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 Riemannian geometry of maximal surface group representations in pseudo-hyperbolic space.
problem Characterize the geometry of maximal surface group representations in pseudo-hyperbolic space.
method Introduced a scalar product on the first cohomology group, leading to a Riemannian metric on the smooth locus.
result Found totally geodesic sub-varieties and orbifold structures in the space of representations.
The paper describes maximal components of character varieties for PSp(4,R) and Sp(4,R).
problem Character varieties of surface group representations into PSp(4,R) and Sp(4,R).
method Using Higgs bundles and mapping class group invariant structures, the paper constructs and parameterizes maximal components as holomorphic fiber bundles over Teichmüller space.
result The quotient of maximal components for PSp(4,R) and Sp(4,R) by the mapping class group is a holomorphic submersion over the moduli space of curves.
Minimal energy local systems on curves are compact components of character varieties.
problem Characterizing local systems on surfaces with minimal energy.
method Study of minimal energy local systems on surfaces of genus g with d punctures.
result Minimal energy local systems form compact components of real relative character varieties.
We show that for any knot there exist only finitely many irreducible metabelian characters in the SL(2,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,C)-character va…
We study the (relative) SL(2,C) character varieties of the four-holed sphere and the action of the mapping class group on it. We describe a domain of discontinuity for this action, and, in the case of real characters, show that this domain of discontinuity may be non-empty on the components where the relative euler cla…
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…
Study shows mapping class group action is ergodic on specific representations.
problem Ergodicity of mapping class group action on specific representations.
method Applied symplectic methods developed by Goldman and Xia.
result The action is ergodic.
Construct model Higgs bundles in exceptional components of Sp(4, R) character variety.
problem Constructing model Higgs bundles in exceptional components of the Sp(4, R) character variety.
method Using the gluing construction for Higgs bundles over connected sums of Riemann surfaces and solutions to the Sp(4, R)-Hitchin equations.
result Provide model Higgs bundles in all 2g-3 exceptional components of the maximal Sp(4, R)-Higgs bundle moduli space.
The paper extends Dunfield's theorem about 3-manifold holonomy maps and volume functions.
problem The birationality of the peripheral holonomy map on character varieties.
method Generalization of Dunfield's rigidity argument using volume functions.
result A conjecture about the volume function on character varieties implies birationality.
Study on skein module dimensions at irreducible representations.
problem Dimension of skein module at irreducible representations.
method Localization of skein module at maximal ideal corresponding to irreducible representation.
result Localization forms a one-dimensional free module over the unreduced coordinate ring.
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…
We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of L-twisted G-Higgs bundles, for the groups G=GL(2,C), SL(2,C) and PSL(2,C). We also determine the twisted Chern class of the regula…
New twisted versions of wild character varieties created.
problem Constructing twisted wild character varieties.
method Creating new versions of wild character varieties.
result Twisted versions of wild character varieties constructed.
Let G be a complex reductive algebraic group (not necessarily connected), let K be a maximal compact subgroup, and let A be a finitely generated Abelian group. We prove that the conjugation orbit space Hom(A,K)/K is a strong deformation retract of the GIT quotient space Hom(A,G)//G. As a corollary, we determine necessa…
Explains how Higgs bundles help study Fuchsian representations on surfaces.
problem Understanding components of character varieties for Fuchsian representations.
method Theory of Higgs bundles, focusing on real groups and their subtleties.
result Characterizes deformation spaces of Fuchsian representations using Higgs bundles.
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
Study geodesic currents on surfaces, proving new properties of character varieties.
problem Characterizing discontinuity sets in Hitchin and maximal character varieties.
method Decompose geodesic currents into measured laminations or positive systole components.
result Bi-Lipschitz equivalence of intersection functions to length functions for currents with positive systole.
For a closed surface S, the Hitchin component Hit_n(S) is a preferred component of the character variety consisting of group homomorphisms from the fundamental group pi_1(S) to the Lie group PSL_n(R). We construct a parametrization of the Hitchin component that is well-adapted to a maximal geodesic lamination on the su…
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.
Decomposes geodesic currents on surfaces into measured laminations or positive systole components.
problem Decomposing geodesic currents on surfaces of finite type.
method Topological decomposition and analysis of intersection functions.
result Currents with positive systole are bilipschitz equivalent to length functions under hyperbolic metrics.
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.
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.
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.
Study character varieties in SL_2 and Kauffman skein algebras for surfaces and 3-manifolds.
problem Understanding the algebraic geometry of character varieties.
method Focus on surface and 3-manifolds groups, construct Reidemeister torsion as a rational volume form.
result Construct the Reidemeister torsion as a rational volume form on the character variety.
Let X be the moduli space of SL(3,C) representations of a free group of rank r. In this paper we describe maximal algebraically independent subsets of certain minimal sets of coordinate functions on X. These subsets locally parametrize the moduli space.
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.
Study on representations of a specific surface group, identifying components with non-maximal Euler class.
problem Characterizing representations of the thrice punctured projective plane group.
method Use of decorated character varieties for non-orientable surfaces, focusing on connected components with non-maximal Euler class.
result Identification of components with different types of simple closed curves under various Euler classes.