Buildings and their Applications in Geometry and Topology

Ji, Lizhen

doi:10.4310/ajm.2006.v10.n1.a5

Cited by 19 publications

(14 citation statements)

References 309 publications

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“…For applications of Tits buildings in geometry and topology, see [208] and the extensive references there.…”

Section: Compactifications and Boundaries Of Symmetric Spacesmentioning

confidence: 99%

A tale of two groups: arithmetic groups and mapping class groups

2012

IRMA Lectures in Mathematics and Theoretical Physics

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SummaryIn this chapter, we discuss similarities, differences and interaction between two natural and important classes of groups: arithmetic subgroups Γ of Lie groups G and mapping class groups Mod g,n of surfaces of genus g with n punctures. We also mention similar properties and problems for related groups such as outer automorphism groups Out(F n ), Coxeter groups and hyperbolic groups. Since groups are often effectively studied by suitable spaces on which they act, we also discuss related properties of actions of arithmetic groups on symmetric spaces and actions of mapping class groups on Teichmüller spaces, hoping to get across the point that it is the existence of actions on good spaces that makes the groups interesting and special, and it is also the presence of large group actions that also makes the spaces interesting. Interaction between locally symmetric spaces and moduli spaces of Riemann surfaces through the example of the Jacobian map will also be discussed in the last part of this chapter. Since reduction theory, i.e., finding good fundamental domains for proper actions of discrete groups, is crucial to transformation group theory, i.e., to understand the algebraic structures of groups, properties of group actions and geometry, topology and compactifications of the quotient spaces, we discuss many different approaches to reduction theory of arithmetic groups acting on symmetric spaces. These results for arithmetic groups motivate some results on fundamental domains for the action of mapping class groups on Teichmüller spaces. For example, the Minkowski reduction theory of quadratic forms is generalized to the action of Mod g = Mod g,0 on the Teichmüller space T g to construct an intrinsic fundamental domain consisting of finitely many cells, * Partially Supported by NSF grant DMS 0905283 2 solving a weaker version of a folklore conjecture in the theory of Teichmüller spaces.On several aspects, more results are known for arithmetic groups, and we hope that discussion of results for arithmetic groups will suggest corresponding results for mapping class groups and hence increase interactions between the two classes of groups. In fact, in writing this survey and following the philosophy of this chapter, we noticed the natural procedure in §5.12 of constructing the Deligne-Mumford compactification of the moduli space of Riemann surfaces from the Bers compactification of the Teichmüller space by applying the general procedure of Satake compactifications of locally symmetric spaces, i.e., how to pass from a compactification of a symmetric space to a compactification of a locally symmetric space by making use of the reduction theory for arithmetic groups.The layout of this chapter is as follows. In §1, the introduction, we discuss some general questions about discrete groups, group actions and transformation group theories. In §2, we summarize results on arithmetic subgroups Γ of semisimple Lie groups G and mapping class groups Mod g,n of surfaces of genus g with n punctures, and their actions on symmetri...

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“…For applications of Tits buildings in geometry and topology, see [208] and the extensive references there.…”

Section: Compactifications and Boundaries Of Symmetric Spacesmentioning

confidence: 99%

A tale of two groups: arithmetic groups and mapping class groups

2012

IRMA Lectures in Mathematics and Theoretical Physics

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“…We will recall briefly some of these buildings below. See the book [2] for detailed definitions and structures of buildings, and the survey [87] for references on many different applications of buildings.…”

Section: The Origin Of Tits Buildingsmentioning

confidence: 99%

“…This chapter can be seen as a sequel of the survey papers [87] and [88] in some sense. Since it emphasizes curve complexes and their applications, it can complement the other two papers.…”

Section: The Origin Of Tits Buildingsmentioning

confidence: 99%

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Curve Complexes Versus Tits Buildings: Structures and Applications

2011

Buildings, Finite Geometries and Groups

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Abstract. Tits buildings ∆ Q (G) of linear algebraic groups G defined over the field of rational numbers Q have played an important role in understanding partial compactifications of symmetric spaces and compactifications of locally symmetric spaces, cohomological properties of arithmetic subgroups and S-arithmetic subgroups of G(Q). Curve complexes C(Sg,n) of surfaces Sg,n were introduced to parametrize boundary components of partial compactifications of Teichmüller spaces and were later applied to understand properties of mapping class groups of surfaces and the geometry and topology of 3-dimensional manifolds. Tits buildings are spherical building. Another important class of buildings consists of Euclidean buildings, for example, the BruhatTits buildings of linear algebraic groups defined over local fields. In this chapter, we summarize and compare some properties and applications of buildings and curve complexes. We try to emphasize their similarities but also point out differences. In some sense, curve complexes are combinations of spherical, Euclidean and hyperbolic buildings. We hope that such a comparison might motivate more questions and at the same time suggest methods to solve them. Furthermore it might introduce buildings to people who study curve complexes and curve complexes to people who study buildings.

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“…For each non-archimedean place p, let X p be the Bruhat-Tits building associated to the reductive group G(k p ) over a local field. It is known that X p has a Tits metric whose restriction to each apartment is isometric to R r , where r is the k p -rank of G, and G(k p ) acts isometrically and properly on X p (see [BS2] Ji5] for definitions of the Bruhat-Tits buildings, the Tits metric, and other properties). In the fololwing, X p is considered as a metric space with respect to the Tits metric.…”

Section: Novikov Conjectures For S-arithmetic Subgroupsmentioning

confidence: 99%

Large scale geometry, compactifications and the integral Novikov conjectures for arithmetic groups

Lau¹,

Xin²,

Yau³

2008

AMS/IP Studies in Advanced Mathematics

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The original Novikov conjecture concerns the (oriented) homotopy invariance of higher signatures of manifolds and is equivalent to the rational injectivity of the assembly map in surgery theory. The integral injectivity of the assembly map is important for other purposes and is called the integral Novikov conjecture. There are also assembly maps in other theories and hence related Novikov and integral Novikov conjectures. In this paper, we discuss several results on the integral Novikov conjectures for all torsion free arithmetic subgroups of linear algebraic groups and all S-arithmetic subgroups of reductive linear algebraic groups over number fields. For reductive linear algebraic groups over function fields of rank 0, the integral Novikov conjecture also holds for all torsion-free S-arithmetic subgroups. Since groups containing torsion elements occur naturally and frequently, we also discuss a generalized integral Novikov conjecture for groups containing torsion elements, and prove it for all arithmetic subgroups of reductive linear algebraic groups over number fields and S-arithmetic subgroups of reductive algebraic groups of rank 0 over function fields.

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Buildings and their Applications in Geometry and Topology

Cited by 19 publications

References 309 publications

A tale of two groups: arithmetic groups and mapping class groups

A tale of two groups: arithmetic groups and mapping class groups

Curve Complexes Versus Tits Buildings: Structures and Applications

Large scale geometry, compactifications and the integral Novikov conjectures for arithmetic groups

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