The Flat Torus Theorem
This is the first of a number of chapters in which we study the subgroup structure of groups F that act properly by semi-simple isometries on CAT(0) spaces X. In this chapter our focus will be on the abelian subgroups of Γ. The Flat Torus Theorem (7.1) shows that the structure of such subgroups is faithfully reflected in the geometry of the flat subspaces in X. One important consequence of this fact is the Solvable Subgroup Theorem (7.8): if Γ acts properly and cocompactly by isometries on a CAT(0) space, then every solvable subgroup of Γ is finitely generated and virtually abelian. In addition to algebraic results of this kind, we shall also present some topological consequences of the Flat Torus Theorem.
KeywordsAbelian Subgroup Finite Index Mapping Class Group Proper Action Dehn Twist
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