Lectures on proper CAT(0) spaces and their isometry groups ...

1 Lectures on proper CAT(0) spacesand their isometry groups (preliminary version)Pierre-Emmanuel CapraceContentsIntroductionvLecture I. Leading examples11. The basics12. The Cartan hadamard theorem23. proper cocompact spaces34. Symmetric spaces45. Euclidean buildings56. Rigidity77. Exercises8 Lecture II. Geometric density111. A geometric relative of Zariski density112. The visual boundary113. Convexity134. A product decomposition theorem145. Geometric density of normal subgroups156. Exercises16 Lecture III. The full isometry group191. Locally compact groups192.

2 The isometry group of an irreducible space193. de Rham decomposition214. Exercises23 Lecture IV. Lattices251. Geometric Borel density252. Fixed points at infinity263. Levi decomposition274. Back to rigidity305. Exercises31 Bibliography33iiiIntroductionCAT(0) spaces , introduced by Alexandrov in the 1950 s, were givenprominence by M. Gromov, who showed that a great deal of the theoryof manifolds of non-positive sectional curvature could be developed withoutusing much more than the CAT(0) condition (see [BGS85]). Since then,CAT(0) spaces have played a central role in geometric group theory, open-ing a gateway to a form of generalized differential geometry encompassingnon-positively curved manifolds as well as large families of singular spacessuch as trees, Euclidean or non-Euclidean buildings, and many other cellcomplexes of non-positive introductions on CAT(0) spaces may be found in the literature, in the books [Bal95] and [BH99].

3 The goal of these Lectures is to presentsome material not covered by those references. While the rigidity of (usuallydiscrete)group actionson non-positively curved space is a standard themeof study in geometric group theory, the main idea we would like to convey isthat, in the locally compact case, thespacesthemselves turn out to be muchmore rigid than one might expect as soon as they admit a reasonable amountof isometries. This phenomenon will be highlighted by placing a specialemphasis on thefullisometry group of a proper CAT(0) space. Taking intoaccount the fact the this isometry group is naturally endowed with a locallycompact group topology which is possibly non-discrete, many structural(and especially rigidity) properties of the underlying space can be derivedby combining results on locally compact groups with (mostly elementary)geometric arguments.

4 A number of results obtained along this approach arepresented in the some of the very basics on CAT(0) spaces will be recalled, a fa-miliarity with the aforementioned standard references is recommended. Wehave chosen to present the results not always in their most general form, butrather in a way that makes their statement simpler and hopefully more en-lightening. More general statements, detailed arguments and further resultsmay be found in the papers [CM09a,CM09b,CM12a]. All the original resultspresented here have been obtained in collaboration with Nicolas author is an research associate at UCLouvain, Bel-gium, supported in part by FNRS grant and by the ERC grant# ILeading examples1.

5 The basicsLet (X, d) be a metric space. Ageodesic mapis an isometric map :I Xof a convex subsetI RtoX, where the real lineRis endowedwith the Euclidean distance. The map is called ageodesic segment( ,line) ifIis a closed interval ( a half-line,I=R). Itshould be noted that the notion of geodesic introduced here is a global one,as opposed to the corresponding notion in differential metric spaceis a metric space (X, d) in which any twopoints are joined by a geodesic The Euclidean space (Rn, dEucl) is a geodesic metric space. More generally, a Riemannian manifold, viewed as a metric spacewith its canonical distance function, is a geodesic metric spacepro-vided it is complete.

6 An incomplete Riemannian manifold need notbe a geodesic metric space. Any metric graph is a geodesic metric (X, d) be a geodesic metric space. Given a triple (x, y, z) X3, aEuclidean comparison trianglefor (x, y, z) is a triple ( x, y, z) of points ofthe Euclidean planeR2such thatd(x, y) =dEucl( x, y),d(y, z) =dEucl( y, z)andd(z, x) =dEucl( z, x). Notice that any triple inXadmits some Euclideancomparison (0) spaceis a geodesic metric space all of whose triple of points(x, y, z) X3satisfy the following condition: given a Euclidean comparisontriangle ( x, y, z) inR2, any pointp Xwhich belongs to some geodesicsegment joiningytozinXsatisfies the inequalityd(x, p) dEucl( x, p),where p R2is the unique point ofR2such thatd(y, p) =dEucl( y, p) andd(p, z) =dEucl( p, z).

7 The following fundamental properties of CAT(0) spaces are straightfor-ward to deduce from the (X, d)be aCAT(0)space. Then:(i)(X, d)isuniquely geodesic, any two points are joined by auniquegeodesic segment.(ii)Xis LEADING EXAMPLES The Euclidean space (Rn, dEucl) is a CAT(0) space. So is any pre-Hilbert space. A complete Riemannian manifoldM, endowed with its canonicaldistance function, is a CAT(0) space if and only ifMhas non-positive sectional curvature. See [BH99, Theorem ]. So is inparticular the real hyperbolic spaceHn. A metric graphXis a CAT(0) space if and only ifXis a short list of examples already illustrates that the category of CAT(0) spaces encompasses both smooth and singular objects.

8 The singular charac-ter expresses itself by the fact that geodesics maybranch, two distinctgeodesic segments may share a common sub-segment of positive are several ways to construct news examples of CAT(0) spacesfrom known subsetYof a CAT(0) space (X, d) is calledconvexif the geodesicsegment joining any two points ofYis entirely contained inY. Clearly, aconvex subset of a CAT(0) space is itself a CAT(0) space when endowedwith the induced key feature of the CAT(0) condition is its stability under Carte-sian products. The proof is left as an (X1, d1)and(X2, d2)beCAT(0) spaces .

9 Thenthe Cartesian productX=X1 X2, endowed with the metricdde ned byd2=d21+d22, is aCAT(0) more exotic constructions, like gluing two CAT(0) spaces alongan isometric convex subset, also preserve the CAT(0) condition. We closethis section with the following noteworthy facts, for which we refer toCor. and in [BH99]. (i) The Cauchy completion of aCAT(0)space is itselfCAT(0).(ii) An ultraproduct ofCAT(0) spaces is itselfCAT(0). In particular, theasymptotic cones of aCAT(0)space areCAT(0).2. The Cartan{ hadamard theoremA fundamental feature of the CAT(0) condition is that it is a local con-dition, as is the condition of being non-positively curved in the realm ofRiemannian geometry.}

10 This matter of fact is made precise by the followingbasic result, for which we refer to [Bal95, Theorem ] and [BH99, Theo-rem ]. (Cartan hadamard ).Let(X, d)be a complete connectedmetric every point ofXadmits some neighbourhood which isCAT(0)whenendowed with the appropriate restriction ofd(we then say that(X, d)islocallyCAT(0)), then there is a unique distance function don the universalcovereXsuch that following two conditions hold: the covering mapeX Xis a local isometry ; (eX, d)is aCAT(0) proper COCOMPACT SPACES3 The metric dcoincides with thelength metric(also calledinner met-ric) induced bydoneX.