Character varieties get a natural Poisson structure.
problem Character varieties of surfaces.
method Natural Poisson structure.
result Character varieties have a Poisson structure.
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48 results for “surface characters”
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…
Researchers describe the dual of cohomology generators for SU(2) character varieties of surfaces.
problem Understanding the cohomology structure of SU(2) character varieties of surfaces.
method Explicit description of Poincaré duals of cohomology generators.
result An explicit description of the Poincaré dual of each generator of the rational cohomology ring.
Anosov representations of surface groups are complex manifolds.
problem Characterizing the geometry of Anosov representations of surface groups.
method Analyzing the character varieties of Anosov representations into $\SL(n , \C)$.
result Character varieties of Anosov representations are complex manifolds of specific dimension.
The paper studies compactifications of SL(2,C) character varieties for punctured surfaces.
problem Compactifying character varieties of punctured surfaces.
method Projective compactifications using ideal triangulations and Komyo's method.
result Boundary divisors are toric varieties and the boundary complex is a sphere.
We describe a family of hyperbolic knots whose character variety contain exactly two distinct components of characters of irreducible representations. The intersection points between the components carry rich topological information. In particular, these points are non-integral and detect the Seifert surface.
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 character variety, and following Boyer and Zhang in the character variety as well. We show that any slope detected on a canoni…
Study of quantum decorated character stacks and their quantizations.
problem Quantization of decorated character stacks and their compatibility with cutting and gluing.
method Using stratified factorization homology, extend Fock and Goncharov's construction to include stacky points.
result Construction of categorical charts and flips on quantum decorated character stacks.
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…
Study of pseudo-Anosov actions on -character variety for genus 2 surfaces.
problem Characterizing pseudo-Anosov actions on -character variety.
method Analysis of mapping class group subgroups and their pseudo-Anosov elements.
result Existence of a subgroup containing infinitely many pseudo-Anosov elements with invariant rational functions.
New proof shows almost all surface group actions are dense.
problem Transitivity of normal subgroups on character varieties.
method Proved almost minimal action of non-trivial normal subgroups.
result Almost all points in character variety have dense orbits.
Causal characters of zero mean curvature surfaces of Riemann type in the Lorentz-Minkowski 3-spacemath.DG
A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and as a corollary, we prove that if …
The paper studies mapping class group actions on character varieties of surfaces.
problem Understanding the dynamics of mapping class group actions on relative -character varieties.
method Definition and proof of simple-stability and primitive-stability of representations.
result Holonomies of hyperbolic cone surfaces are simple-stable and primitive-stable.
Study character varieties of arborescent knots and hyperbolic knots.
problem Computing and understanding -character varieties of knots.
method Described a procedure for computing -character varieties of arborescent knots, clarified facts about arborescent tangles, and studied character varieties of hyperbolic knots.
result Found that each hyperbolic knot with essential surfaces has a 1-dimensional character variety.
A natural family of affine cubic surfaces arises from SL(2)-characters of the 4-holed sphere and the 1-holed torus. The ideal locus is a tritangent plane which is generic in the sense that the cubic curve at infinity consists of three lines pairwise intersecting in three double points. We show that every affine cubic s…
A Kauffman bracket on a surface is an invariant for framed links in the thickened surface, satisfying the Kauffman skein relation and multiplicative under superposition. This includes representations of the skein algebra of the surface. We show how an irreducible representation of the skein algebra usually specifies a …
Paper describes a new method for character varieties of surface groups.
problem Characterizing surface groups into SL_n(C).
method Rational WKB method applied to higher complex structures.
result Generalizes known flat SL_2(C) connections to higher dimensions.
We determine the characters of SL(2) representations of groups and surface groups.
Positive representations of surface groups in PO(p,q) form connected components of character varieties.
problem Characterizing representations of surface groups in special orthogonal groups PO(p,q).
method Using Anosov representations and root versus weight collar lemmas.
result Connected components of character varieties are formed by -positive Anosov representations.
Refines Haupt's theorem for surface characters.
problem Characterization of holomorphic 1-forms on surfaces.
method Refines Haupt's theorem with divisor data.
result Necessary and sufficient conditions for surface characters.
We use Bonahon-Wong's trace map to study character varieties of the once-punctured torus and of the 4-punctured sphere. We clarify a relationship with cluster algebra associated with ideal triangulations of surfaces, and we show that the Goldman Poisson algebra of loops on surfaces is recovered from the Poisson structu…
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.
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.
In 1983 Culler and Shalen established a way to construct essential surfaces in a 3-manifold from ideal points of the -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…
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…
The paper provides coordinates for -web diagrams on surfaces.
problem Quantizing the character variety of closed surfaces.
method Introducing explicit coordinates for non-elliptic web diagrams in terms of a submonoid of .
result Explicit coordinates for non-elliptic web diagrams on a closed surface yield a parametrization by a submonoid of , with dimensions matching the character variety.
Rigidity of mapping class group on surfaces with punctures.
problem Studying the rigidity of mapping class groups on surfaces with punctures.
method Analyzing the relative automorphism group of the SL(2, C)-character variety, fixing monodromies along punctures.
result The relative automorphism group is a finite extension of the mapping class group and rigid.
We compute both natural and smooth models for the character varieties of the two component double twist links, an infinite family of two-bridge links indexed as . For each , the component(s) of the character variety containing characters of irreducible representations are birational to…
A gauge theoretic description of the Morgan-Shalen compactification of the $\SL$ character variety of the fundamental group of a hyperbolic surface is given in terms of a natural compactification of the moduli space of Higgs bundles via the Hitchin map.
We show that the algebraic automorphism group of the SL(2,C) character variety of a closed orientable surface with negative Euler characteristic is a finite extension of its mapping class group. Along the way, we provide a simple characterization of the valuations on the character algebra coming from measured laminatio…
Monodromy map from differential systems to character variety is generically immersive for complex -representations.
problem Characterizing when the monodromy map is immersive for differential systems.
method Analyzing the space of -differential systems on a compact Riemann surface and the character variety of -representations.
result The monodromy map is an immersion at the generic point when the complex dimension of is at least three.
We prove that the balanced Chekhov-Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this observation a classification of the irreducible representations of the balanced Che…
Given a 3 manifold M with torus boundary and an ideal triangulation, Yoshida and Tillmann give different methods to construct surfaces embedded in M from ideal points of the deformation variety. Yoshida builds a surface from twisted squares whereas Tillmann produces a spun-normal surface. We investigate the relation be…
Introduces Fock bundles for studying surface group character varieties.
problem Character varieties of surface groups without fixed complex structures.
method Introduces Fock bundles as smooth principal bundles with special adjoint-valued 1-forms, constructs canonical connections, and solves non-linear PDEs.
result Explicit solutions for Fock bundles in the Fuchsian locus map to the Hitchin component.
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.
Anosov deformations created for surfaces with specific properties.
problem Creating proper slices in character varieties for specific surface groups.
method Constructing proper slices in character varieties for surface groups into SL(3,R).
result Representations are Borel Anosov.
Study of measured laminations on surfaces using Newton polytopes and Poisson brackets.
problem Understanding the space of measured laminations on surfaces from a valuative perspective.
method Introducing Newton polytopes for character variety functions, defining tangent spaces, and identifying symplectic structures.
result Trace functions have unit coefficients at the extremal points of their Newton polytopes.
Extending Culler-Shalen theory, Hara and the second author presented a way to construct certain kinds of branched surfaces in a -manifold from an ideal point of a curve in the -character variety. There exists an essential surface in some -manifold known to be not detected in the classical $\o…
Culler-Shalen theory extended to arbitrary characteristic.
problem Understanding essential surfaces in 3-manifolds over arbitrary fields.
method Using character varieties and group actions on trees over arbitrary algebraically closed fields.
result Examples show changes in the -polynomial and essential surfaces in characteristic 2.
New coordinates for SL3-web graphs on surfaces defined by Fock-Goncharov.
problem Characterizing functions on SL3-character varieties of surfaces.
method Introduced tropical Fock-Goncharov coordinates based on Knutson-Tao rhombus inequalities and congruence conditions.
result Tropical coordinates naturally index the commutative algebra of functions on SL3-character varieties.
For an oriented surface of genus g with b boundary components, we construct a rational map from a subset of C^{6g-6+3b} onto an open algebraic subset of the PSL(2,C)-character variety as an analogue of the Fenchel-Nielsen coordinates. After taking the quotient by an action of a finite group, we obtain a parametrization…
The study examines dynamics on SU(2)-representation varieties for surfaces and non-orientable surfaces.
problem Dynamics of group actions on SU(2)-representation varieties of surfaces and non-orientable surfaces.
method Description and analysis of group actions generated by Dehn twists on SU(2)-representation varieties.
result Explicit invariant rational functions on SU(2)-representation varieties for specific cases of surfaces and non-orientable surfaces.
New surfaces found in 5D space.
problem Constructing smooth embedded special Legendrian surfaces in \(\mathbb S^5\).
method Combining implicit function theorem, loop algebra-valued meromorphic connections, and character variety analysis.
result First genus > 1 embedded special Legendrian surfaces in \(\mathbb S^5\).
The paper classifies ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia.
problem Classifying ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia.
method Maximum principle applications, classification based on causal character of rulings.
result Planes are the only cylindrical stationary surfaces. For non-cylindrical surfaces, classification depends on the causal character of the rulings.
Abstract framework for two meromorphic forms on punctured surfaces.
problem Developing a framework for two meromorphic forms on punctured Riemann surfaces.
method Abstract framework with Teichmüller regularity, degeneration detection, and pushability.
result Existence of a surface carrying two meromorphic differentials realizing any prescribed restricted pair.
A differential topology proof that the character variety of the genus two surface is homeomorphic to math.GT
Proof that character variety of genus 2 surface is .
problem Character variety structure of genus 2 surface.
method Differential topology, algebraic topology, representations.
result Character variety is homeomorphic to .
The paper solves a geometric P=W conjecture for SL(2,C) using Thurston's compactification.
problem Addressing the geometric P=W conjecture for SL(2,C) in projective compactifications of character varieties of closed surfaces.
method Using Thurston's compactification of Teichmüller space and new results, the paper constructs a projective compactification of the SL(2,C)-character variety of any closed surface of genus g>1.
result The boundary divisors are toric varieties and the dual intersection complex is a sphere.
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 th…