Framework for reconstructing knot exterior surface complexes.
problem Understanding the mapping class group action on knot exteriors.
method Organizing rigidity questions, using a reduction principle, and classifying automorphisms.
result Common image-kernel bookkeeping and reconstruction framework for knot exteriors.
New framework for quantum invariants of 3-manifolds using homology.
problem Quantum invariants of 3-manifolds.
method Homological construction of mapping class group representations extended to cobordisms.
result Construction of a ( 2 + 1 ) (2+1) ( 2 + 1 ) -dimensional TQFT assigning twisted homology to surfaces. The paper constructs braiding structures for a specific subfactor.
problem The challenge is to understand the braiding structures of a Jones-Wassermann subfactor.
method The approach involves constructing braiding structures on the multi-interval Jones-Wassermann subfactor planar algebra.
result The braiding structures induce a projective unitary representation of the balanced superelliptic mapping class group.
The paper disproves the existence of certain subgroups with nontrivial rational abelianization.
problem The existence of finite-index subgroups with nontrivial rational abelianization in handlebody groups.
method Proved that meridian multitwists vanish in H 1 ( Γ ; Q ) H_1(Γ; \mathbb{Q}) H 1 ( Γ ; Q ) and showed that H 1 ( Γ ; Q ) = 0 H_1(Γ; \mathbb{Q}) = 0 H 1 ( Γ ; Q ) = 0 for specific subgroups. result No finite-index subgroups of the handlebody group contain nontrivial rational abelianization.
Compute central extension of mapping class group from stated skein algebra
problem Compute central extension of mapping class group from stated skein algebra
method Compute central extension of mapping class group from stated skein algebra
result Compute central extension of mapping class group from stated skein algebra
The diameter function is a topological Morse function.
problem The relationship between systole and diameter functions on Teichmüller space.
method Mapping class group-equivariant topological Morse function approach.
result The diameter function on Teichmüller space is a topological Morse function.
New combinatorial structures for Teichmüller spaces with Thurston's metric are explored.
problem Understanding the combinatorial structures of Teichmüller spaces with Thurston's metric.
method Analyzing the unit tangent and cotangent spheres of Teichmüller space, proving formulas for dimensions and codimensions of faces.
result The combinatorial structure of unit spheres in Teichmüller spaces is independent of the underlying point and is isomorphic to the extended mapping class group.
Two elements generate all mappings of a nonorientable surface.
problem Generating the mapping class group of a nonorientable surface.
method Proving two elements generate the mapping class group for g ≥ 13 g \geq 13 g ≥ 13 . result The mapping class group of a nonorientable surface of genus g ≥ 13 g \geq 13 g ≥ 13 can be generated by exactly two elements. Introduces K \mathbb{K} K -framings for surfaces, generalizing quadratic forms.
problem Generalizing quadratic forms to commutative rings with unit.
method Introduces K \mathbb{K} K -framings and maps based loops to homology classes. result Bijection between K \mathbb{K} K -framings and twisted cocycles for surfaces with positive genus. The study examines Morse diagrams and their behavior under Murasugi sums, leading to contact structure classifications.
problem Understanding Morse diagrams and their behavior under Murasugi sums.
method Examination of combinatorial Morse structures, open book decompositions, and contact structures.
result Diagrammatic criterion for detecting overtwisted contact structures and classification of Morse diagrams for one-holed torus pages.
Proves a conjecture about matrix orders for pseudo-Anosov maps.
problem Proving a conjecture about matrix orders for pseudo-Anosov maps.
method Proved the conjecture for prime r and Johnson kernel elements.
result Proved the conjecture for prime r and Johnson kernel elements.
Study of SL(2,R) representations on a once-punctured torus, showing Cantor set spectrum.
problem Characterizing SL(2,R) representations on a once-punctured torus.
method Introduction of spectrum as a subset of projective measured laminations, analysis of dynamics of cocycles.
result Spectrum of a generic representation on a once-punctured torus is a Cantor set.
Study of Dehn twists on a disc with 3 points, solving conjugacy problem.
problem Solving conjugacy problem for Dehn twists on a disc with 3 marked points.
method Explicit description of orbits of Dehn twists on the Dynnikov plane, relating to homology dynamics.
result Explicit solution to conjugacy problem for Dehn twists, presenting an untwisting algorithm.
Study mapping class group action on cohomology of SL_n character variety.
problem Mapping class group action on cohomology of SL_n character variety.
method Relative endoscopic decomposition and Looijenga's results.
result Reduced problem to cohomology of a finite cover of surface.
Defines complex for infinite-type surfaces with non-planar ends.
problem No finite analog for surfaces with non-planar ends.
method Defined a complex Γ k ( S ) Γ_k(S) Γ k ( S ) for infinite-type surfaces. result Automorphism group of Γ k ( S ) Γ_k(S) Γ k ( S ) is isomorphic to the extended mapping class group. Proves Grothendieck-Teichmüller group acts on specific mapping class groups.
problem Proving the action of G T ^ \widehat{GT} GT on specific mapping class groups. method Analyzes the Grothendieck-Teichmüller group and mapping class groups.
result Proves G T ^ \widehat{GT} GT acts on Γ ^ g , 0 \widehatΓ_{g,0} Γ g , 0 and Γ ^ g , 1 \widehatΓ_{g,1} Γ g , 1 for all g > 0 g>0 g > 0 . This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.
problem Classifying and determining the length of the shortest filling pairs on a specific type of surface.
method Classifying and determining the length of the shortest filling pairs on a specific type of surface.
result The paper classifies and determines the length of the shortest minimal filling pairs on a genus two surface.
The paper computes Fenchel-Nielsen coordinates for fixed points of cyclic actions on Teichmüller space.
problem Computing Fenchel-Nielsen coordinates for cyclic actions on Teichmüller space.
method Developed algorithms to describe Fenchel-Nielsen coordinates of fixed points of cyclic subgroups of Mod(S_g) on Teich(S_g).
result Computed Fenchel-Nielsen coordinates for cyclic subgroups of orders 10, 8, and 4 in Mod(S_2).
New invariant detects more elements in 4D diffeomorphism group.
problem Detecting more elements in the mapping class group of 4D handlebodies.
method Defined and computed a new invariant ( W 3 ) m (W_3)_m ( W 3 ) m for π 0 D i f f ( a t u r a l m S 1 i m e s D 3 , ∂ ) π_0\mathrm{Diff}(
atural_m S^1 imes D^3,\partial) π 0 Diff ( a t u r a l m S 1 im es D 3 , ∂ ) . result Infinitely many non-isotopic separating 3-balls found.
Nontrivial boundary Dehn twist found on K3#K3 manifold.
problem Proving nontriviality of a Dehn twist on a specific 4-manifold.
method Algebraic criterion and equivariant topological K-theory to show non-isotopy.
result Boundary Dehn twist is nontrivial in the smooth mapping class group.
The paper studies exotic diffeomorphisms of 4-manifolds with b_+ = 2.
problem Examining exotic diffeomorphisms in 4-manifolds with a specific topological invariant.
method Utilizes Seiberg-Witten invariants for 1-parameter families of 4-manifolds and a gluing formula for connected sums.
result Proves that the mapping class group of certain 4-manifolds is not finitely generated and surjects to Z^∞.
Algorithm solves word problem in mapping class group quickly.
problem Word problem in mapping class group of a surface.
method Quasi-linear time algorithm (O(n log^3(n))).
result Solves word problem efficiently.
Mapping class group subgroups yield quasi-isometric curve complex.
problem Understanding the curve complex through coset intersections.
method Proving quasi-isometry and combinatorial equivalence of curve complex and coset intersection complex.
result Automorphism group of coset intersection complex is the extended mapping class group.
Study invariant measures on measured laminations for subgroups of mapping class group.
problem Classify invariant Radon measures on space of measured laminations for subgroups of mapping class group.
method Geometric approach, focusing on recurrent measured laminations, explicitly constructing ergodic measures.
result Show uniquely ergodic for divergence-type subgroups, generalize results for full mapping class group.
Improved lower bounds for faithful linear representations of mapping class groups.
problem Finding the minimum dimension of faithful linear representations of mapping class groups.
method Using finer study of commutation relations and specific pants decompositions to show representations must kill certain subgroups.
result Established lower bounds of 4 g − 3 4g - 3 4 g − 3 for faithful representations of mapping class groups of genus g ≥ 7 g \ge 7 g ≥ 7 . The curve graph and related graphs are hyperbolic and have quasi-tree fibers.
problem Understanding the structure of the curve graph and related graphs.
method Analyzing a sequence of graphs with Lipschitz maps and proving hyperbolicity and quasi-tree properties.
result The graphs in the sequence are hyperbolic and have quasi-tree fibers, leading to bounds on asymptotic dimension and acylindrical actions.
Compute group cohomology for mapping class group with non-symplectic coefficients.
problem Compute group cohomology for mapping class group with non-symplectic coefficients.
method Compute the invariant subspace of the rational group ring of a surface, truncated by powers of the augmentation ideal, under the action of the mapping class group.
result First group cohomology computation for the mapping class group with non-symplectic coefficients.
This paper studies a deformation retraction of Teichmüller space and its analogy with well-rounded retractions.
problem Understanding the well-rounded deformation retraction of Teichmüller space.
method Examining the mapping class group-equivariant deformation retraction of Teichmüller space onto a CW complex and comparing it to well-rounded retractions of other spaces.
result The well-rounded deformation retraction of Teichmüller space is analogous to well-rounded retractions of other spaces.
The paper finds pseudo-Anosov-like maps on an infinite ladder surface.
problem Exploring dynamics on infinite surfaces.
method Lifts Penner-type pseudo-Anosov maps from a closed surface to an infinite ladder surface.
result Existence and properties of pseudo-Anosov-like maps on the infinite ladder surface.
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.
Schmutz Schaller and Thurston's approaches are dual.
problem Mapping class group-equivariant deformation retractions of Teichmüller space.
method Comparing Schmutz Schaller's and Thurston's methods.
result Schmutz Schaller and Thurston's approaches are dual.
The paper connects function theory, dynamics, and ergodic theory via Thurston's theory.
problem Function theory on Teichmüller space and dynamics of mapping class groups.
method Utilizes Thurston's theory and Sullivan's theory on discrete subgroups of hyperbolic space.
result Establishes connections between function theory, dynamics, and ergodic theory.