Rank 2 Affine Manifolds in Genus 3

亏格二中Teichmüller曲线在亏格三中的回响

Echoes in genus three of Teichmüller curves in genus two

本文分类了亏格三模空间中由亏格二本原Teichmüller曲线经覆盖构造得到的曲线，描述了相关仿射群在模二同调上的作用，并分类了$\mathcal H(2, 2)$中作为亏格二覆盖的铺砖曲面的$\mathrm{SL}_2(\mathbb Z)$-轨道。

Rank gradient, cost of groups and the rank versus Heegaard genus problem

We study the growth of the rank of subgroups of finite index in residually finite groups, by relating it to the notion of cost. As a by-product, we show that the `Rank vs. Heegaard genus' conjecture on hyperbolic 3-manifolds is incompatible with the `Fixed Price problem' in topological dynamics.

Hyperbolic volume, Heegaard genus and ranks of groups

Some conjectures about Heegaard genera and ranks of fundamental groups of 3-manifolds are formulated, and it is shown that they imply new statements about hyperbolic volume.

A topological bound on the Cantor-Bendixson rank of meromorphic differentials

In translation surfaces of finite area (corresponding to holomorphic differentials), directions of saddle connections are dense in the unit circle. On the contrary, saddle connections are fewer in translation surfaces with poles (corresponding to meromorphic differentials). The Cantor-Bendixson rank of their set of directions is a measure of descriptive set-theoretic complexity. Drawing on a previous work of David Aulicino, we prove a sharp upper bound that depends only on the genus of the underlying topological surface. The proof uses a new geometric lemma stating that in a sequence of three …

Knot 4--genus and the rank of classes in W(Q(t))

To a Seifert matrix of a knot K one can associate a matrix w(K) with entries in the rational function field, Q(t). The Murasugi, Milnor, and Levine-Tristram knot signatures, all of which provide bounds on the 4-genus of a knot, are determined by w(K). More generally, the minimal rank of a representative of the class represented by w(K) in the Witt group of hermitian forms over Q(t) provides a lower bound for the 4-genus of K. Here we describe an easily computed new bound on the minimal rank of the class represented by w(K). Furthermore, this lower bound is complete modulo torsion in the Witt g…

Turaev-Viro TQFT and the Rank versus Genus Conjecture

This paper presents a way to estimate the Heegaard genus of a $3$-manifold using the Turaev-Viro state sum TQFT. The Turaev-Viro state sum TQFT is derived from the modular category associated to the quantum group $U_q(\mathfrak{sl}_2)$, which is unitary for some $q$ by Wenzl. Hence by Turaev and Virelizier the corresponding TQFT is unitary. We modify a proof by Garoufalidis to give a lower bound of the Heegaard genus using a unitary TQFT, and then use the software Regina to provide some known counterexamples to the rank versus genus conjecture.

Constructions of surface bundles with rank two fundamental groups

We give a construction of hyperbolic 3-manifolds with rank two fundamental groups and report an experimental search to find such manifolds. Our manifolds are all surface bundles over the circle with genus two surface fiber. For the manifolds so obtained, we then examine whether they are of Heegaard genus two or not. As a byproduct, we give an infinite family of fibered knots of genus two in the 3-sphere whose knot groups are of rank two.

Volumes of the sets of translation surfaces with small saddle connections in rank one affine submanifolds

We prove some estimates of the volumes of the sets of translation surfaces of unit area having several independent small saddle connections in a rank one affine submanifold.

Quantum invariants can provide sharp Heegaard genus bounds

Using Seifert fibered three-manifold examples of Boileau and Zieschang, we demonstrate that the Reshetikhin-Turaev quantum invariants may be used to provide a sharp lower bound on the Heegaard genus which is strictly larger than the rank of the fundamental group.

实射影三维空间中纽结的几何

Geometry of knots in real projective $3$-space

本文研究实射影三维空间中纽结的几何分类问题，提出空间弯曲手术和亏格概念，并给出仿射性与类别的判据，揭示其纽结理论不同于经典三维空间。

The topology of the zero locus of a genus 2 theta function

Mess showed that the genus 2 Torelli group $T_2$ is isomorphic to a free group of countably infinite rank by showing that genus 2 Torelli space is homotopy equivalent to an infinite wedge of circles. As an application of his computation, we compute the homotopy type of the zero locus of any classical genus 2 theta function in $\mathfrak{h}_2 \times \mathbb{C}^2$, where $\mathfrak{h}_2$ denotes rank 2 Siegel space. Specifically, we show that the zero locus of any such function is homotopy equivalent to an infinite wedge of 2-spheres.

关于有限同调秩和的光滑仿射曲面的正确下降

On Proper Descent of Smooth Affine Surfaces with Finite Homotopy Rank-Sum

本文研究了光滑复仿射曲面在有限可缩映射下的同调性质下降问题，提出有限同调秩和性质并证明其有效性，同时分类了被复代数2-扭矩支配的曲面。