A survey of length series identities for surfaces, % 3-manifolds and representation varieties

Moduli spaces of hyperbolic 3-manifolds and dynamics on character varieties

The space AH(M) of marked hyperbolic 3-manifold homotopy equivalent to a compact 3-manifold with boundary M sits inside the PSL_2(C)-character variety X(M) of π_1(M). We study the dynamics of the action of Out(π_1(M)) on both AH(M) and X(M). The nature of the dynamics reflects the topology of M. The quotient AI(M)=AH(M)/Out(π_1(M)) may naturally be thought of as the moduli space of unmarked hyperbolic 3-manifolds homotopy equivalent to M and its topology reflects the dynamics of the action.

Ideal points of character varieties, algebraic non-integral representations, and undetected closed essential surfaces in 3-manifolds

Closed essential surfaces in a three-manifold can be detected by ideal points of the character variety or by algebraic non-integral representations. We give examples of closed essential surfaces not detected in either of these ways. For ideal points, we use Chesebro's module-theoretic interpretation of Culler-Shalen theory. As a corollary, we construct an infinite family of closed hyperbolic Haken 3-manifolds with no algebraic non-integral representations into PSL(2, C), resolving a question of Shanuel and Zhang.

On representation varieties of 3-manifold groups

We prove universality theorems ("Murphy's Laws") for representation schemes of fundamental groups of closed 3-dimensional manifolds. We show that germs of SL(2,C)-representation schemes of such groups are essentially the same as germs of schemes of over rational numbers.

正特征代数簇上的形式流形结构

Formal Manifold Structures on Positive Characteristic Varieties

本文实施Sullivan提出的l进形式化纲领，证明正特征代数簇上的l进形式流形结构及其阿贝尔化伽罗瓦对称性，并给出单连通代数簇l进同伦等价于有限CW复形的充要条件。

Hasse-Weil zeta functions of ${\rm SL}_2$-character varieties of closed orientable hyperbolic $3$-manifolds

It is proved that the Hasse-Weil zeta functions of the canonical components of the ${\rm SL}_2$ (${\rm PSL}_2$)-character varieties of closed orientable complete hyperbolic $3$-manifolds of finite volume are equal to the Dedekind zeta functions of their trace fields (invariant trace fields). When the closed $3$-manifold is arithmetic, the special value at $s=2$ of the Hasse-Weil zeta function of the canonical component of the ${\rm PSL}_2$-character variety is expressed in terms of the hyperbolic volume of the manifold up to rational numbers.

Reidemeister torsion, hyperbolic three-manifolds, and character varieties

This is a survey on Reidemeister torsion for hyperbolic three-manifolds of finite volume. Torsions are viewed as topological invariants and also as functions on the variety of representations in $\operatorname{ SL}_2(\mathbb C)$. In both cases, the torsions may also be computed after composing with finite dimensional representations of $\operatorname{ SL}_2(\mathbb C)$. In addition the paper deals with the torsion of the adjoint representation as a function on the variety of $\operatorname{ PSL}_{n+1}(\mathbb C)$-characters, using that the first cohomology group with coefficients twisted by th…

Characteristic varieties of graph manifolds and quasi-projectivity of fundamental groups of algebraic links

The present paper studies the structure of characteristic varieties of fundamental groups of graph manifolds. As a consequence, a simple proof of Papadima's question is provided on the characterization of algebraic links that have quasi-projective fundamental groups. The type of quasi-projective obstructions used here are in the spirit of Papadima's original work.

闭3-流形的绞纱模在特征簇上定义线丛

Skein modules of closed 3 manifolds define line bundles over character varieties

本文利用Frobenius态射将闭3-流形在奇数阶单位根处的绞纱模解释为SL2特征概形上凝聚层的全局截面空间，并证明当特征概形约化时该凝聚层为线丛。

The Meyer functions for projective varieties and their application to local signatures for fibered 4-manifolds

We study a secondary invariant, called the Meyer function, on the fundamental group of the complement of the dual variety of a smooth projective variety. This invariant have played an important role when studying the local signatures of fibered 4-manifolds from topological point of view. As an application of our study, we define a local signature for generic non-hyperelliptic fibrations of genus 4 and 5 and compute some examples.

Closed surfaces and character varieties

The powerful character variety techniques of Culler and Shalen can be used to find essential surfaces in knot manifolds. We show that module structures on the coordinate ring of the character variety can be used to identify detected boundary slopes as well as when closed surfaces are detected. This approach also yields new number theoretic invariants for the character varieties of knot manifolds.

Seifert流形的Skein模与特征簇

Skein modules and character varieties of Seifert manifolds

本文证明闭Seifert纤维化3-流形的Kauffman括号skein模有限生成当且仅当流形不可约且非Haken，并计算其维数等于特征簇的点数。

Representation varieties detect essential surfaces

Extending Culler-Shalen theory, Hara and the second author presented a way to construct certain kinds of branched surfaces in a $3$-manifold from an ideal point of a curve in the $\operatorname{SL}_n$-character variety. There exists an essential surface in some $3$-manifold known to be not detected in the classical $\operatorname{SL}_2$-theory. We prove that every connected essential surface in a $3$-manifold is given by an ideal point of a rational curve in the $\operatorname{SL}_n$-character variety for some $n$.

Characteristic Subsurfaces, Character Varieties and Dehn Filling

We give new bounds for the distance between two exceptional filling slopes for a 1-cusped hyperbolic 3-manifold in several different situations. The distance between a reducible slope and a slope that produces a manifold with finite fundamental group is at most 2. The distance between a reducible slope and one that produces a very small manifold is also at most 2. The distance between a reducible slope and one which produces a manifold with a pi_1 injective torus is at most 4. The methods used involve both characteristic submanifold theory and the theory of the PSL(2,C) character variety.

$Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ 在代数簇拓扑流形结构上的 Galois 对称性

Galois Symmetry of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on Topological Manifold Structures of Varieties

本文提出profinite正规结构集的定义，证明了$Gal(\overline{\mathbb{Q}}/\mathbb{Q})$在单连通复代数簇拓扑流形结构上的Galois作用可通过阿贝尔化分解并典范延拓，回答了Sullivan的问题。

$PSL_2(\mathbb{C})$-character varieties and Seifert fibered cosmetic surgeries

We study small Seifert possibly chiral cosmetic surgeries on not necessarily null-homologous knot in rational homology spheres. Using $PSL_2(\mathbb{C})$-character variety theory we give a sharp bound on the number of slopes producing the same small Seifert manifold if the ambient manifold satisfies some representation theoretic conditions.

Character varieties of higher dimensional representations and splittings of 3-manifolds

In 1983 Culler and Shalen established a way to construct essential surfaces in a 3-manifold from ideal points of the $SL_2$-character variety associated to the 3-manifold group. We present in this article an analogous construction of certain kinds of branched surfaces (which we call essential tribranched surfaces) from ideal points of the $SL_n$-character variety for a natural number $n$ greater than or equal to 3. Further we verify that such a branched surface induces a nontrivial presentation of the 3-manifold group in terms of the fundamental group of a certain 2-dimensional complex of grou…