Innovations in Incidence Geometry
- Innov. Incidence Geom.
- Volume 10, Number 1 (2009), 33-80.
From symmetric spaces to buildings, curve complexes and outer spaces
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 of a semisimple linear algebraic group defined over , (2) a parametrization by the simplices of of the boundary components of the Borel-Serre partial compactification of the symmetric space associated with , which gives the Borel-Serre compactification of the quotient of by every arithmetic subgroup of , (3) and a realization of by a truncated submanifold of . We then explain similar results for the curve complex of a surface , Teichmüller spaces , truncated submanifolds , and mapping class groups of surfaces. Finally, we recall the outer automorphism groups of free groups and the outer spaces , construct truncated outer spaces , and introduce an infinite simplicial complex, called the core graph complex and denoted by , and we then parametrize boundary components of the truncated outer space by the simplices of the core graph complex . 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.
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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]
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