Tilings and finite energy retractions of locally symmetric spaces

Saper, Leslie

doi:10.1007/pl00000369

Cited by 35 publications

(49 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Other progress has been recently made by Bullock [Bul00], Bullock and Connell [BC06], and Yasaki [Yas05b,Yas05a] in the case of groups of Qrank 1. In particular, Yasaki uses the tilings of Saper [Sap97] to construct an explicit retract for the unitary group SU(2, 1) over the Gaussian integers. His method also works for Hilbert modular groups, although further refinement may be needed to produce a regular cell complex.…”

Section: A6 Complements and Open Problemsmentioning

confidence: 99%

Modular Forms, a Computational Approach

Stein

2007

Graduate Studies in Mathematics

264

289

View full text Add to dashboard Cite

show abstract

Section: A6 Complements and Open Problemsmentioning

confidence: 99%

Modular Forms, a Computational Approach

Stein

2007

Graduate Studies in Mathematics

264

289

View full text Add to dashboard Cite

show abstract

“…For general symmetric spaces X = G/K and arithmetic groups acting on them, coarser equivariant tilings (or decompositions) are known (see [371] for precise statements of the results and references). Such an equivariant decomposition is important for the Arthur-Selberg trace formula but does not give rise to a well-defined fundamental domain.…”

Section: Equivariant Tilings Of Symmetric Spacesmentioning

confidence: 99%

“…This is related to Γ-equivariant tilings of X mentioned before in §4.10. The central tile in [371] is a cocompact deformation retract of X and is equal to X T , which corresponds to X(ε) in the case of the upper half plane H 2 above. This gives another cofinite model of EΓ different from the one in Proposition 4.69.…”

Section: The Universal Spaces Eγ and Eγ Via The Borel-serre Partial Cmentioning

confidence: 99%

See 1 more Smart Citation

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

2012

IRMA Lectures in Mathematics and Theoretical Physics

View full text Add to dashboard Cite

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...

show abstract

“…This picture can essentially be ascertained from [6], [29]; the fact that K\G/Γ has finite Gromov-Hausdorff distance from cP is asserted in [10]. However, one needs a key estimate about the "coarse isotropy."…”

Section: Picture From Reduction Theorymentioning

confidence: 99%

Coarse Obstructions to Positive Scalar Curvature in Noncompact Arithmetic Manifolds

Chang¹

2001

J. Differential Geom.

View full text Add to dashboard Cite

ABSTRACT. Block and Weinberger show that an arithmetic manifold can be endowed with a positive scalar curvature metric if and only if its Q-rank exceeds 2. We show in this article that these metrics are never in the same coarse class as the natural metric inherited from the base Lie group. Furthering the coarse C * -algebraic methods of Roe, we find a nonzero Dirac obstruction in the Ktheory of a particular operator algebra which encodes information about the quasi-isometry type of the manifold as well as its local geometry.

show abstract

Tilings and finite energy retractions of locally symmetric spaces

Cited by 35 publications

References 36 publications

Modular Forms, a Computational Approach

Modular Forms, a Computational Approach

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

Coarse Obstructions to Positive Scalar Curvature in Noncompact Arithmetic Manifolds

Contact Info

Product

Resources

About