4 Amenability and non-positive curvature
4.1 CAT(0) spaces
Definition 4.1 A CAT(0) space is complete metric space \(X\) such that is
- geodesic 2 for any \(x,y,z\in X\) and \(m\) midpoint of a geodesic segment between \(x\) and \(z\),
\[d(z,m)^2\leq 1/2(d(z,x)^2+d(z,y)^2)-1/4d(x,y)^2\]
For three points \(x,y,z\) in a geodesic metric space \(X\) a geodesic triangle \(\Delta(x,y,z)\) is the union of three geodesic segments \([x,y], [y,z]\) and \([z,x]\). A comparison triangle \(\Delta(\overline{x},\overline{y},\overline{z})\) is a triangle in \(\mathbf{R}^2\) such that \(d(\overline{x},\overline{y})=d(x,y)\), \(d(\overline{x},\overline{z})=d(x,z)\) and \(d(\overline{z},\overline{y})=d(z,y)\). For a point \(p\) on a side of \(\Delta(x,y,z)\), we denote by \(\overline{p}\) the corresponding point on \(\Delta(\overline{x},\overline{y},\overline{z})\)
Proposition 4.1 Let \(X\) a complete geodesic metric space. The following are equivalent:
- \(X\) is CAT(0).
- For any geodesic triangle \(\Delta(x,y,z)\) with comparison triangle \(\Delta(\overline{x},\overline{y},\overline{z})\), for \(p,q\) on sides of \(\Delta\),
\[d(\overline{p},\overline{q})\leq d(p,q).\]
Lemma 4.1 In a CAT(0) space, geodesic segments are unique.
From now on, for two points \(x,y\) in a CAT(0) space \(X\), we denote by \([x,y]\) the unique geodesic segment between \(x\) and \(y\).
Lemma 4.2 Any CAT(0) space is contractible.
Example 4.1
- Euclidean spaces and Hilbert spaces
- trees with their geodesic distances
- Any simply connected Riemannian manifold of non-positive sectional curvature
- The space of definite positive matrices
Proposition 4.2 In a CAT(0) space, any bounded subset has a circumcenter.
Theorem 4.1 (Cartan fixed point theorem) Let \(X\) be a CAT(0) space and \(G\) be a group of isometries of \(X\). If \(G\) has a bounded orbit then \(G\) has a fixed point.
Corollary 4.1 Let \(G\) be a bounded subgroup of \(GL_n(\mathbf{R})\) then \(G\) is conjugated to the subgroup of orthogonal transformations.
Definition 4.2 Let \(X\) be a CAT(0) space. A subset \(C\subset X\) is convex if for \(x,y\in C\), \([x,y]\subset C\).
Proposition 4.3 In a CAT(0) space, balls are convex.
Proposition 4.4 Let \(X\) be a CAT(0) space. The distance function is convex, that if \(c,c'\colon[0,1]\to X\) are two parametrizations of geodesic segments with constant speed then for all \(t\in[0,1]\),
\[d(c(t),c'(t))\leq (1-t)d(c(0),c'(0))+td(c(1),c'(1)).\]
Proposition 4.5 Let \(C\) be a closed convex subspace of a CAT(0) space. Then for \(x\in X\), there is a unique \(p\in C\) such that \(d(x,p)=\inf_{c\in C} d(x,c)\). This point is called the projection of \(x\) to \(C\).