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Geometric Retracts of Siegel's Upper Half Space

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Geometric Retracts of Siegel's Upper Half Space Geometric retracts of Siegel's upper half space by 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 in 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 !-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 coho- mology 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 = KnG is a finite-dimensional contractible nonpositively curved space. The subgroup of Z-points Γ = G(Z) has a properly discon- tinuous right action on KnG given by (Kx; γ) 7! Kxγ for x 2 G; γ 2 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) = dimRKnG − 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 KnG for which •W is a Γ-equivariant strong deformation retract of KnG; • 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 Soul´ein his 1973 thesis [20]. Afterwords Ash [2], motivated by group cohomology, expanded Soul´e'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 Teichm¨uellerspace 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 := SO2RnSL2R is well known to coincide with the hyperbolic upper half space H. We see a point SO2R·A 2 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 := fx + iy : y > 0 jxj ≤ 1=2; x2 + y2 ≥ 1g: 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 fx + iy 2 F : x2 + y2 = 1g, 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 2 S2 n 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 2 X2, we do nothing and leave it stationary, i.e.
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