Innovations in Incidence Geometry

From symmetric spaces to buildings, curve complexes and outer spaces

Lizhen Ji

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In this article, we explain how spherical Tits buildings arise naturally and play a basic role in studying many questions about symmetric spaces and arithmetic groups, why Bruhat-Tits Euclidean buildings are needed for studying S-arithmetic groups, and how analogous simplicial complexes arise in other contexts and serve purposes similar to those of buildings.

We emphasize the close relationships between the following: (1) the spherical Tits building Δ(G) of a semisimple linear algebraic group G defined over , (2) a parametrization by the simplices of Δ(G) of the boundary components of the Borel-Serre partial compactification X¯BS of the symmetric space X associated with G, which gives the Borel-Serre compactification of the quotient of X by every arithmetic subgroup Γ of G(), (3) and a realization of X¯BS by a truncated submanifold XT of X. We then explain similar results for the curve complex C(S) of a surface S, Teichmüller spaces Tg, truncated submanifolds Tg(ε), and mapping class groups Modg of surfaces. Finally, we recall the outer automorphism groups Out(Fn) of free groups Fn and the outer spaces Xn, construct truncated outer spaces Xn(ε), and introduce an infinite simplicial complex, called the core graph complex and denoted by CG(Fn), and we then parametrize boundary components of the truncated outer space Xn(ε) by the simplices of the core graph complex CG(Fn). This latter result suggests that the core graph complex is a proper analogue of the spherical Tits building.

The ubiquity of such relationships between simplicial complexes and structures at infinity of natural spaces sheds a different kind of light on the importance of Tits buildings.

Article information

Innov. Incidence Geom., Volume 10, Number 1 (2009), 33-80.

Received: 6 September 2007
Accepted: 25 March 2008
First available in Project Euclid: 28 February 2019

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Zentralblatt MATH identifier

Primary: 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 20E06: Free products, free products with amalgamation, Higman-Neumann- Neumann extensions, and generalizations 20E42: Groups with a $BN$-pair; buildings [See also 51E24] 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]

Tits building symmetric space curve complex outer space arithmetic group mapping class group outer automorphism group of free group


Ji, Lizhen. From symmetric spaces to buildings, curve complexes and outer spaces. Innov. Incidence Geom. 10 (2009), no. 1, 33--80. doi:10.2140/iig.2009.10.33.

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  • M. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. 121 (1985), 429–461.
  • W. Ballmann, Manifolds of nonpositive sectional curvature and manifolds without conjugate points, in Proc. of International Congress of Mathematicians, (Berkeley, Calif., 1986), pp. 484–490, Amer. Math. Soc., 1987.
  • ––––, On the Dirichlet problem at infinity for manifolds of nonpositive curvature, Forum Math. 1 (1989), 201–213.
  • W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, Progr. Math. vol.\@ 61, Birkhäuser Boston, 1985.
  • W. Ballmann and F. Ledrappier, The Poisson boundary for rank one manifolds and their cocompact lattices, Forum Math. 6 (1994), 301–313.
  • J. Behrstock, B. Kleiner, Y. Minsky and L. Mosher, Geometry and rigidity of mapping class groups, preprint. arXiv:0801.2006.
  • M. Bestvina, The topology of ${\rm Out}(F\sb n)$, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 373–384, Higher Ed. Press, 2002.
  • M. Bestvina and M. Feighn, The topology at infinity of ${\rm Out}(F\sb n)$, Invent. Math. 140 (2000), 651–692.
  • A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. 75 (1962), 485–535.
  • A. Borel and L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Math. Theory Appl., Birkhäuser Boston, 2006.
  • A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491.
  • ––––, Cohomologie d'immeubles et de groupes $S$-arithmétiques, Topology 15 (1976), 211–232.
  • M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren Math. Wiss. 319, Springer-Verlag, Berlin, 1999.
  • K. Brown, Buildings, Springer Monogr. Math.. Springer-Verlag, 1998.
  • ––––, Cohomology of Groups, Corrected reprint of the 1982 original. Grad. Texts in Math. 87, Springer-Verlag, 1994.
  • M. Culler, Finite groups of outer automorphisms of a free group, in Contributions to group theory, pp. 197–207, Contemp. Math., vol.\@ 33, Amer. Math. Soc., 1984.
  • M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), 91–119.
  • Y. Guivarch, L. Ji and T. C. Taylor, Compactifications of Symmetric Spaces, Progr. Math. 156, Birkhäuser Boston, 1998.
  • U. Hamenstädt, Geometry of the mapping class groups III: Quasi-isometric rigidity, preprint. arXiv:math/0512429.
  • J. Harer, The cohomology of the moduli space of curves, in Theory of moduli, pp. 138–221, Lecture Notes in Math. vol. 1337, Springer, 1988.
  • W. Harvey, Boundary structure of the modular group, in Riemann surfaces and related topics, pp. 245–251, Ann. of Math. Stud. 97, Princeton Univ. Press, 1981.
  • A. Hatcher and K. Vogtmann, The complex of free factors of a free group, Quart. J. Math. Oxford Ser. 49 (1998), 459–468.
  • N. Ivanov, Mapping Class Groups, in Handbook of geometric topology, pp. 523–633, North-Holland, 2002.
  • N. Ivanov and L. Ji, Infinite topology of curve complex and non-Poincaré duality of Teichmüller modular groups, Enseign. Math. 54 (2008), 381–395.
  • L. Ji, Buildings and their applications in geometry and topology, Asian J. Math. 10 (2006), 11–80.
  • ––––, Integral Novikov conjectures and arithmetic groups containing torsion elements, Commun. Anal. Geom. 15 (2007), 509-533.
  • ––––, Arithmetic Groups and Their Generalizations: What, Why and How, AMS/IP Stud. Adv. Math. 43, Amer. Math. Soc., Providence, RI; Intern. Press, Cambridge, MA, 2008.
  • ––––, A summary of some work of Gregory Margulis, Pure Appl. Math. Q. 4 (2008), 1–69.
  • ––––, The integral Novikov conjectures for linear groups containing torsion elements, J. Topol. 1 (2008), 306–316.
  • ––––, Steinberg representations, and duality properties of arithmetic groups, mapping class groups and outer automorphism groups of free groups, in Representation Theory, pp. 71–98, Contemp. Math., vol.\@ 478, Amer. Math. Soc., Providence, RI, 2009.
  • L. Ji and S. Wolpert, A cofinite universal space for proper actions for mapping class groups, in In the Tradition of Ahlfors-Bers, V, eds.\@ M. Bonk, J. Gilman, H. Masur, Y. Minsky and M. Wolf, Contemp. Math., vol.\@ 510, pp. 151–163, Amer. Math. Soc., 2010.
  • B. Kleiner and B. Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Publ. Math. Inst. Hautes Études Sci. 86 (1997), 115–197.
  • S. Krstic and K. Vogtmann, Equivariant outer space and automorphisms of free-by-finite groups, Comment. Math. Helv. 68 (1993), 216–262.
  • B. Leeb, A Characterization of Irreducible Symmetric Spaces and Euclidean Buildings of Higher Rank by Their Asymptotic Geometry, Bonner Math. Schriften 326, Universität Bonn, Mathematisches Institut, Bonn, 2000.
  • E. Leuzinger, An exhaustion of locally symmetric spaces by compact submanifolds with corners, Invent. Math. 121 (1995), 389–410.
  • G. Mostow, Strong rigidity of locally symmetric spaces, Annals of Math. Stud., No. 78, Princeton University Press, 1973.
  • G. Mostow and T. Tamagawa, On the compactness of arithmetically defined homogeneous spaces, Ann. of Math. 76 (1962), 446–463.
  • D. Mumford, A remark on Mahler's compactness theorem, Proc. Amer. Math. Soc. 28 (1971), 289–294.
  • G. Prasad, Lattices in semisimple groups over local fields, in Studies in algebra and number theory, pp. 285–356, Adv. in Math. Suppl. Stud., vol. 6, Academic Press, 1979.
  • L. Saper, Tilings and finite energy retractions of locally symmetric spaces, Comment. Math. Helv. 72 (1997), 167–202.
  • R. Spatzier, An invitation to rigidity theory, in Modern dynamical systems and applications, pp. 211–231, Cambridge Univ. Press, 2004.
  • J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Math., Vol. 386, Springer-Verlag, 1974.
  • K. Vogtmann, Automorphisms of free groups and outer space, Geom. Dedicata 94 (2002), 1–31.
  • ––––, The cohomology of automorphism groups of free groups, in Proceedings of the International Congress of Mathematicians, Vol. II, pp. 1101–1117, Eur. Math. Soc., 2006.
  • T. White, Fixed points of finite groups of free group automorphisms, Proc. Amer. Math. Soc. 118 (1993), 681–688.
  • S. Wolpert, Geometry of the Weil-Petersson completion of Teichmüller space, in Surveys in differential geometry, Vol. VIII, pp. 357–393, Surv. Differ. Geom., VIII, Int. Press, 2003.