A Summary of the Work of Gregory Margulis

Ji, Lizhen

doi:10.4310/pamq.2008.v4.n1.a1

Cited by 5 publications

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“…Indeed her work in [22] is a major contribution towards it (see Section 1.2). This conjecture is also reported by Lizhen Ji in his summary [12,Section 7] of the work of Margulis. The main feature here is that "Γ is a lattice" is a conclusion, not an assumption.…”

Section: Introductionsupporting

confidence: 73%

Arithmeticity of discrete subgroups containing horospherical lattices

Benoist¹,

Miquel²

2020

Duke Math. J.

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Section: Introductionsupporting

confidence: 73%

Arithmeticity of discrete subgroups containing horospherical lattices

Benoist¹,

Miquel²

2020

Duke Math. J.

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“…For more results about mapping class groups, see also the books [124] [334] [335]. For more references on arithmetic groups, see [198] [39] [411].…”

Section: Summaries Of Properties Of Groups and Spaces In This Chaptermentioning

confidence: 99%

“…8 In [43], for a compact locally symmetric space of negative sectional curvature, its invariant metric is characterized by the property that it has minimal entropy among all negatively curved metrics of the same volume on it. For other generalizations of Mostow strong rigidity, see the references in [198] and [205]. (16) In [89] and [21], rank rigidity says that any finite volume irreducible nonpositively curved Riemannian manifold of rank at least 2 is a locally symmetric space.…”

Section: Properties Of Actions Of Arithmetic Groups γ Onmentioning

confidence: 99%

“…More precisely, the famous arithmeticity theorem of Margulis (see [278], and [198] and the references there) says that if G is a semisimple Lie group of rank at least two and Γ is an irreducible lattice of G, then Γ is an arithmetic subgroup with respect to a suitable Q-structure on G. Among rank 1 semisimple Lie groups, the question of arithmeticity of lattices is open only for SU(n, 1) when n is at least 4. For a survey of some geometric constructions of lattices in SU(3, 1), see [341], and for some constructions of lattices, in SO(n, 1), for example by reflections, see [410].…”

Section: Generalizations Of Arithmetic Groups: Non-arithmetic Latticesmentioning

confidence: 99%

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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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Introduction

Bavafa-Toosi¹

2019

Introduction to Linear Control Systems

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A Summary of the Work of Gregory Margulis

Cited by 5 publications

References 136 publications

Arithmeticity of discrete subgroups containing horospherical lattices

Arithmeticity of discrete subgroups containing horospherical lattices

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

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