Characterizes totally elliptic surface group representations into Lie groups.
problem Understanding totally elliptic surface group representations into Lie groups.
method Characterization of representations into PSL2R and PSL2C by their mapping properties. result They are either into a compact subgroup or Deroin--Tholozan representations.
Polynomial representations found in surface braid and mapping class groups.
problem Homological representations of surface braid and mapping class groups.
method Study of homological representation functors and short exact sequences.
result Many homological representation functors are polynomial.
New representations for surface groups expand known Anosov classes.
problem Understanding new types of representations for surface groups.
method Introducing and studying simple Anosov representations of closed hyperbolic surface groups.
result Simple Anosov representations strictly contain Anosov representations.
New method connects surface group representations to Higgs bundles with smooth structures.
problem Mapping surface group representations to smooth Higgs bundles.
method Establishes a connection between surface group representations and Higgs bundles with smooth spectral data.
result For every nonelementary representation, there exists a Riemann surface structure making the associated Higgs bundle smooth.
New representations from finite groups for surface mapping classes.
problem Understanding representations of mapping class groups from finite group data.
method Using Drinfeld doubles of finite groups to describe representations of mapping class groups of surfaces.
result These representations have finite images and non-trivial restrictions to Torelli groups for certain surfaces and finite groups.
New representation for braid groups and surface braid groups, extending Lawrence-Krammer-Bigelow.
problem Constructing representations for braid groups and surface braid groups.
method Pro-nilpotent tower of representations, starting with the original LKB representation.
result 3-variable enrichment of the Lawrence-Krammer-Bigelow representation.
Classifies finite orbits of mapping class group actions on surface group representations.
problem Finite orbits of mapping class group actions on surface group representations.
method Classification based on surface genus and representation properties.
result Finite orbits correspond to homomorphisms with finite image for genus at least two, and to finite or special dihedral representations for genus one.
Study k-positive surface group representations and their degenerations.
problem Understanding the behavior of surface group representations under degenerations.
method Introduced k-positive representations and studied their degenerations using a limit theorem for positively ratioed representations.
result Degenerations of k-positive representations can lead to limits that are at least (k-3)-positive and irreducible limits are (k-1)-positive.
Introduces representations of surface groups into Lie groups preserving order.
problem Representations of surface groups into Lie groups that preserve the group order.
method Relates order preserving representations to weakly maximal representations and uses geometric and causal structure.
result Order preserving representations into Lie groups of Hermitian type are faithful with discrete image and form a closed set.
Study topological components of surface group representations into SL(2,R) and PSL(2,R).
problem Understanding the structure of representation spaces of surface groups.
method Utilized the signature formula to determine connected components of representation spaces.
result Determined the number of connected components for different types of holonomies.
Anosov representations uniquely determined by boundary maps for surface groups to SL(3,R).
problem Uniqueness of Anosov representations determined by boundary maps.
method Analysis of boundary maps and representation properties.
result Anosov representations uniquely determined by boundary maps if not reducible.
Study of low dimensional representations of mapping class groups of surfaces, focusing on genus ≥ 7.
problem Classifying (2g+1)-dimensional complex linear representations of mapping class groups. method Using twisted 1-cohomology groups and Morita's computation, a complete classification is given for g≥7. result No irreducible linear representations of dimension 2g+1 for g≥7. 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 Θ-positive Anosov representations. Study on hyperconvex representations of surface groups and their geometric properties.
problem Understanding the geometry of hyperconvex representations of surface groups.
method Holomorphic extension of Ahlfors--Bers map and analysis of limit sets.
result Limit set has Hausdorff dimension 1 if and only if representation is in PSL(d,R).
New representations solve a gap in projective structure proof.
problem Prove every non-elementary surface group representation is a projective structure holonomy.
method Define pentagon representations and show they are non-elementary and not Schottky decomposable.
result Repair Gallo-Kapovich-Marden proof by showing pentagon representations arise as holonomies.
Collar lemma proven for certain surface group representations.
problem Proving a collar lemma for specific surface group representations.
method Using partial hyperconvexity properties and Anosov representations.
result 'Positivity properties' hold for partially hyperconvex representations.
We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation.
Surface groups have solvable representations in SL(2,R).
problem Representing surface groups without simple loops.
method Torsion-free group of upper-triangular matrices in SL(2,R).
result No simple loop in the kernel of representations.
Study maximal representations of surface groups into SO(2,n), proving geometric and dynamical properties.
problem Maximal representations of surface groups into SO(2,n) and their geometric properties.
method Combining Higgs bundle theory, Anosov representations, and pseudo-Riemannian geometry.
result Holonomies of certain geometric structures and unique minimal surfaces preserved.
This study finds a special class of representations that dominate others in a complex hyperbolic group.
problem Domination of surface-group representations in complex hyperbolic groups.
method Analysis of T-bent representations and their domination by discrete and faithful representations. result A discrete and faithful representation exists that dominates a given T-bent representation in the Bergman translation length spectrum. Classifies representations up to dimension 3g-3 for surface mapping class groups.
problem Classifying representations of mapping class groups up to a certain dimension.
method Direct sum of a 2g or 2g+1 dimensional representation and a trivial one.
result Any representation up to dimension 3g-3 is a direct sum of a 2g or 2g+1 dimensional representation and a trivial one.
Research classifies geometric structures on manifolds using surface group representations.
problem Classifying geometric structures on manifolds related to surface group actions on character varieties.
method Surveying results on surface groups, focusing on compact and non-compact target groups, discussing various representations and their dynamics.
result Dichotomy in dynamics of character varieties based on compactness of target groups.
Explains Klein surfaces and their fundamental groups, linking them to real algebraic curves.
problem Understanding the fundamental group of Klein surfaces.
method Topological classification, introduction of fundamental group, study of representations.
result Relates the fundamental group of Klein surfaces to the representation variety of Riemann surfaces.
Study proves surface group quotients have non-virtually solvable profinite completions.
problem Understanding profinite completions of surface group quotients.
method Quantum representations of mapping class groups.
result Profinite completions are not virtually solvable, with infinitely many finite simple quotients.
New Θ-positive representations of surface groups discovered.
problem Generalizing Lusztig's total positivity to surface groups.
method Introducing Θ-positivity and proving properties of Θ-positive representations. result Discrete and faithful Θ-positive representations exist and form open sets in representation varieties. The paper provides a spinorial representation for submanifolds in Lie groups.
problem Representation of submanifolds in Lie groups.
method Spinorial representation approach.
result Spinorial proof of the Fundamental Theorem for submanifolds in Lie groups.
New method finds Fuchsian representations dominating others in surface group representations.
problem Finding Fuchsian representations that dominate others in surface group representations.
method Straightening the pleated plane and applying strip deformations.
result There exists a Fuchsian representation that strictly dominates a given non-Fuchsian representation.
Algorithm decides if curves on a surface are simple.
problem Deciding if curves on a surface are simple.
method Combining TQFT representations and LERF for surface groups.
result Representation theoretic algorithm for deciding curve simplicity.
Study Heisenberg homology on surface configurations, revealing new representations of mapping class groups.
problem Homology of surface configurations with Heisenberg group representations.
method Analysis of unordered configurations in a surface, using Heisenberg group actions and representations.
result Obtained genuine and projective representations of mapping class groups from Heisenberg group actions.
New representations for surface groups in PU(2,1) are stable and larger than convex cocompact ones.
problem Characterizing representations of surface groups in PU(2,1).
method Introducing simple-stable representations and proving their properties.
result The set of conjugacy classes of simple-stable representations is a domain of discontinuity strictly larger than convex cocompact representations.
Nonorientable surfaces have a faithful linear representation of a specific dimension.
problem Understanding the structure of hyperelliptic mapping class groups of nonorientable surfaces.
method Proved the existence of a faithful linear representation of dimension g2−1. result The hyperelliptic mapping class group of a nonorientable surface of genus g≥4 has a faithful representation into GL(g2−1,R). Global fixed points in low-dimensional surface group space correspond to trivial representations.
problem Understanding global fixed points in surface group deformation spaces.
method Direct analysis of the deformation space, focusing on the trivial representation.
result Global fixed points in low-dimensional surface group deformation spaces correspond to the trivial representation of the pure mapping class group.
Study TQFT representations for surfaces with boundary, showing irreducibility at prime roots of unity and Zariski density for transcendental parameters.
problem Understanding TQFT representations of mapping class groups with boundary.
method Examined TQFT representations for surfaces with boundary associated with SU(2) gauge group or $U_q(\Sl(2))$ quantum group. result At prime roots of unity, representations are irreducible; for transcendental parameters, the image of mapping class groups is Zariski dense.
Researchers parametrize spaces of positive representations for Lie groups.
problem Tackling spaces of positive representations for Lie groups.
method Generalizing Lusztig's total positivity, they introduce spaces of positive framed representations and parametrize them.
result The number of connected components of the space of framed positive representations agrees with the number of positive representations.
Study on random representations of surface groups into SU(n), focusing on asymptotic expansions.
problem Understanding random representations of surface groups into special unitary groups.
method Use of a symplectic form on moduli space, establishing asymptotic expansions for trace values.
result Existence of large n asymptotic expansions for expected values of trace of elements under random representations.
The paper constructs deformations of group representations and applies them to earthquake deformations on specific surface groups.
problem Deformations of group representations and earthquake deformations on specific surface groups.
method Constructs deformations based on codimension 1 hypersurfaces and multi-hypersurfaces, showing commutativity of deforming along disjoint hypersurfaces.
result Shows earthquake deformations on Anosov surface groups in SO(n,1). Deforms surface groups to be Zariski dense in SL(n,R)
problem Finding Zariski dense surface groups in SL(n,R)
method Deforming K-integral representations of surface groups result Generalizes Long and Thistlethwaite's method to SL(n,R)
We determine the number of connected components of the moduli space for representations of a surface group in the general linear group.
Geometric intersection detection in mapping class groups is linked to representation injectivity.
problem Detecting geometric intersection in mapping class group representations.
method Linking geometric intersection detection to representation injectivity.
result Injectivity of representations up to center is implied by geometric intersection detection, and vice versa.
Convex-cocompact groups in infinite hyperbolic space are deformable.
problem Understanding deformability of convex-cocompact groups in infinite hyperbolic spaces.
method Proving convex-cocompact representations form an open set and using bending to deform them.
result Deformable convex-cocompact representations of surface groups not conjugate to exotic PSL(2,R) representations.
Finite orbit representations of surface groups are finite.
problem Understanding representations of surface groups with finite orbit under mapping class group action.
method Analyzing representations of fundamental groups of surfaces and free groups under the action of mapping class groups and automorphism groups.
result Finite image representations of surface groups with finite orbit under mapping class group action.
Paper uses index theorem to relate symplectic bundle signature to surface group representation in real symplectic group.
problem Relating symplectic bundle signature to surface group representations in real symplectic group.
method Using Atiyah-Patodi-Singer index theorem.
result Obtained a formula for the signature of a flat symplectic vector bundle over a surface with boundary.
We present results connecting crossratios, representations of surface groups in SL(n,R) and in an infinite dimensional group related to the group of diffeomorphisms of the circle. More precisely, we show that representations of a surface group in SL(n,R) can be interpreted as crossratios satisfying …
Study mapping class groups' action on surface homology.
problem Characterize mapping class groups' homology representation.
method Precise characterization of induced homology representation.
result Characterized the image of the homology representation.
Maximal and Borel Anosov representations in Sp(4,R) are proven to be Hitchin.
problem Characterizing representations of surface groups into Sp(4,R) that are Borel Anosov and maximal. method Proving representations are Hitchin if they have maximal Toledo invariant and are Borel Anosov.
result Maximal and Borel Anosov representations in Sp(4,R) are Hitchin. Study on mapping class groups for low genus surfaces.
problem Arithmetic image of homological representations for low genus mapping class groups.
method Examined centralizers of finite groups acting on surfaces with genus at most 3.
result Complete answer to the arithmetic image question for low genus surfaces.
This is a survey of Magnus representations with particular emphasis on their applications to mapping class groups and monoids (groups) of homology cobordisms of surfaces. In the first half, we begin by recalling the basics of the Fox calculus and overview Magnus representations for automorphism groups of free groups an…
Study domination between non-Fuchsian surface group representations and anti-de Sitter geometry.
problem Domination problem between non-Fuchsian representations of closed surface groups.
method Analysis of branched harmonic immersions and construction of anti-de Sitter 3-manifolds.
result Found that representations admitting branched harmonic immersions dominate other representations, and constructed large families of branched anti-de Sitter 3-manifolds.