arXiv research

We will construct twisted versions of the wild character varieties.

We will give several descriptions of some basic examples of wild character varieties, including a discussion of links to work of Sibuya, Calabi and Euler, amongst others.

A family of new algebraic Poisson varieties will be constructed, generalising the complex character varieties of Riemann surfaces. Then the well-known (Poisson) mapping class group actions on the character varieties will be generalised.

Algorithm describes Fourier transform of Stokes data at infinity.

Study of wild mapping class groups on complex reflection groups.

Short survey based on talk given at the Institut Henri Poincare January 17th 2012, during program on surface groups. The aim was to describe some background results before describing in detail (in subsequent talks) the results of [Boa11c] related to wild character varieties and irregular mapping class groups.

In previous work a relation between a large class of Kac-Moody algebras and meromorphic connections on global curves was established---notably the Weyl group gives isomorphisms between different moduli spaces of connections, and the root system is also seen to play a role. This involved a modular interpretation of many…

Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking…

The theory of Hitchin systems is something like a "global theory of Lie groups", where one works over a Riemann surface rather than just at a point. We'll describe how one can take this analogy a few steps further by attempting to make precise the class of rich geometric objects that appear in this story (including the…

Character varieties get a natural Poisson structure.

Quantum character varieties unify four construction methods.

Character variety of Borromean link solved, Alexander polynomial formula found.

Computes dimensions of representation and character varieties for 2 and 3-dimensional orbifolds.

Researchers describe character varieties for Hopf links, proving geometric properties.

Character variety of Whitehead link described in detail.

Proves Torelli group action is ergodic on Lie group character varieties.

We study some basic properties of the variety of characters in PSL(2,C) of a finitely generated group. In particular we give an interpretation of its points as characters of representations. We construct 3-manifolds whose variety of characters has arbitrarily many components that do not lift to SL(2,C). We also study t…

Study character varieties of hyperbolic 3-manifolds using bundle methods.

Study of pseudo-Anosov actions on $SU(2)$-character variety for genus 2 surfaces.

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…

Study of SU(2,1) character varieties on one-holed torus.

Character varieties of knot groups into SU(3) are stratified and analyzed.

New compactification for character varieties with good topological properties.

Study character varieties of arborescent knots and hyperbolic knots.

Study character variety of a special 3-manifold, finding a polynomial formula.

We describe the (P)SL(2,C) character varieties of all 2-bridge knots and the diagonal character varieties for all 2-bridge links in terms of a set of polynomials defined using Farey recursion.

Let $Γ$ be a finitely generated group and $G$ a real form of $\mathrm{SL}_n(\mathbb{C})$. We propose a definition for the $G$-character variety of $Γ$ as a subset of the $\mathrm{SL}_n(\mathbb{C})$-character variety of $Γ$. We consider two anti-holomorphic involutions of the $\mathrm{SL}_n(\mathbb{C})$ character variet…

Minimal action of mapping class group on character variety.

Anosov representations of surface groups are complex manifolds.

In this paper we consider some families of links, including (-2,2m+1,2n)-pretzel links and twisted Whitehead links. We calculate the character varieties of these families, and determine the number of irreducible components of these character varieties.

We describe a birational map between subvarieties in the character varieties of mutative 3-manifolds. By studying the birational map, one can decide in certain circumstances whether a mutation surface is detected by an ideal point of the character variety.

The paper studies compactifications of SL(2,C) character varieties for punctured surfaces.

Study shows how certain knots and tori are detected by ideal points in character varieties.

Study of twisted $L^2$-torsion on 3-manifold character varieties.

Authors determine the SL(2,C) character variety of a specific knot without computer aid.

Compactifies character varieties for group actions.

New method to compute tangle sums in character varieties.

We show that for any knot there exist only finitely many irreducible metabelian characters in the $SL(2,\mathbb{C})$-character variety of the knot group, and the number is given explicitly by using the determinant of the knot. Then it turns out that for any 2-bridge knot a section of the $SL(2,\mathbb{C})$-character va…

We extend Culler and Shalen's construction of detecting essential surfaces in 3-manifolds to 3-orbifolds. We do so in the setting of the $\mathrm{SL}_2(\mathbb{C})$ character variety, and following Boyer and Zhang in the $\mathrm{PSL}_2(\mathbb{C})$ character variety as well. We show that any slope detected on a canoni…

Researchers describe the dual of cohomology generators for SU(2) character varieties of surfaces.

We study $\mathrm{SL}_2(\mathbb{F})$-character varieties of knots over algebraically closed fields $\mathbb{F}$. We give a sufficient condition in terms of the double branched cover of a $2$-bridge knot (or, equivalently, of its Alexander polynomial) on the characteristic of $\mathbb{F}$, an odd prime, for the $\mathrm…

A marked surface is a compact oriented surface equipped with some pairwise disjoint arcs embedded in its boundary. In this paper, we extend the notion of character varieties to marked surfaces, in such a way that they have a nice behaviour for the operation of gluing two boundary arcs together. These stated character v…

The paper calculates the motive of a specific knot's character variety.

Classifies invariant measures on specific character varieties.

In this paper we define the adjoint Reidemeister torsion as a differential form on the character variety of a compact oriented 3-manifold with toral boundary, and prove it defines a regular volume form. Then we show that the torsion form can vanish only at singular points of the character variety. In fact, if the singu…

These lecture notes concern the algebraic geometry of the character variety of a finitely generated group in SL(2,C) from the point of view of skein modules. We focus on the case of surface and 3-manifolds groups and construct the Reidemeister torsion as a rational volume form on the character variety.

Method determines SL(2,C) character variety for Montesinos knots.

It has been an open question whether all boundary slopes of hyperbolic knots are strongly detected by the character variety. The main result of this paper produces an infinite family of hyperbolic knots each of which has at least one strict boundary slope that is not strongly detected by the character variety.