Study shows pure mapping classes can generate pseudo-Anosov mapping classes with certain conditions.
problem Understanding when pure mapping classes generate pseudo-Anosov mapping classes.
method Analyzing products of a given mapping class and powers of pure mapping classes, deriving an explicit constant.
result Almost all pure mapping classes generate pseudo-Anosov mapping classes when their powers exceed a certain constant.
New insights into mapping class groups' cohomology.
problem Understanding the first integral cohomology of mapping class groups.
method Semi-direct product decomposition and simplicial homology computation.
result Computed the first integral cohomology group for surfaces of genus at least 2.
Study of pure mapping class groups on infinite graphs.
problem Classifying graphs with specific mapping class groups.
method Completely classified graphs with pure mapping class groups.
result Established semidirect product decomposition and computed first integral cohomology.
Homomorphisms between pure mapping class groups are classified for certain genus surfaces.
problem Classifying homomorphisms between pure mapping class groups for specific genus surfaces.
method Proving every multitwist-preserving map is induced by a multi-embedding, then applying to classify homomorphisms.
result All homomorphisms between pure mapping class groups for g≥4 and g′≤6⋅2g−4 are classified. Classifies surfaces for pure mapping class groups with automatic continuity.
problem Determining surfaces for which pure mapping class groups have automatic continuity.
method Completely classified orientable infinite-type surfaces and specific cases of surfaces with finite ends.
result Classification of surfaces for automatic continuity of pure mapping class groups.
Study of mapping class groups on infinite graphs, focusing on their large-scale geometry.
problem Understanding the large-scale geometry of mapping class groups on infinite graphs.
method Using coarse geometry techniques, classify coarsely bounded groups and compute asymptotic dimension.
result Identify conditions for global and local coarsely bounded pure mapping class groups of infinite rank graphs.
In this paper, we calculate the p-torsion of the Farrell cohomology for low genus pure mapping class groups with punctures, where p is an odd prime. Here, `low genus' means g=1,2,3; and `pure mapping class groups with punctures' means the mapping class groups with any number of punctures, where the punctures are not al…
Classifies pure mapping class groups based on surface properties.
problem Classifying pure mapping class groups based on surface properties.
method Complete classification without additional assumptions.
result PMap(Σ) is globally CB if and only if Σ is the Loch Ness monster surface.
Proves transitivity of a specific class of quadratic polynomials.
problem Transitivity of pure Hurwitz classes of post-critically finite quadratic polynomials.
method Uses mapping classes of the sphere with finitely many marked points.
result Establishes transitivity for pure Hurwitz classes of post-critically finite quadratic polynomials.
New subgroup behavior in genus-2 mapping class group identified.
problem Understanding subgroups in genus-2 mapping class group.
method Analyzing purely pseudo-Anosov subgroups as convex cocompact.
result Finitely-generated, purely pseudo-Anosov subgroups are convex cocompact.
The paper shows pure mapping class groups are linear in certain cases.
problem Understanding linearity of pure mapping class groups.
method Analyzing presentations of pure mapping groups and comparing them to braid groups and Artin groups.
result Pure mapping class groups are linear in specific cases.
Study cohomology of mapping class groups for big genus surfaces.
problem Characterize the first cohomology of pure mapping class groups for genus one and zero surfaces.
method Analyzes the first integral cohomology of pure mapping class groups for infinite type genus one surfaces and genus zero surfaces.
result For genus one surfaces, the first integral cohomology is trivial. For genus zero surfaces, there is an uncountable family of non-trivial homomorphisms to Z.
Study Farrell cohomology for non-orientable surfaces, classifying subgroup conjugacy.
problem Determine the p-primary component of Farrell cohomology for non-orientable surfaces. method Classify subgroups of order p using topological equivalence adapted to surfaces with marked points. result Determine the p-primary component of Farrell cohomology for non-orientable surfaces. Study calculates integral cohomology of non-orientable infinite type surfaces.
problem Computing the first integral cohomology group of non-orientable infinite type surfaces.
method Alexander method, isomorphism to automorphism group, topological rigidity of curve graph, semi-direct product structure.
result First integral cohomology group computed for non-orientable infinite type surfaces.
Study mapping class groups of infinite type surfaces with noncompact boundaries.
problem Classify pure mapping class groups of infinite type surfaces.
method Developed a method to cut surfaces into simpler ones and combined recent results.
result Complete classification of perfect and uniformly perfect pure mapping class groups.
Paper computes rational cohomology of spin hyperelliptic mapping class groups.
problem Computing rational cohomology of spin hyperelliptic mapping class groups.
method Computes the G-invariant part of the rational cohomology of the pure braid group. result Includes rational cohomology of spin hyperelliptic mapping class groups of genus g. This paper presents a new exact sequence for orbifold braid groups and mapping class groups.
problem Understanding the relationship between orbifold braid groups and mapping class groups.
method Developed an exact sequence and used presentations of orbifold mapping class groups to determine the kernel.
result The kernel of the orbifold braid group is non-trivial and provides a new presentation.
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.
A spine is constructed for a non-orientable surface's decorated Teichmüller space.
problem Constructing a spine for a non-orientable surface's decorated Teichmüller space.
method Building on Harer's work, constructing a spine and computing its dimension, showing equivariance with the pure mapping class group.
result A spine is constructed with minimal dimension for a punctured non-orientable surface.
This paper presents a number of problems about mapping class groups and moduli space. The paper will appear in the book "Problems on Mapping Class Groups and Related Topics", ed. by B. Farb, Proc. Symp. Pure Math. series, Amer. Math. Soc.
This paper presents fifteen problems about mapping class groups. It is an expanded and updated version of the author's preprint "Ten problems on the mapping class groups". The paper will appear in the book "Problems on Mapping Class Groups and Related Topics", ed. by B. Farb, Proc. Symp. Pure Math. series, Amer. Math. …
The study classifies normal subgroups of mapping class groups of surfaces with Cantor subsets.
problem Understanding the structure of normal subgroups in mapping class groups of surfaces with specific subsets.
method Proves two structure theorems: purity and inertia, characterizing normal subgroups.
result Characterizes finite-type normal subgroups of mapping class groups of surfaces with Cantor subsets.
New normal subgroups found in mapping class groups.
problem Finding normal subgroups in mapping class groups.
method New windmill apparatus tailored to projection complexes.
result Normal subgroups isomorphic to non-free right-angled Artin groups.
Paper shows mapping class groups are not extremely amenable except for specific cases.
problem Determining when mapping class groups are extremely amenable.
method Utilized Kechris-Pestov-Todorčević machinery.
result Big mapping class groups are not extremely amenable unless the surface is a sphere or a once-punctured sphere.
The paper solves the Andreadakis problem for specific groups using inner automorphisms.
problem Solving the Andreadakis problem for specific groups.
method Generalizing tools from [Dar19b] to study subgroups of IAn, focusing on the behavior of the Andreadakis problem with inner automorphisms.
result The Andreadakis equality holds for the pure braid group and the mapping class group of the n-punctured sphere.
The study finds abundant normal generators for mapping class groups.
problem Understanding normal generation in mapping class groups.
method Analyzing restrictions on invariant subsurfaces and Teichmüller spaces.
result Reducible mapping classes can normally generate mapping class groups based on their asymptotic translation lengths.
Continuous epimorphisms between certain mapping class groups are induced by homeomorphisms.
problem Understanding continuous epimorphisms between specific mapping class groups.
method Analyzing subgroups of mapping class groups of infinite-genus 2-manifolds with no planar ends.
result Continuous epimorphisms are induced by homeomorphisms for the specified subgroups.
In the study of the relation between the mapping class group M of a surface and the theory of finite-type invariants of homology 3-spheres, three subgroups of the mapping class group play a large role. They are the Torelli group, the Johnson subgroup K and a new subgroup L, which contains K, defined by a choice of a La…
Study on cohomology of spin hyperelliptic mapping class groups.
problem Cohomology of spin hyperelliptic mapping class groups.
method Study of G-invariant part of rational cohomology of pure braid groups. result Independence of cohomology dimensions in low degrees and formulas for dimensions.
In this paper we prove that groups as in the title are convex cocompact in the mapping class group.
Explicitly generates right-angled Artin subgroups from mapping classes.
problem Generating right-angled Artin subgroups from mapping classes.
method Explicit constant N depending on the collection of pure mapping classes, showing Nth powers generate the subgroup.
result Explicitly generated subgroups are undistorted.
Study shows mapping class group dimension for surfaces with punctures.
problem Determining the geometric dimension of mapping class groups of surfaces with punctures.
method Proved cocompact classifying space for proper actions with dimension equal to virtual cohomological dimension.
result Proper geometric dimension of mapping class groups of orientable surfaces with punctures.
New examples of surface bundles found over surfaces.
problem Finding compact atoroidal surface bundles.
method Type-preserving homomorphism from knot complement to mapping class group.
result Infinitely many commensurability classes of surface subgroups.
Marginal MAP inference involves making MAP predictions in systems defined with latent variables or missing information. It is significantly more difficult than pure marginalization and MAP tasks, for which a large class of efficient and convergent variational algorithms, such as dual decomposition, exist. In this work,…
Characterizes pseudo-Anosov mapping classes using cluster algebra techniques.
problem Characterize pseudo-Anosov mapping classes purely in terms of shear coordinates.
method Uses cluster algebraic generalization and tropical cluster transformations.
result Algebraic entropies of cluster transformations match topological entropy.
Study shows periodic cohomology of non-orientable surface mapping class groups for odd primes.
problem Investigating periodic cohomology of non-orientable surface mapping class groups for odd primes.
method Using Yagita invariant, cohomology classes, Nielsen realization theorem, and properties of cyclic subgroups of order p.
result The p-period of Ngk is bounded below by 4 when Ngk has p-periodic cohomology, g⩾3 and k⩾0. 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 aim of this paper is to introduce a group containing the mapping class groups of all genus zero surfaces. Roughly speaking, such a group is intended to be a discrete analogue of the diffeomorphism group of the circle. One defines indeed a {\it universal mapping class group of genus zero}, denoted $\B$. The latter i…
Associated to a Thurston map f:S2→S2 with postcritical set P are several different invariants obtained via pullback: a relation on the set of free homotopy classes of curves in S2−P, a linear operator on the free R-module generated by these homotopy classes of curves, a virtual endomorphism on the pur…
Study on dimensions of mapping class groups of non-orientable surfaces.
problem Determining dimensions of mapping class groups for non-orientable surfaces.
method Analyzing cohomological, geometric, and virtual dimensions.
result Equal dimensions for Ng when geq4,5. Study shows certain subgroups of fibered 3-manifolds are convex cocompact.
problem Understanding subgroups of fibered 3-manifolds in mapping class groups.
method Used the Birman exact sequence to show convex cocompactness.
result Finitely generated pseudo-Anosov subgroups are convex cocompact.
Study of wild mapping class groups and their cabled braids.
problem Understanding the structure of wild mapping class groups and their cabled versions.
method Define and study generalizations of pure g-braid groups, establish product decompositions, and introduce fission trees. result Obtain cabled versions of braid groups, related to braid operads.
Classifies homomorphisms from braid groups to mapping class groups of nonorientable surfaces.
problem Classifying homomorphisms from braid groups to mapping class groups of nonorientable surfaces.
method Classifies homomorphisms based on the properties of Dehn twists and crosscap transpositions.
result Every homomorphism is either cyclic or maps generators to distinct Dehn twists or crosscap transpositions.
The study counts arcs on hyperbolic surfaces, providing asymptotic growth formulas.
problem Counting arcs on hyperbolic surfaces with boundaries and cusps.
method Asymptotic analysis of pure mapping class group orbits and arc lengths.
result The number of arcs of bounded length is asymptotically proportional to L6g−6+2(n+p). Study big mapping class groups and their co-Hopfian property, finding new examples and proving injective homomorphisms results.
problem Characterizing co-Hopfian property in big mapping class groups of infinite-type surfaces.
method Constructing examples, proving properties, exploring injective homomorphisms.
result First examples of injective endomorphisms of mapping class groups of infinite-type surfaces that fail to be surjective.
Study kernels of mapping class group representations on surface configuration spaces.
problem Understanding kernels of mapping class group representations on surface configuration spaces.
method Relate kernels to a natural twisted intersection pairing and analyze specific examples.
result Identify subrepresentations and find faithful representations for certain configurations.
Constructs a universal Cannon-Thurston map for a new curve complex.
problem Mapping class groups and their boundaries.
method Using Birman exact sequence, proves hyperbolicity, constructs map.
result Universal Cannon-Thurston map to surviving curve complex boundary.
In this paper we apply the theory of finitely generated FI-modules developed by Church, Ellenberg and Farb to certain sequences of rational cohomology groups. Our main examples are the cohomology of the moduli space of n-pointed curves, the cohomology of the pure mapping class group of surfaces and some manifolds of hi…