arXiv math/01080922001-08-13math.DS

Exceptional Discrete Mapping Class Group Orbits in Moduli Spaces

Joseph P. Previte, Eugene Z. Xia

备注：: 7 pages, 1 figure

英文摘要：

Let $M$ be a four-holed sphere and $Γ$ the mapping class group of $M$ fixing $\partial M$. The group $Γ$ acts on the space ${\mathcal M}_{\mathcal B}(SU(2))$ of SU(2)-gauge equivalence classes of flat SU(2)-connections on $M$ with fixed holonomy on $\partial M$. We give examples of flat SU(2)-connections whose holonomy groups are dense in SU(2), but whose $Γ$-orbits are discrete in ${\mathcal M}_{\mathcal B}(SU(2))$. This phenomenon does not occur for surfaces with genus greater than zero.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Rates of mixing for the Weil-Petersson geodesic flow I: no rapid mixing in non-exceptional moduli spaces

We show that the rate of mixing of the Weil-Petersson flow on non-exceptional (higher dimensional) moduli spaces of Riemann surfaces is at most polynomial.

2015-10-09math.DS

区间平移映射的典型弱混合性与例外谱性质

Typical Weak Mixing and Exceptional Spectral Properties for Interval Translation Mappings

Universidad Nacional de Colombia（哥伦比亚国立大学）；Institut für Mathematik Universität Zürich（苏黎世大学数学研究所）；IMPA（巴西高等科学研究学院）；Aix Marseille Université CNRS Centrale Marseille Institut de Mathématiques de Marseille I2m - Umr 7373（艾克斯-马赛大学、法国国家科学研究中心、中央马赛学院、马赛数学研究所）；National Research University Higher School of Economics（俄罗斯国立高等经济大学）

本文研究了区间平移映射的弱混合性，证明了典型Bruin-Troubetzkoy映射的弱混合性，并扩展到其他类型映射，同时构造了非弱混合的无限类型Bruin-Troubetzkoy映射例子。

2026-03-19math.DS

Rates of mixing for the Weil-Petersson geodesic flow II: exponential mixing in exceptional moduli spaces

We establish exponential mixing for the geodesic flow $φ_t\colon T^1S\to T^1S$ of an incomplete, negatively curved surface $S$ with cusp-like singularities of a prescribed order. As a consequence, we obtain that the Weil-Petersson flows for the moduli spaces ${\mathcal M}_{1,1}$ and ${\mathcal M}_{0,4}$ are exponentially mixing, in sharp contrast to the flows for ${\mathcal M}_{g,n}$ with $3g-3+n>1$, which fail to be rapidly mixing. In the proof, we present a new method of analyzing invariant foliations for hyperbolic flows with singularities, based on changing the Riemannian metric on the pha…

2016-05-29math.DGmath.DS

Integrability of moduli and regularity of Denjoy counterexamples

We study the regularity of exceptional actions of groups by $C^{1,α}$ diffeomorphisms on the circle, i.e. ones which admit exceptional minimal sets, and whose elements have first derivatives that are continuous with concave modulus of continuity $α$. Let $G$ be a finitely generated group admitting a $C^{1,α}$ action $ρ$ with a free orbit on the circle, and such that the logarithms of derivatives of group elements are uniformly bounded at some point of the circle. We prove that if $G$ has spherical growth bounded by $c n^{d-1}$ and if the function $1/α^d$ is integrable near zero, then under som…

2020-05-06math.DSmath.GT

Dynamics of the mapping class group on the moduli of a punctured sphere with rational holonomy

Let $M$ be a four-holed sphere and $Γ$ the mapping class group of $M$ fixing the boundary $\partial M$. The group $Γ$ acts on $M_B(SL(2,C)) = Hom_B^+(pi_1(M),SL(2,C))/SL(2,C)$ which is the space of completely reducible $SL(2,C)$-gauge equivalence classes of flat $SL(2,C)$-connections on $M$ with fixed holonomy $B$ on $\partial M$. Let $B \in (-2,2)^4$ and $M_B$ be the compact component of the real points of $M_B(SL(2,C))$. These points correspond to SU(2)-representations or $SL(2,R)$-representations. The $Γ$-action preserves $M_B$ and we study the topological dynamics of the $Γ$-action on $M_B…

2003-11-21math.DS

Hyperelliptic Components of the Moduli Spaces of Quadratic Differentials with Prescribed Singularities

Moduli spaces of quadratic differentials with prescribed singularities are not necessarily connected. We describe here all cases when they have a special hyperelliptic connected component. We announce the general classification theorem: up to the four exceptional cases in low genera the strata of meromorphic quadratic differentials are either connected, or have exactly two connected components, and one of the two components is hyperelliptic.

2005-03-07math.DSmath.GT

更多相关论文收起更多论文

扭曲全纯1-形式的模空间与膨胀曲面分层上的不变测度

Invariant measures on moduli spaces of twisted holomorphic 1-forms and strata of dilation surfaces

针对黎曼曲面上扭曲全纯1-形式的模空间，通过映射类群系数上同调的新颖计算，构造了分层上的SL(2,R)-不变Lebesgue类测度，即Masur-Veech测度的类似物。

2025-07-14math.DSmath.GT

Counting closed geodesics in Moduli space

We compute the asymptotics, as R tends to infinity, of the number of closed geodesics in Moduli space of length at most R, or equivalently the number of pseudo-Anosov elements of the mapping class group of translation length at most R.

2011-03-19math.DSmath.GT

Surface group representations in ${\rm SL}_2({\mathbb C})$ with finite mapping class orbits

Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental group of the surface. For surfaces of genus at least two, such orbits correspond to homomorphisms with finite image. For genus one, they correspond to the finite or special dihedral representations. We also obtain an analogous result for bounded orbits in the moduli space.

2021-04-10math.DSmath.GT

Moduli spaces of complex affine and dilation surfaces

We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech, we show that the the moduli space of affine surfaces with fixed genus and with cone points of fixed complex order is a holomorphic affine bundle over the moduli space of Riemann surfaces. Similarly, the moduli space of dilation surfaces is a covering space of the moduli space of Riemann surfaces. We classify the connected components of the moduli space of dilation surfaces and show that any component is an orbifold K(G,1) where G is the framed mapping class group of Calderon-Salter.

2022-04-11math.AGmath.DSmath.GT

Mapping Class Group d'un germe de courbe plane singulière

We prove that every topological conjugation between two germs of singular holomorphic curves in the complex plane is homotopic to another conjugation which extends homeomorphically to the exceptional divisors of their minimal desingularizations. As an application we give an explicit presentation of a finite index subgroup of the mapping class group of the germ of such a singularity.

2010-04-15math.AGmath.DSmath.GT