## Innovations in Incidence Geometry

### From symmetric spaces to buildings, curve complexes and outer spaces

Lizhen Ji

#### Abstract

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 $Δℚ(G)$ of a semisimple linear algebraic group $G$ defined over $ℚ$, (2) a parametrization by the simplices of $Δℚ(G)$ 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 $G(ℚ)$, (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

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

Dates
Accepted: 25 March 2008
First available in Project Euclid: 28 February 2019

https://projecteuclid.org/euclid.iig/1551323103

Digital Object Identifier
doi:10.2140/iig.2009.10.33

Mathematical Reviews number (MathSciNet)
MR2665194

Zentralblatt MATH identifier
1264.20031

#### Citation

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. https://projecteuclid.org/euclid.iig/1551323103

#### References

• 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.