BULLETIN New Series OF THE AMERICAN MATHEMATICAL SOCIETY Volume Numb er Xxxx XXXX Pages S XX BALLMANN BRIDSONHAEFLIGER EBERLEIN BRUCE KLEINER The study of nonp ositively curved spaces go es back to the discovery of hyp erb olic space the work of Hadamard around and Car tans work in the s These spaces play a signicant role in many areas Lie group theory combinatorial and geometric group theory dynamical systems harmonic maps and vanishing theorems geometric top ology Kleinian group theory and Teichmuller theory In some of these contexts for instance in dynamics and in harmonic map theory nonp ositive curvature turns out to b e the right condition to make things work smo othly while in others such as Lie theory manifold top ology and Teichmuller theory the basic ob jects of study happ en to b e nonp ositively curved spaces With so many closely related in terdep endent elds nonp ositive curvature has b een a very active topic in the last twenty years To get an idea of the scop e of the activity consider some of the highlights Harmonic maps Cor GS KS MSY manifolds and Kleinian groups MS Gab CJ Ota Ota Min Min Can McM Kap Gab GMT Structure theory and rigidity BS BBE BBS EH BB Lee High dimensional top ology FH FJ CGM Hyp erb olic groups quasiconformal geometryanalysis Gro Pan Sel RS BM Bowa Bowb BP BP HK Geometriccombinatorial group theory Gro DJ CD BM KLa KLb Dynamics Cro Ota BCG BFK One p oint of view which has b een quite inuential in recent years is that it is fruitful to work with synthetic conditions which are equiva lent to nonp ositive sectional curvature in the Riemannian case rather than sectional curvature itself Though this idea and the analog for spaces with lower curvature b ounds go es back to A D Alexandrov it was Gromov Gro who brought it to the attention of a much wider Received by the editors March c copyright holder 1 audience in the s The most p opular alternate condition is a triangle comparison inequality which says that suciently small geo desic tri angles are at least as thin as corresp onding Euclidean triangles The precise version runs as follows We say that a metric space X is a 1 Hadamard space or C AT space if X is complete Every two p oints in X are joined by a geo desic segment a path whose length equals the distance b etween its endp oints 0 0 0 2 Whenever x y z X x y z R are isometric triples d x y X 0 0 0 0 0 0 z x w lies on y z d z x d x y d y z d d X X R R R 0 a geo desic segment joining y to z and w is the corresp onding 0 0 0 0 p oint on the Euclidean segment w y d w y then y z d X R 0 0 w x d w x d X R We say that a metric space Z is an Alexandrov space with nonpositive curvature or simply a nonpositively curved space if for every z Z there is an r so that the closed ball B z r is a Hadamard space with resp ect to the induced metric Work of Cartan Car implies that a Riemannian manifold has nonp ositive sectional curvature i it denes an Alexandrov space with nonp ositive curvature A Riemann ian manifold is a Hadamard space i it is a Hadamard manifold a complete simply connected manifold with nonp ositive sectional curva ture Working with Alexandrov spaces rather than nonp ositively curved manifolds has several advantages Most of the foundational material carries over to the more general setting with only minor mo dications and the pro ofs which are no longer allowed to use ob jects that de p end on smo oth structure the LeviCivita connection Jacobi elds sometimes b ecome simpler In this more general context one still has a tangent cone asso ciated with each p oint of a Hadamard space X and a tangent cone at innity the Tits cone C X named after T Jacques Tits these tangent cones are Hadamard spaces and they play a leading role in the theory The Tits cone is almost never iso metric to a Riemannian manifold even when X itself is a Riemannian manifold Many imp ortant examples of Hadamard spaces esp ecially examples from geometric group theory are nonRiemannian For in stance simplicial trees and more generally R trees and Euclidean and hyp erb olic Tits buildings A connected nite complex built by glu ing together Euclidean triangles along their edges denes a space with 1 The letters C A and T stand for Cartan Alexandrov and Top onogov The term Hadamard space was intro duced in Bal since Hadamard spaces are gener alizations of Hadamard manifolds nonp ositive curvature i each vertex link when endowed with the an gle metric is a graph with no cycles of length this construction already gives an abundance of interesting examples and is imp ortant in the theory of small cancellation groups The b o oks under review have several common themes All three base their development on the geometry of geo desics and distance functions and for the most part they work toward results which apply to a broad class of nonp ositively curved spaces rather than fo cussing on sp ecial classes of spaces The basic ob ject of study is a Hadamard space X with an isometric group action X X one typically gets such an action by taking the deck group action for the universal covering of a nonp ositively curved space Under appropriate conditions one gets a strong relationship b etween algebraic structure in ab elian sub groups pro duct structure centralizers and geometric structure in X at convex subspaces pro duct structure convex subspaces with Eu clidean factors Results of this typ e go back to GW LY Eb e The pro ofs dep end crucially on the fact that the curvature is allowed to b e zero ie at subspaces are allowed in fact many of the most striking results in the sub ject rely on rigid b ehavior asso ciated with at subspaces Another key isssue in the b o oks is the global b ehavior of geo desics this comes into play in three closely related guises the geo desic ow GX the b oundary at innity and the Tits b oundary The geo desic ow GZ of a metric space Z is the set of unit sp eed lo cally geo desic paths in Z top ologized by uniform convergence on compact subsets and equipp ed with the R action R GZ GZ dened by pre comp osition with translation in R When Z is a complete Riemannian manifold the R action R GZ GZ is equivalent to the usual geo desic ow R SZ SZ on the unit tangent bundle SZ The geo desic ow of a complete nonp ositively curved space Z relates directly with the fundamental group each p erio dic orbit of the ow determines a nontrivial conjugacy class in the fundamental group and two p erio dic orbits determine distinct conjugacy classes unless the corresp onding 1 1 M b ound a lo cally isometric map S L M of a maps S at cylinder into M If X is a Hadamard space then one can use the asymptotic b ehavior of geo desics to dene a boundary at innity X 1 as follows One says that two unit sp eed geo desic rays X 1 and X are asymptotic if the distance d t t 2 X 1 2 is b ounded indep endent of t The relation of b eing asymptotic is an equivalence relation on the set of unit sp eed rays As a set the b ound ary at innity X is the collection of asymptote classes of unit sp eed 1 geo desic rays to dene a top ology on this set one observes that X 1 can b e identied with the set of unit sp eed rays leaving any given base p oint p X and the latter has a natural top ology the top ology of uniform convergence on compact sets nally one veries that the top ology this induces on X is indep endent of the choice of basep oint 1 p When X is an ndimensional Hadamard manifold X is homeo 1 n1 morphic to S The isometry group of X has an induced action on X by homeomorphisms When X X is the deck group action 1 for the universal covering of a compact nonp ositively curved space Z the induced action X X is one of the key to ols for ana 1 1 lyzing the geo desic ow of Z and the structure of It is also used in the pro of of rigidity theorems like Mos BCG Pan BP To dene the Tits b oundary of a Hadamard space one starts with the set X and denes the distance b etween the asymptote classes of two 1 unit sp eed rays and to b e the asymptotic angle of divergence 1 2 1 ie arcsin where lim d t t this limit always ex t!1 1 2 2 t ists for unit sp eed rays in Hadamard spaces and dep ends only on the asymptote classes of the rays This distance function which is called the Tits angle metric usually denes a dierent top ology on the set X than the one mentioned ab ove the top ology usually do es not 1 have a countable basis the metric space it denes is denoted X The T Tits b oundary X registers asymptotically Euclidean structure in X T k k 1 Euclidean subspaces F X pro duce round spheres S X and T under mild assumptions a partial converse holds A common ob jective of the b o oks by Ballmann and Eb erlein is the rank rigidity theorem This is a structure theorem for complete nite volume nonp ositively curved Riemannian manifolds M and is the cen 2 terpiece of the theory It says that if M X X X is the 0 1 k de Rham decomp osition of the universal cover of M X is the Eu 0 clidean factor then each X is either an irreducible symmetric space i of noncompact typ e of rank at least two or else it contains a geo desic which do es not b ound a at half plane a subset isometric to 2 fx y R j y g in X The theorem was rst proved in this pre i cise form in EH slightly extending the results in the earlier pap ers Bal BBE BBS BS The theorem when combined with ear lier work Bal has many implications for manifolds M as ab ove with no other known pro ofs p erio dic orbits of the geo desic ow are dense when the fundamental group is nitely generated