

(iii) there exists > 0 such that syst1A ≥ for all A ∈ W 2g Proof. Set Λ = AZ and define δmin = systL Λ. Let `1, . . . , `k be distinct minimal volume Λ-rational lagrangians. We set ìj··· = ì ∩ `j ∩ · · · . We choose k minimal such that

`1...k = 0 and `1···ˆi···k 6= 0 for 1 ≤ i ≤ k. The generalized Hadamard inequality 2.2 yields the estimates

δmind(`2 + ··· + `k) 1 ≤ d(`1 + ··· + `k) ≤ d(`1 ∩ (`2 + ··· + `k)) k Y ≤ δmin d(`i ∩ (`i+1 + ··· + `k)). 1≤i≤k−1

We claim that for j = 1, . . . , k − 1

the subspace `j+1,...,k + `j ∩ (`j+1 + ··· + `k) is a rational lagrangian. (6.1)

Indeed by Lemma 2 we see `j+1,...,k is the radical of `j+1 + ··· + `k; in particular

ωx vanishes identically on the totally isotropic subspace `j ∩ (`j+1 + ··· + `k) for every x ∈ `j+1 + ··· + `k. So (6) is totally isotropic. From Lemma 3 and the standard dimension formula from linear algebra it follows that (6) is g-dimensional, hence lagrangian.

Consequently d(`j+1,...,k + .`j ∩ (`j+1 + ··· + `k)) ≥ δmin. Together with the Hadamard inequality we get d(`j,j+1,...,k) δmin ≤ d(`j ∩ (`j+1 + ··· + `k)). d(`j+1,...,k)

22 Chapter 6. Cocompactness of Wg and altogether we ﬁnd

k k−1 Y d(`j...k) 2 2 1 ≤ d(`1 + ··· + `k) ≤ δminδmin = δmin = (systL Λ) . d(`j+1,...,k) j=1,...,k−1

This proves (i). 2g To prove (ii) we show % = 1 suﬃces. Take A ∈ hg and set Λ = AZ . Let x be a ⊥ primitive nonzero lattice vector in Λ and suppose ` is a Λ-rational lagrangian with ` * x , or equivalently x∈ / `. Then ` ∩ x⊥ is a (g − 1)-dimensional Λ-rational totally isotropic ⊥ 2g ⊥ subspace. Since ` + x = R , the Hadamard inequality and Lemma 4 yields d(` ∩ x ) ≤ ⊥ 2 ⊥ d(`)d(x ) = d(`)||x|| . Let z1, . . . , zg−1 be a Z-basis for (` ∩ x ) ∩ Λ. Now x, z1, . . . , zg−1 ⊥ 0 are linearly independant (since x∈ / ` ∩ x ). Therefore ` := spanR{x, z1, . . . , zg−1} is a Λ-rational lagrangian contained in x⊥. From the Hadamard inequality we ﬁnd

0 3 d(` ) ≤ ||x||d(z1, . . . , zg−1) ≤ ||x|| d(`).

So if ||x|| < 1, then d(`0) < d(`). Therefore any symplectic lattice Λ admitting a nonzero ⊥ lattice element x with ||x|| < 1 has LminΛ ⊂ x . This proves (ii). Taking = 1 one deduces (iii) immediately from (ii).

Finally we have

Proposition 17. (i) detg is bounded on =W and

(g) (ii) there exists > 0 such that syst1 Y ≥ for all Y ∈ =(W ∩ F).

Proof. We establish (i) by showing separately detg is bounded below and above on =(W ∩F).

First a trivial computation: a point Ω ∈ hg with Iwasawa coordinates X + iY uniquely ! YYX determines a positive definite symmetric symplectic matrix A = . XYXYX + Y −1 ! Y Consequently the first g columns of A generate a Λ-rational lagrangian subspace, XY 2g p 2 say À, with Λ = AZ . But one readily computes vol(À, Λ) = |detY | |det(Ig + X )|, where Ig is the g × g identity matrix. For A (or equivalently Ω) in W , Prop 16 gives a lower bound |det=A| ≥ |det(I +

23 Chapter 6. Cocompactness of Wg

Since 0. This finds |detg| is bounded from below on =(W ∩ F). To find an upper bound for |detg| on =W , we argue as follows. A symplectic lattice 2g ∗ ∗ Λ = AZ is self-dual, ie. Λ and Λ are isometric. Explicitly one can take Λ = JΛ. Since t t −1 ∗ t −1 2g A ∈ Sp2gR if and only if JAJ = A , we can explicitly take Λ = JΛ = A Z . Let us also make the observation that W ∗ = W , ie. Λ ∈ W if and only if Λ∗ ∈ W . This is obvious because J is simultaneously an isometry and symplectomorphism. t −1 −1 If A ∈ W ⊂ hg has Iwasawa coordinates X + iY , then A = A has Iwasawa coordinates (−X) + iY −1. Hence we find, as before, that 1 ≤ |detY |−1p|det(I + X2)|.

This establishes that |detg| is also bounded from above on =W . This proves (i).

We establish (ii) by contradiction. Suppose we had a sequence (Ak) in W ∩ F with (g) syst Yk → 0, where Yk := =Ak and Xk :=

(g) ||y˜k|| ≤ ||yk|| + ||vk|| = syst1 Yk + ||Xkyk|| (g) ≤ syst1 Yk(1 + ||Xk||∞), where || · ||∞ denotes the usual operator sup-norm.

Since the Xk vary over a compact set, the term 1 + ||Xk||∞ is bounded, and we (g) ﬁnd syst1 (1 + ||Xk||∞) → 0 as k → ∞. But this means the sequence (Ak) ∈ W has (2g) (g) syst1 Ak → 0, which contradicts (iii) of the previous proposition 16. Therefore syst1 is bounded away from zero on =(W ∩ F).

The above proposition together with 6 yields

Proposition 18. For g ≥ 1, the retract Wg is cocompact in hg.

24 Chapter 7

Codimension of Wg

Let us begin by describing the maximal R-split tori in Sp2gR. We shall be rather formal throughout this section because we are looking to derive the precise formula 7.1. From the theory of reductive algebraic groups one knows that all maximal R-split tori are conjugate in Sp2gR. With respect to the standard symplectic basis

e1, . . . , eg, f1, . . . , fg the standard maximal R-split torus has the form