Reduction theory for mapping class groups and applications to moduli spaces

Leuzinger, Enrico

doi:10.1515/crelle.2010.086

Cited by 6 publications

(18 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By [11], Lemma 5, outer cones are -invariant. This allows one to define outer cones in moduli space : for X ∈ Thick ε M (S) we set C O(X ) := π(C O(X )) whereX is any lift of X .…”

Section: Proposition 2 Yields Thementioning

confidence: 90%

See 1 more Smart Citation

A Gauss–Bonnet formula for moduli spaces of Riemann surfaces

Leuzinger

2015

Geom Dedicata

Self Cite

View full text Add to dashboard Cite

show abstract

“…By [11], Lemma 5, outer cones are -invariant. This allows one to define outer cones in moduli space : for X ∈ Thick ε M (S) we set C O(X ) := π(C O(X )) whereX is any lift of X .…”

Section: Proposition 2 Yields Thementioning

confidence: 90%

“…Given σ ∈ C (S) we denote by S σ the corresponding cut surface, i.e., the result of cutting S along (nonintersecting) circles from the isotopy classes α ∈ σ . In [11] I proved.…”

Section: This Set Is -Invariant and Its Quotient Thick ε M (S) Is Tmentioning

confidence: 93%

A Gauss–Bonnet formula for moduli spaces of Riemann surfaces

Leuzinger

2015

Geom Dedicata

Self Cite

View full text Add to dashboard Cite

show abstract

“…As mentioned in the summary earlier in this subsection, an analogue of the reduction theory in [371], i.e., the Γ-equivariant tiling of X recalled in §4.10, also holds for Mod g,n [257].…”

Section: Fundamental Domains and Rough Fundamental Domainsmentioning

confidence: 92%

“…(See [257] [129].) (20) The eventually distance minimizing (EDM) geodesics of Mod g,n \T g,n in the Teichmüller metric can be classified and the boundary of the DeligneMumford compactification of Mod g,n \T g,n can be described in terms of equivalence classes of these geodesics.…”

Section: Properties Of Actions Of Mapping Class Groups Mod Gn On Teimentioning

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

The Asymptotic Schottky Problem

Leuzinger

2010

Geom. Funct. Anal.

Self Cite

View full text Add to dashboard Cite

Let M g denote the moduli space of compact Riemann surfaces of genus g and let A g be the moduli space of principally polarized abelian varieties of dimension g. Let J : M g → A g be the map which associates to a Riemann surface its Jacobian. The map J is injective, and the image J g := J(M g ) is contained in a proper subvariety of A g when g ≥ 4. The classical and long-studied Schottky problem is to characterize the Jacobian locus J g in A g . In this paper we address a large scale version of this problem posed by Farb and called the coarse Schottky problem: What does J g look like "from far away", or how "dense" is J g in the sense of coarse geometry? The large scale geometry of A g is encoded in its asymptotic cone, Cone ∞ (A g ), which is a Euclidean simplicial cone of real dimension g. Our main result asserts that the Jacobian locus J g is "coarsely dense" in A g , which implies that the subset of Cone ∞ (A g ) determined by J g actually coincides with this cone. The proof shows that the Jacobian locus of hyperelliptic curves is coarsely dense in A g as well. We also study the boundary points of the Jacobian locus J g in A g and in the Baily-Borel and the Borel-Serre compactification. We show that for large genus g the set of boundary points of J g in these compactifications is "small".

show abstract

Reduction theory for mapping class groups and applications to moduli spaces

Cited by 6 publications

References 26 publications

A Gauss–Bonnet formula for moduli spaces of Riemann surfaces

A Gauss–Bonnet formula for moduli spaces of Riemann surfaces

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

The Asymptotic Schottky Problem

Contact Info

Product

Resources

About