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) sh
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This is the first of a number of chapters in which we study the subgroup structure of groups r that act properly by semi-simple isometries on CAT(O) spaces X. In this chapter our focus will be on the abelian subgroups of r. 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 r acts properly and cocompactly by isometries on a CAT(O) space, then every solvable subgroup of r 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. Both the Flat Torus Theorem and the Solvable Subgroup Theorem were discovered in the setting of smooth manifolds by Gromoll and Wolf [GW71] and, independently, by Lawson and Yau [LaY72]. Our proofs are quite different to the original ones. Throughout this chapter we shall work primarily with proper actions by semisimple isometries rather than cocompact actions. Working in this generality has a number of advantages: besides the obvious benefit of affording more general results, one can exploit the fact that when restricting an action to a subgroup one again obtains an action of the same type. (In particular this facilitates induction arguments.)
The Flat Torus Theorem Recall that, given a group of isometries r, we write Min(r) for the set of points which are moved the minimal distance Iy I by every y E r. 7.1 Flat Torus Theorem. Let A be a free abelian group of rank n acting properly by semi-simple isometries on a CAT(O) space X. Then:
naeA Min(a) is non-empty and splits as a product Y x lEn.
(1)
Min(A) =
(2)
Every element a E A leaves Min(A) invariant and respects the product decomposition; a acts as the identity on the first factor Y and as a translation on the second factor lEn.
(3)
The quotient of each n-flat {y} x lEn by the action of A is an n-torus.
M. R. Bridson et al., Metric Spaces of Non-Positive Curvature © Springer-Verlag Berlin Heidelberg 1999
The Flat Torus Theorem
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(4)
If an isometry ofX normalizes A, then it leaves Min(A) invariant and preserves the product decomposition.
(5)
If a subgroup Isom(X) normalizes A, then a subgroup of finite index in r centralizes A. Moreover, ifr is finitely generated, then r has a subgroup of finite index that contains A as a direct factor.
rc
Proof. We shall prove parts (1), (2) and (3) by induction on the rank of A. As the action of A is proper and by semi-simple isometries, each non-trivial element of A is a hyperbolic isometry. Suppose that A ~ 7L,n and choose generators ai, ... , an. We have seen (6.8) that Min(al) splits as Z x lEI where al acts trivially on the
first factor and acts as a translation of amplitude lall on the second factor. Every a E A commutes with al and therefore preserves the subspace Z x lEI with its decomposition, acting by translation on the factor lEI (see (6.8)). We claim that the subgroup N C A formed by the eleme
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