Classifies orbit closures of mapping class group action on affine character variety.
problem Classifying orbit closures of Mod(Σ)-action on χ(Aff(C)). method Classification up to exceptions of Mod(Σ)-action on affine character variety. result The only obstruction for a non-abelian representation to be the holonomy of a branched affine structure is to be Euclidean and not have positive volume.
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…
The paper extends character varieties to marked surfaces and discovers their triangular decompositions.
problem Character varieties on marked surfaces and their properties.
method Triangulations of marked surfaces and cohomological groups.
result Stated character varieties admit triangular decompositions.
Study character varieties of surfaces using cluster algebras and Poisson structures.
problem Character varieties of surfaces and their Poisson structures.
method Use Bonahon-Wong's trace map and cluster algebras associated with ideal triangulations.
result Recover Goldman Poisson algebra from cluster algebra structure and show automorphisms.
For any complex affine reductive group G and a fixed choice of maximal compact subgroup K, we show that the G-character variety of a free group strongly deformation retracts to the corresponding K-character space, which is a real semi-algebraic set. Combining this with constructive invariant theory and classical topolo…
Monodromy map from differential systems to character variety is generically immersive for complex G-representations.
problem Characterizing when the monodromy map is immersive for differential systems.
method Analyzing the space of g-differential systems on a compact Riemann surface and the character variety of G-representations. result The monodromy map is an immersion at the generic point when the complex dimension of G is at least three. We extend the definition of algebraic entropy to endomorphisms of affine varieties. We calculate algebraic entropy of the action of elements of mapping class groups on various character varieties, and show that it is equal to a quantity we call the spectral radius, a generalization of the dilatation of a Pseudo-Anosov …
Characterizes character varieties of generalized torus knot groups.
problem Understanding the structure of character varieties for generalized torus knot groups.
method Analyzes the path-connectedness and counts irreducible components of character varieties for specific groups.
result The G-character varieties of generalized torus knot groups are path-connected. Holomorphic map connects Hitchin components to character varieties.
problem Complex affine spheres and their properties.
method Mapping class group equivariant holomorphic map from Hitchin components to character varieties.
result Holomorphic map includes holonomies of SL(3,C)-opers.
Let G be a complex affine algebraic reductive group, and let K be a maximal compact subgroup of G. Fix elements h_1,...,h_m in K. For n greater than or equal to 0, let X (respectively, Y) be the space of equivalence classes of representations of the free group of m+n generators in G (respectively, K) such that for each…
In this note, we embed the set of all Fricke characters of a free group F -- the set of all characters of representations of F into SL(2,C) -- as an irreducible affine variety V in complex affine space of dimension 2^n-1. Using the Horowitz generating set as the indeterminates, we show that the ideal I of all polynomia…
Study of dynamics on cubic surfaces and their connection to Painlevé 6 Equation.
problem Understanding the dynamics of automorphism groups on cubic surfaces.
method Analyzing holomorphic automorphisms and character varieties.
result Several open questions about the dynamics of automorphism groups.
We prove that for any affine variety S defined over Q there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety Hom(G, PO(3))//PO(3). The subset U contains all real points of S . As an application we construct new examples of…
We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection (work of Calabi and of Cheng-Yau) mentioning its relation with the Hilbert metr…
The Whitehead link complement's geometry and topology are studied using ideal triangulations and tropical geometry.
problem Understanding the geometry and topology of the Whitehead link complement and its Dehn surgeries.
method Ideal triangulations, spun-normal surfaces, and tropical geometry.
result All boundary curves of the Whitehead link complement are strongly detected by its character variety.
Character varieties get a natural Poisson structure.
problem Character varieties of surfaces.
method Natural Poisson structure.
result Character varieties have a Poisson structure.
Quantum character varieties unify four construction methods.
problem No specific problem stated; unification of approaches.
method Four different approaches to construction.
result Unified understanding of quantum character varieties.
Character variety of Borromean link solved, Alexander polynomial formula found.
problem Character variety of Borromean link
method Determined irreducible SL(2,C) character variety, found Alexander polynomial formula
result Formula for twisted Alexander polynomial on character variety
Flawed groups are shown to include all finitely generated groups isomorphic to free products of nilpotent groups.
problem Characterizing flawed groups and understanding their topological properties.
method Analyzing finitely presented groups and their deformation retracts onto subspaces of character varieties.
result All finitely generated groups isomorphic to free products of nilpotent groups are flawed.
The paper confirms a conjecture linking knot invariants to cubic surfaces.
problem Classifying knots using their fundamental groups and character varieties.
method Analyzing the algebraic structure of the peripheral system of knots and extending a Brumfiel-Hilden conjecture.
result The Brumfiel-Hilden conjecture holds for a wide class of knots, including torus and 2-bridge knots.
Character varieties of 2-bridge knots and links explained using Farey recursion.
problem Understanding character varieties of 2-bridge knots and links.
method Using Farey recursion to define polynomials for character varieties.
result Character varieties described in terms of polynomials defined by Farey recursion.
Computes dimensions of representation and character varieties for 2 and 3-dimensional orbifolds.
problem Computing dimensions of representation and character varieties for orbifolds.
method Analyzes varieties of representations and characters of hyperbolic and non-hyperbolic orbifolds, provides tools for computation.
result Shows that the dimension of a specific component of characters equals half the dimension of the variety of characters of the boundary.
Researchers describe character varieties for Hopf links, proving geometric properties.
problem Character variety geometry of Hopf links with n twists. method Geometric descriptions of irreducible and totally reducible representations.
result Complete geometric description of SU(2)-character variety for r=2. Study character varieties for real forms of SL_n(C).
problem Characterize representations of groups in real forms of SL_n(C).
method Define and analyze G-character varieties, study involutions, and examples.
result Irreducible representations fixed by involutions are conjugate to real forms.
Detects essential surfaces in knots using character variety intersections.
problem Identifying essential surfaces in knot theory.
method Analyzing intersections in the character variety of hyperbolic knots.
result Intersection points in the character variety detect Seifert surfaces.
Character variety of Whitehead link described in detail.
problem Character variety of Whitehead link.
method Nice description of a Zariski open subset.
result Zariski open subset of the character variety described.
Efficient method computes mSL(3,C)-character varieties for two-generator groups.
problem Computing mSL(3,C)-character varieties for two-generator groups. method Efficient method for two-generator groups.
result Efficient computation of mSL(3,C)-character varieties. Proves Torelli group action is ergodic on Lie group character varieties.
problem Ergodicity of Torelli group action on compact character varieties.
method Proof based on properties of connected, semi-simple Lie groups.
result Torelli group action on compact character varieties is ergodic.
Study character varieties of odd classical pretzel knots.
problem Determine mSL(2,C)-character varieties for odd classical pretzel knots. method Compute A-polynomial for each knot.
result Character varieties of odd classical pretzel knots computed.
Study SL2(F)-character varieties of knots over fields of odd characteristic.
problem Character varieties of knots over fields of odd characteristic exhibit ramification phenomena.
method Investigate sufficient conditions for ramification using double branched covers and Alexander polynomials.
result Character varieties of knots over fields of odd characteristic can present ramification phenomena.
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…
Detecting essential surfaces in 3-manifolds and orbifolds using character varieties.
problem Detecting essential surfaces in 3-manifolds and orbifolds.
method Extending Culler and Shalen's construction to 3-orbifolds using SL2(C) character variety. result Any slope detected on a canonical component of the (P)SL2(C) character variety of a one-cusped hyperbolic 3-manifold with symmetries must be the slope of a symmetric surface. Quantum trace maps abelian character varieties to SL2 character varieties.
problem Classifying irreducible representations of Chekhov-Fock algebras.
method Non-commutative deformation of algebraic morphisms.
result Induces birational morphism between torus and SL2 character variety.
Finite automorphisms of SL(2,C) character varieties identified.
problem Character varieties of surfaces with negative Euler characteristic.
method Algebraic automorphisms and mapping class groups studied.
result Finite extension of mapping class group identified.
Study character varieties of hyperbolic 3-manifolds using bundle methods.
problem Character varieties of hyperbolic 3-manifolds in once-punctured torus bundles.
method Restrict characters to the fibre and analyze branched covering maps.
result Infinite family of hyperbolic once-punctured bundles with unbounded genus.
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.
Defines Reidemeister torsion form on 3-manifold character varieties.
problem Character variety singularities and vanishing torsion.
method Differential form on character variety, Alexander polynomial, Culler-Shalen theory.
result Torsion vanishes only at singular points, related to reducibility and Euler characteristic.
Study of pseudo-Anosov actions on SU(2)-character variety for genus 2 surfaces.
problem Characterizing pseudo-Anosov actions on SU(2)-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.
Study of SU(2,1) character varieties on one-holed torus.
problem Characterize representations of mapping class group on SU(2,1) character variety.
method Explicit description of SU(2,1) character variety, use of Farey graph adaptation, and mapping class group action analysis.
result Description of an open domain of discontinuity for mapping class group action.
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…
Character varieties of knot groups into SU(3) are stratified and analyzed.
problem Characterizing representations of knot groups into SU(3).
method Stratification of character variety into reducible, 2D+1D, and irreducible representations.
result Homotopy equivalence between SU(3) and SL(3,C) character varieties.
Study character varieties of even pretzel knots, determining their mSL(2,C)-representations.
problem Determine character varieties of even classical pretzel knots.
method Compute irreducible mSL(2,C)-representations of even classical pretzel knots. result Clarify the steps to compute the A-polynomial of even classical pretzel knots.
New compactification for character varieties with good topological properties.
problem Compactification of character varieties with good topological properties.
method Announced a new compactification with interpretations of ideal points.
result Relates to Weyl chamber length compactification and applies to maximal and Hitchin representations.
Study character varieties of arborescent knots and hyperbolic knots.
problem Computing and understanding mSL(2,C)-character varieties of knots. method Described a procedure for computing mSL(2,C)-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.
Study character variety of a special 3-manifold, finding a polynomial formula.
problem Character variety of a specific 3-manifold.
method Determined irreducible SL(2,C) character variety, derived twisted Alexander polynomial.
result Formula for twisted Alexander polynomial of SL(2,C) representations.
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.
Minimal action of mapping class group on character variety.
problem Character variety of Deroin-Tholozan representations.
method Geometric perspective using symplectic structure.
result Infinite mapping class group orbits are dense.
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.