Study subgroup actions on mapping class groups using Heisenberg representations.
problem Untwisting representations of mapping class groups on Heisenberg subgroups.
method Restrict and analyze twisted representations of mapping class groups to Heisenberg subgroups.
result Untwisting representations on Torelli group for any Heisenberg representation.
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.
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.
Method produces faithful representations of Garside groups and torus knot groups.
problem Representations of Garside groups and torus knot groups.
method Method to produce faithful representations in GL2(C[t±1, q±1])
result Explicit construction of representations for G(n,m) and torus knot 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.
Researchers study Prym representations for handlebody groups, focusing on cyclic cases.
problem Understanding Prym representations for handlebody groups and their images.
method Restrict Prym representations to handlebody groups and twist groups, focusing on cyclic cases.
result Determined the image of Prym representations in the cyclic case for handlebody groups.
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.
Researchers describe unitary representations of mixed braid groups.
problem Understanding unitary representations of mixed braid groups.
method Explicitly describe unitary representations on cohomology of Abelian branched covers.
result Image of the representation is generated by complex reflections and related to the multivariate Burau representation.
Surveying Hitchin representations of Fuchsian groups.
problem Understanding representations of Fuchsian groups.
method Survey and conjectural geometric description.
result Conjectural geometric picture of an augmented Hitchin component.
The paper explores mapping class group quotients by Dehn twists and their representations.
problem Finite quotients and representations of mapping class groups by powers of Dehn twists.
method Construction of finite quotients using representations with Zariski dense images into semisimple Lie groups, and Long and Moody's method.
result The Fibonacci TQFT representation is a specialization of the Jones representation in genus 2.
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 hyperconvex representations of hyperbolic groups in complex flag manifolds.
problem Characterizing hyperconvex representations of hyperbolic groups.
method Geometric and topological analysis of representations in mPSL(d,C). result Virtual isomorphism to Kleinian groups and flag manifold Hausdorff dimension restriction.
Characterizes Coxeter groups with convex cocompact representations in projective space.
problem Understanding representations of Coxeter groups as convex cocompact reflection groups.
method Investigates representations of Coxeter groups into GL(n,R) as geometric reflection groups in projective space.
result Characterizes Coxeter groups that admit convex cocompact representations and describes the spaces of such representations.
Extending braid group representations to singular braid monoids and groups.
problem Extending braid group representations to singular braid monoids and groups.
method Investigating the extension of representations from braid groups to singular braid monoids and groups, and computing defects.
result Constructing a linear representation of the singular braid group that is an extension of the Lawrence-Krammer-Bigelow representation and computing its defect.
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. Cube complexes allow hyperbolic groups to have Anosov representations.
problem Understanding representations of hyperbolic groups in higher dimensions.
method Analyzing representations of hyperbolic groups acting on CAT(0) cube complexes.
result Generic representations of certain groups are Anosov.
The paper calculates Alexander polynomials for knots using finite group representations.
problem Calculating Alexander polynomials for knots using specific group representations.
method Defined twisted Alexander polynomials associated with regular representations of finite groups.
result Several formulas for the twisted Alexander polynomial are provided.
Unified study of homological representations of mapping class groups.
problem Fundamental relationships between different homological representations.
method Analysis of twisted homology of configuration spaces.
result Unified account of non-degenerate pairings, embeddings, and isomorphisms.
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.
Constructs 2-representations and 2-vector bundles for Lie 2-groups.
problem No strict model for string 2-group, so coherent model is constructed.
method Builds category of 2-representations and equivariant 2-vector bundles.
result Explicit formulas for 2-representations and 2-vector bundles.
We investigate a family of (reducible) representations of Artin's braid groups corresponding to a specific solution to the Yang-Baxter equation. The images of the braid groups under these representations are finite groups, and we identify them precisely as extensions of extra-special 2-groups. The decompositions of the…
New method for group representation presentations.
problem Constructing presentations for group representations.
method Developed a new technique for generating relations.
result Important for calculating Torelli group's homology.
Theory of Θ-positive representations for real closed fields.
problem Generalizing positive representations to real closed fields.
method Developing theory for Fuchsian groups to linear groups over real closed fields.
result Theory encompasses many generalizations of positive or Anosov representations.
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 MPR to measure and optimize representation across intersectional groups in retrieval.
problem Harmful stereotypes, cultural erasure, and social disparities in image search and retrieval.
method Develops MPR metric, practical estimation methods, theoretical guarantees, and optimization algorithms.
result Optimizing MPR yields more proportional representation across multiple intersectional groups, often with minimal retrieval accuracy compromise.
Quantum theory uses modular group representations to assign invariants to 3-manifolds.
problem Assigning invariants to 3-manifolds via modular group representations.
method Projective representations of the modular group derived from a noncommutative torus.
result Computed traces and determinants of matrices associated with modular group elements.
The paper explores representations of graph manifolds to Seifert motion groups.
problem Existence of faithful representations of graph manifolds to Seifert motion groups.
method Discussion and proof of non-existence of certain representations.
result Graph manifolds can have virtually no faithful representations to the Seifert motion group.
Discrete PU(1,1) representations of hyperelliptic groups are proven.
problem Characterizing PU(1,1) representations of hyperelliptic groups.
method Proving representations are basic if and only if they are discrete and faithful.
result A conjecture by S. Anan'in and E. Bento Gonçalves is partially proven.
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. Identifies Anosov representations of hyperbolic triangle groups in SL(3,R).
problem Classifying Anosov representations of hyperbolic triangle groups into SL(3,R).
method Proving representations are Anosov if they lie in the Hitchin component or the Barbot component, with specific conditions for eigenvalues.
result Anosov representations in SL(3,R) have non-convex boundary maps.
The paper constructs representations of flat virtual braids by free group automorphisms.
problem Representing flat virtual braids by automorphisms of free groups.
method Construction of representations of flat virtual braid groups FVBn by automorphisms of free groups of rank 2n. result Established conditions of faithfulness and properties of the kernel for n≥3. New representations of hyperbolic 3-manifold groups into larger groups.
problem Finding representations of hyperbolic 3-manifold groups into larger matrix groups.
method Holonomy representations from projective deformations of hyperbolic structures.
result First examples of strongly dense representations into SL(4,R) and SU(3,1). For groups of a topological origin, such as braid groups and mapping class groups, an important source of interesting and highly non-trivial representations is given by their actions on the twisted homology of associated spaces; these are known as homological representations. Representations of this kind have proved th…
In this article we introduce order preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order preserving representations to weakly maximal representations, introduced in arXiv:1305.2620, we show that order preserving representations into Lie groups of Hermit…
Study classifies Lie group representations with non-empty boundary orbit space.
problem Classifying representations of Lie groups with non-empty boundary orbit space.
method Detailed calculations based on previous work.
result Classification of Lie group representations with non-empty boundary orbit space.
The paper explores mapping class groups and their quantum field theory representations.
problem Understanding finite dimensional representations of mapping class groups.
method Survey of topological quantum field theory aspects.
result Discussion of finite dimensional representations in quantum field theory.
Develops theory of Anosov representations for Fuchsian groups, showing stability and analytical properties.
problem Understanding geometrically finite Fuchsian groups and their representations.
method Theory of Anosov representations, type-preserving deformations, limit maps, relative Anosov and dominated representations.
result Cusped Hitchin representations are Borel Anosov, stable under deformations, and limit maps vary analytically.
Entropy rigidity theorem for cusped Hitchin representations.
problem Entropy rigidity for Hitchin representations of cusped groups.
method Introduction of (1,1,2)-hypertransverse groups and transverse representations.
result Hausdorff dimension of conical limit set agrees with simple root entropy.
Study local structure of knot group representations into SL(n,C).
problem Understanding the local structure of knot group representations.
method Analysis of tangent cone and use of Luna's slice theorem.
result Local structure of representation variety at diagonal representations.
We study the TQFT mapping class group representations for surfaces with boundary associated with the SU(2) gauge group, or equivalently the quantum group $U_q(\Sl(2))$. We show that at a prime root of unity, these representations are all irreducible. We also examine braid group representations for transcendental valu…
The paper studies conjugating complex representations into real ones.
problem Understanding representations of surface groups into complex Lie groups.
method Analyzes representations of finitely generated groups into PGL(k,C) and determines conjugacy conditions. result Identifies representations in the larger variety that are conjugate in PGL(k,C) to a representation in PGL(k,R). The paper defines new representations and groups related to virtual links.
problem Defining and studying new representations of virtual braid groups and link groups.
method Introducing virtually symmetric representations, virtual link groups, and marked Gauss diagrams.
result Established equivalence of many known representations to virtually symmetric representations.
Proves Burau representation is faithful for 4-strand braids.
problem Proving the Burau representation is faithful for 4-strand braids.
method Used ideas from Moody, Long, Long-Paton, and Bigelow.
result Burau representation is faithful for n = 4.
Study projective representations of infinite-dimensional Hilbert-Lie groups.
problem Characterize and classify representations of Hilbert-Lie groups.
method Use covariance with respect to one-parameter groups of automorphisms and implement perturbation theory.
result Explicit determination of central extensions for projective representations.
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.
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.