Providing an accessible approach to a special case of the Rank Theorem, the present text considers the exact finiteness properties of S-arithmetic subgroups of split reductive groups in positive characteristic when S contains only two places. While the proof of the general Rank Theorem uses an involved reduction theory due to Harder, by imposing the restrictions that the group is split and that S has only two places, one can instead make use of the theory of twin buildings.

1 Basic Definitions and Properties 1

1.1 Metric Spaces 1

      1.1.1 Geodesics 1

      1.1.2 Products and Joins 2

      1.1.3 Model Spaces 3

      1.1.4 CAT(k)-Spaces 3

      1.1.5 Polyhedral Complexes 4

      1.1.6 Links 6

      1.1.7 Visual Boundary 7

1.2 Spherical Geometry 7

      1.2.1 Spherical Triangles 8

      1.2.2 Decomposing Spherical Simplices 9

      1.2.3 Spherical Polytopes with Non-obtuse Angles 11

1.3 Finiteness Properties 13

1.4 Simplicial Morse Theory 17

1.5 Number Theory 18

      1.5.1 Valuations 19

      1.5.2 Discrete Valuations 20

      1.5.3 Local Fields 21

      1.5.4 S-Integers 22

1.6 Affine Varieties and Linear Algebraic Groups 23

      1.6.1 Affine Varieties 23

      1.6.2 Linear Algebraic Groups 24

      1.6.3 S-Arithmetic Groups 29

1.7 Buildings 29

      1.7.1 Spherical Coxeter Complexes 30

      1.7.2 Euclidean Coxeter Complexes 32

      1.7.3 General Coxeter Complexes 34

      1.7.4 Buildings 34

      1.7.5 Twin Buildings 38

1.8 Buildings and Groups 40

      1.8.1 BN-Pairs 40

      1.8.2 Twin BN-Pairs 41

      1.8.3 BN-Pairs of Groups over Local Fields 41

1.9 Affine Kac-Moody Groups 42

2 Finiteness Properties of G(Fq[t]) 45

2.1 Hemisphere Complexes 46

2.2 Metric Codistance 48

2.3 Height: A First Attempt 49

2.4 Zonotopes 52

2.5 Height 56

2.6 Flat Cells and the Angle Criterion 59

2.7 Secondary Height: The Game of Moves 61

      2.7.1 Proof of Proposition 2.31 Using Spherical Geometry 67

      2.7.2 Proof of Proposition 2.31 Using Coxeter Diagrams 69

2.8 The Morse Function 70

2.9 More Spherical Subcomplexes of Spherical Buildings 72

2.10 Descending Links 73

2.11 Proof of the Main Theorem for G(Fg[t]) 78

3 Finiteness Properties of G(Fq[t,t-1]) 81

3.1 Height 82

3.2 Flat Cells and the Angle Criterion 85

3.3 The Morse Function 86

3.4 Beyond Twin Apartments 87

3.5 Descending Links 92

3.6 Proof of the Main Theorem for G(Fg[t,t-1]) 96

A Adding Places 99

References 101

Index of Symbols 107

Index 109

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