Characterizes a general range decreasing group homomorphism.
problem Understanding range decreasing group homomorphisms in the entire mapping group.
method Characterization of a general range decreasing group homomorphism.
result Computes a particular class of homomorphisms and identifies all range decreasing group homomorphisms on specific mapping groups.
The hyperelliptic mapping class group has been studied in various contexts within topology and algebraic geometry. What makes this study tractable is that there is a surjective map from the hyperelliptic mapping class group to a mapping class group of a punctured sphere. The more general family of superelliptic mapping…
The article explores the mapping class group using unicellular maps and provides filtrations.
problem Understanding the structure of the mapping class group.
method Using unicellular maps and surgeries, the article describes the mapping class group.
result Provides filtrations of the mapping class group.
Classifies manifolds with dense conjugacy classes in their mapping class groups.
problem Classifying manifolds based on conjugacy classes in their mapping class groups.
method Analyzing connected orientable 2-manifolds and their mapping class groups.
result Mapping class groups of certain manifolds have dense conjugacy classes.
Explicit presentations found for asymptotically rigid mapping class groups.
problem Understanding the structure of asymptotically rigid mapping class groups.
method Using a graph of groups structure, we compute explicit presentations.
result Computed explicit presentations for asymptotically rigid mapping class groups of surfaces.
We show that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups, induces an infinite loop map on the classifying spaces of the stable groups after plus construction. The proof uses automorphisms of free groups with boundaries which play the role of mapping class groups o…
New descriptions of a subgroup in mapping class groups.
problem Understanding the normal subgroup generated by genus n n n bounding pair maps. method Using Chillingworth and Casson-Morita invariants.
result Two descriptions of the subgroup generated by genus n n n bounding pair maps. Smooth C 1 C^1 C 1 contact maps are always smooth in rigid Carnot groups.
problem Smoothness of C 1 C^1 C 1 contact maps in rigid Carnot groups. method Analyzing C ∞ C^\infty C ∞ -rigid Carnot groups to show C 1 C^1 C 1 -contact maps are smooth. result Smooth C 1 C^1 C 1 contact maps are always smooth in rigid Carnot groups. Harmonic maps studied in sub-Riemannian geometry for Lie groups.
problem Defining and characterizing harmonic maps in sub-Riemannian settings.
method Generalization of Riemannian harmonic maps to sub-Riemannian manifolds and Lie groups.
result Conditions for sub-Riemannian harmonic maps and their classification.
We show that finite index subgroups of the handlebody group are rigid in their ambient mapping class group: any injective map of a finite index subgroup of the genus g g g handlebody group into the genus g g g mapping class group is conjugation by a mapping class group element. On the other hand, we construct an injection …
In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show th…
Overview of infinite surface mapping class groups.
problem Understanding mapping class groups of infinite surfaces.
method Survey of recent research findings.
result Recent developments in mapping class groups of infinite surfaces.
Study of hyperelliptic mapping class groups with applications and profinite completions.
problem Understanding hyperelliptic mapping class groups and their properties.
method Defined and studied hyperelliptic mapping class groups, applied theory to counterexamples, and examined profinite completions.
result Found a counterexample to a conjecture about mapping class groups and extended congruence subgroup property.
Unified framework for studying Torelli group and congruence subgroup maps.
problem Understanding maps between Torelli group and congruence subgroup.
method Unified algorithms for computing Rochlin invariants and maps.
result Unified proofs and explicit evaluations of maps on Dehn twists.
Article provides conditions for embedding braid groups into mapping class groups.
problem Embedding braid groups into mapping class groups.
method Necessary and sufficient condition for embedding finite index subgroups.
result Conditions for embedding braid groups into mapping class groups.
This thesis introduces big mapping class groups and their structure.
problem Understanding mapping class groups of infinite-type surfaces.
method Systematic introduction and analysis of structure and topological generation.
result Key differences from finite-type mapping class groups.
Finite presentations for mapping class groups of surfaces and surfaces with points/boundaries.
problem Finding finite presentations for balanced superelliptic mapping class groups.
method Construct finite presentations for corresponding liftable mapping class groups in a different generating set.
result Finite presentations for balanced superelliptic mapping class groups of various surfaces.
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.
Three elements generate balanced superelliptic mapping class groups.
problem Generating balanced superelliptic mapping class groups.
method Proving groups are generated by three elements through normalizers and liftable mapping class groups.
result Balanced superelliptic mapping class groups are generated by three elements.
We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that ma…
Study the relationship between orbit braid group and equivariant mapping class group on surfaces.
problem Understanding the relationship between mapping class groups and braid groups with group actions.
method Using the fibration F 0 G M i g h t a r r o w F ( M / G , n ) \mathcal{F}_{0}^{G}M
ightarrow F(M/G,n) F 0 G M i g h t a r r o w F ( M / G , n ) and exact sequence. result The conclusion is closely connected with the braid group of the quotient space.
Proves asymptotic mapping class groups of Cantor manifolds are of type F_infinity.
problem Finiteness properties of asymptotic mapping class groups.
method General theorem deducing asymptotic mapping class groups of Cantor manifolds are of type F_infinity under certain hypotheses.
result Asymptotic mapping class groups of Cantor manifolds are of type F_infinity.
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.
Study maps surface configurations to Heisenberg homologies for mapping class groups.
problem Understanding Mapping Class Groups of punctured surfaces.
method Action of mapping classes on Heisenberg homologies of surface configurations.
result Representations of Mapping Class Groups derived from Heisenberg homologies.
New subgroups of mapping class groups constructed for infinite-type surfaces.
problem Constructing new subgroups of mapping class groups for infinite-type surfaces.
method Utilization of special homeomorphisms called shift maps and multipush maps.
result Countably (and uncountably in certain cases) many non-conjugate embeddings of subgroups into mapping class groups.
The paper proves mapping class groups of closed surfaces are simply connected at infinity.
problem Understanding connectivity at infinity for mapping class groups of surfaces.
method Proved a general simply connected at infinity result for finitely presented groups.
result All mapping class groups of closed surfaces of genus ≥ 3 are simply connected at infinity.
A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasi-trees. The groups we can handle include non-elementary (relatively) hyperbolic groups, rank 1 CAT(0) groups, mapping class groups and Out(Fn). As an application, we show that mapping clas…
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.
Computes mapping class groups of 4-manifolds with boundary.
problem Computing mapping class groups for 4-manifolds with boundary.
method Topological and smooth methods applied to compact, simply connected 4-manifolds.
result Description of topological and stable smooth mapping class groups.
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.
We show that the mapping class group of a compact orientable surface with higher complexity has the following extreme rigidity in the sense of measure equivalence: if the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernel. Moreover, we describe all lattice emb…
We prove that both the hyperelliptic mapping class group and the extended hyperelliptic mapping class group are generated by two torsion elements. We also compute the index of the subgroup of the hyperelliptic mapping class group which is generated by involutions and we prove that the extended hyperelliptic mapping cla…
The paper computes the mapping class group of certain 6-manifolds.
problem Computing the mapping class group of specific 6-manifolds.
method Algebraic properties and homology groups of the mapping class group were determined.
result The mapping class group is residually finite and virtually torsion-free.
The paper defines and proves the existence of train track maps on graphs of groups.
problem Understanding homotopy equivalences in graphs of groups.
method Developed the theory of train track maps on graphs of groups, defining maps and homotopy equivalences.
result Any homotopy equivalence of a graph of groups may be represented by a relative train track map under certain conditions.
Maps between classifying spaces for certain groups are studied, with rational cohomology results.
problem Analyzing maps between classifying spaces for specific groups.
method Study of maps Θ Θ Θ between null-components of homomorphism and based map spaces. result Surjectivity of map Θ Θ Θ in rational cohomology for certain groups, and non-surjectivity for others. New findings on generating mapping class groups with specific torsion elements.
problem Understanding the structure of mapping class groups through torsion elements.
method Analyzing the orders of torsion elements required to generate mapping class groups of surfaces of varying genus.
result For specific genera, mapping class groups can be generated by two torsion elements of particular orders.
Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.
problem Understanding asymptotic dimension of big mapping class groups of infinite type surfaces.
method Analyzing big mapping class groups with coarsely bounded generating sets and essential shifts.
result Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.
Extended logarithm for solvable elements in mapping class groups.
problem Logarithm of Johnson map extension to solvable elements.
method Extension to exponential solvable elements in mapping class groups using solvable Lie groups.
result Solvability of extended logarithm.
Study the first homology group for a specific mapping class group.
problem Calculating the first homology group for a specific mapping class group.
method Determined using coefficients in H 1 ( N ; Z ) H_1(N;\mathbb{Z}) H 1 ( N ; Z ) for a non-orientable surface. result Determined the first homology group for the mapping class group of a specific surface.
Invites geometers to Garside theory for mapping class groups.
problem None explicitly stated, but related to geometric group theory.
method Garside theory applied to mapping class groups.
result No specific key result mentioned in the abstract.
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 . The study characterizes subgroups of mapping tori of free groups.
problem Characterizing subgroups of mapping tori of free groups.
method Using canonical maximal sub-mapping tori and relative hyperbolicity.
result Characterizes locally quasi-convex hyperbolic groups.
Two elements generate extended mapping class groups of certain surfaces.
problem Generating extended mapping class groups with specific elements.
method Analyzing finite order elements and isotopy classes of homeomorphisms.
result Extended mapping class groups of certain surfaces are generated by two elements of finite order.
Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.
problem Classifying non-linear proper Fredholm maps between Hilbert spaces.
method Using stable homotopy groups of spheres to classify maps up to proper homotopy.
result Determines the non-trivial kernel of the map from stable homotopy groups to non-linear proper Fredholm maps.
Proves congruence subgroup property for mapping class groups of hyperbolic surfaces.
problem Residual finiteness of hyperbolic groups and congruence subgroup property for mapping class groups.
method Assumption of residual finiteness of hyperbolic groups leads to proof of congruence subgroup property.
result Congruence subgroup property for mapping class groups of hyperbolic surfaces.
New examples of subgroups in mapping class groups are found.
problem Understanding subgroups in mapping class groups.
method Constructing new families of parabolically geometrically finite subgroups.
result These subgroups are undistorted in Mod( S S S ). Study shows mapping class groups are one-ended for surfaces with at least one end.
problem Analyzing the number of ends in mapping class groups of surfaces.
method Proving the associated translatable curve graph is one-ended, quasi-isometric to the mapping class group.
result Mapping class groups are one-ended for surfaces with at least one end of discrete type.
Characterizes and analyzes the large scale geometry of big mapping class groups of surfaces.
problem Analyzing the large scale geometry of big mapping class groups of surfaces with a unique maximal end.
method Building on previous work, the paper characterizes and analyzes the large scale geometry of big mapping class groups of surfaces with a unique maximal end.
result Proves that any locally CB big mapping class group is CB generated and gives an explicit criterion for determining which big mapping class groups are CB generated.