Topology 44 (2005) 959–1058 www.elsevier.com/locate/top Tree-graded spaces and asymptotic cones ofgroups ଁ Cornelia Dru¸tua, Mark Sapirb,∗ with an Appendix by Denis Osinb and Mark Sapir aUFR de Mathématiques, Université de Lille-1 59655 Villeneuve d’Ascq, France bDepartment of Mathematics, Vanderbilt University, Nashville, TN 37240, USA Abstract We introduce a concept oftree-graded metric space and we use it to show quasi-isometry invariance ofcertain classes ofrelatively hyperbolic groups, to obtain a characterization ofrelatively hyperbolic groups in terms oftheir asymptotic cones, to find geometric properties ofCayley graphs ofrelatively hyperbolic groups, and to construct the first example ofa finitely generated group with a continuum ofnon- 1-equivalent asymptotic cones. Note that by a result ofKramer, Shelah, Tent and Thomas, continuum is the maximal possible number ofdifferentasymptotic cones ofa finitely generated group, provided that the Continuum Hypothesis is true. ᭧ 2005 Elsevier Ltd. All rights reserved. Contents 1. Introduction ......................................................................................... 960 1.1. Open problems ................................................................................. 968 1.2. Plan ofthe paper ................................................................................ 969 2. Tree-graded spaces ................................................................................... 970 2.1. Properties oftree-graded spaces ................................................................... 970 2.2. Modifying the set of pieces ....................................................................... 976 ଁ The first author is grateful to the CNRS of France for granting her the délégation CNRS status in the 2003–2004 academic year. The research ofthe second author was supported in part by NSF grant DMS 0072307 and by the US–Israeli BSF grant 1999298. We worked on the present paper during the first author’s visits to Vanderbilt University, and the second author’s visits to University of Lille-1 and to the IHES. The authors are grateful to the administration and faculty of these institutions for their support and hospitality. ∗ Corresponding author. E-mail addresses: [email protected] (C. Dru¸tu), [email protected] (M. Sapir). 0040-9383/$ - see front matter ᭧ 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.top.2005.03.003 960 C. Dru¸tu, M. Sapir / Topology 44 (2005) 959–1058 2.3. Geodesics in tree-graded spaces ................................................................... 978 2.4. Cut-points and tree-graded spaces ................................................................. 978 3. Ultralimits and asymptotic cones ....................................................................... 980 3.1. Preliminaries ................................................................................... 980 3.2. Ultralimits ofasymptotic cones are asymptotic cones ................................................. 983 3.3. Another definition ofasymptotic cones ............................................................. 985 3.4. Simple triangles in ultralimits ofmetric spaces ...................................................... 986 4. A characterization ofasymptotically tree-graded spaces .................................................... 992 5. Quasi-isometric behavior .............................................................................. 1004 5.1. Asymptotically tree-graded spaces ................................................................. 1004 5.2. Asymptotically tree-graded groups ................................................................. 1007 6. Cut-points in asymptotic cones ofgroups ................................................................ 1011 6.1. Groups with central infinite cyclic subgroups ........................................................ 1012 6.2. Groups satisfying a law .......................................................................... 1013 7. Fundamental groups ofasymptotic cones ................................................................ 1017 7.1. Preliminaries on nets ............................................................................ 1017 7.2. Construction ofthe group ........................................................................ 1021 7.3. Tree-graded asymptotic cones ..................................................................... 1024 7.4. Free products appearing as fundamental groups of asymptotic cones .................................... 1030 7.5. Groups with continuously many non-homeomorphic asymptotic cones .................................. 1032 8. Asymptotically tree-graded groups are relatively hyperbolic ................................................. 1034 8.1. Weak relative hyperbolicity ....................................................................... 1035 8.1.1. A characterization ofhyperbolicity ......................................................... 1035 8.1.2. Generalizations ofalready proven results and new results ....................................... 1036 8.1.3. Hyperbolicity of Cayley(G, S ∪ H) ........................................................ 1038 8.2. The BCP property............................................................................... 1045 8.3. The Morse lemma............................................................................... 1047 8.4. Undistorted subgroups and outer automorphisms ofrelatively hyperbolic groups .......................... 1049 Acknowledgements ..................................................................................... 1052 Appendix A. Relatively hyperbolic groups are asymptotically tree-graded. By Denis Osin and Mark Sapir ............ 1052 References ............................................................................................ 1056 Further reading ........................................................................................ 1058 1. Introduction An asymptotic cone ofa metric space is, roughly speaking, what one sees when one looks at the space from infinitely far away. More precisely, any asymptotic cone of a metric space (X, dist) corresponds to an ultrafilter , a sequence ofobservation points e = (en)n∈N from X and a sequence ofscaling constants d = (dn)n∈N diverging to ∞. The cone Con(X; e, d) corresponding to e and d is the -limit ofthe sequence ofspaces with basepoints (X, dist/dn,en) (see Section 3 for precise definitions). In particular, if X is the Cayley graph ofa group G with a word metric then the asymptotic cones of X are called asymptotic cones of G. The concept ofasymptotic cone was essentially used by Gromov in [28] and then formally introduced by van den Dries and Wilkie [54]. C. Dru¸tu, M. Sapir / Topology 44 (2005) 959–1058 961 Asymptotic cones have been used to characterize important classes ofgroups: • A finitely generated group is virtually Abelian ifand only ifits asymptotic cones are isometric to the Euclidean space Rn [28,44]. • A finitely generated group is virtually nilpotent ifand only ifits asymptotic cones are locally compact [28,54,25]. • A finitely generated group is hyperbolic ifand only ifits asymptotic cones are R-trees [30]. In [20] it is shown moreover that asymptotic cones ofnon-elementary hyperbolic groups are all isometric to the complete homogeneous R-tree ofvalence continuum. The asymptotic cones ofelementary groups are isometric to either a line R (ifthe group is infinite) or to a point. In particular, every hyperbolic group has only one asymptotic cone up to isometry. Asymptotic cones ofquasi-isometric spaces are bi-Lipschitz equivalent. Thus the topology ofan asymp- totic cone ofa finitely generated group does not depend on the choice ofthe generating set. This was used in [33,34] to prove rigidity results for fundamental groups of Haken manifolds, in [37] to prove rigidity for cocompact lattices in higher rank semisimple groups, and in [23] to provide an alternative proofofthe rigidity fornon-cocompact lattices in higher rank semisimple groups. For a survey ofresults on quasi-isometry invariants and their relations to asymptotic cones see [25]. The power ofasymptotic cones stems fromthe factthat they capture both geometric and logical properties ofthe group, since a large subgroup ofthe ultrapower G ofthe group G acts transitively by isometries on the asymptotic cone Con(G; e, d). Logical aspects ofasymptotic cones are studied and used in the recent papers by Kramer et al. [38,39]. One ofthe main properties ofasymptotic cones ofa metric space X is that geometry offinite config- urations ofpoints in the asymptotic cone reflects the “coarse” geometry ofsimilar finite configurations in X. This is the spirit ofGromov–Delzant’s approximation statement [19] and ofthe applications of R-trees to Rips–Sela theory ofequations in hyperbolic groups and homomorphisms ofhyperbolic groups [48]. This was also used in Dru¸tu’s proofofhyperbolicity ofgroups with sub-quadratic isoperimetric inequality [24]. By a result ofGromov [30] ifall asymptotic cones ofa finitely presented group are simply connected then the group has polynomial isoperimetric function and linear isodiametric function. Papasoglu proved in [45] that groups having quadratic isoperimetric functions have simply connected asymptotic cones. In general, asymptotic cones ofgroups are not necessarily simply connected [53]. In fact, if a group G is not finitely presented then its asymptotic cones cannot all be simply connected [30,25]. A higher-dimensional version