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.
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.
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 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.
The paper provides presentations for mapping class groups and cluster automorphism groups of surfaces.
problem Presentations of mapping class groups of surfaces stabilizing boundaries.
method Gave presentations of mapping class groups of marked surfaces stabilizing boundaries.
result Presented cluster automorphism groups of cluster algebras from surfaces.
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.
Proves homology of mapping class groups for infinite-type surfaces.
problem Homology of mapping class groups for infinite-type surfaces.
method Modification of Mather's argument and homological stability result.
result Homology of mapping class groups determined for binary tree surfaces.
New stability theorem for nonorientable surfaces mapping class groups.
problem Stability of homology groups of mapping class groups of nonorientable surfaces.
method Galatius--Kupers--Randal-Williams framework of cellular E2-algebras. result New best known stability range for homology of nonorientable surfaces.
Infinitely many unique ways to generate a surface's mapping class group.
problem Identifying unique ways to generate mapping class groups.
method Analyzing Nielsen equivalence classes of two-element generators.
result There are infinitely many unique ways to generate a surface's mapping class group.
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.
Simplified presentations for non-orientable surface mapping class groups.
problem Presenting the mapping class group of non-orientable surfaces.
method Provided simpler infinite presentations.
result Simplified presentations for the group.
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.
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.
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.
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.
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.
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.
Characterizes multitwists in surface mapping class groups.
problem Understanding multitwists in surface mapping class groups.
method Characterization through braid relations.
result Characterized multitwists satisfying braid relations.
Surface Houghton groups are studied for their mapping class properties.
problem Understanding the mapping class properties of surface Houghton groups.
method Analyzing the asymptotic rigidity and monodromy homeomorphisms of fibered components.
result Surface Houghton groups are of type Fn−1 but not of type FPn. 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.
This paper provides an infinite presentation for a subgroup of mapping class groups of non-orientable surfaces.
problem Finite presentations for mapping class groups of non-orientable surfaces.
method Using Stukow's finite presentation and Birman exact sequences.
result An infinite presentation for the twist subgroup of the mapping class group of a compact non-orientable surface.
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.
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.
The paper explores non-amenability in infinite-type surfaces and graphs.
problem Determining non-amenability in mapping class groups of infinite-type surfaces and graphs.
method Analyzes mapping class groups of infinite-type surfaces and graphs, provides examples and exhibits classes of groups.
result Completely determines non-amenability of mapping class groups of infinite-type surfaces and graphs.
Study shows certain mapping class groups cannot be realized as subgroup of homeomorphisms.
problem Proving non-realizability of specific mapping class groups.
method Analyzing compactly supported and full mapping class groups of surfaces with genus 3 or order 6 symmetries.
result Proven non-realizability of mapping class groups for surfaces with genus 3 or order 6 symmetries.
The paper explores the twisted Rokhlin property in mapping class groups of surfaces.
problem Classifying surfaces whose mapping class groups have the twisted Rokhlin property.
method Generalizing the Rokhlin property to the twisted version, the authors classify surfaces based on their mapping class groups' properties.
result The mapping class groups of connected orientable infinite-type surfaces without boundaries have the twisted Rokhlin property, while those of other surfaces do not.
Birman-Lubotzky-McCarthy proved that any abelian subgroup of the mapping class groups for orientable surfaces is finitely generated. We apply Birman-Lubotzky-McCarthy's arguments to the mapping class groups for non-orientable surfaces. We especially find a finitely generated group isomorphic to a given torsion-free sub…
Study of CB generating sets for infinite-type surfaces.
problem Understanding CB generating sets for infinite-type surfaces.
method Constructing CB generating sets for specific infinite-type surfaces.
result Examples of surfaces with and without CB generating sets.
New groups found that don't virtually algebraically fiber, related to mapping class group orbits.
problem Existence of finite orbits for higher Prym representations of the mapping class group.
method Study of surface-by-surface and surface-by-free groups.
result Existence of free-by-free and free-by-surface groups that do not algebraically fiber.
Study on self-similar surfaces and their mapping class groups generated by involutions.
problem When do big mapping class groups of self-similar surfaces generated by involutions?
method Investigation of self-similar surfaces with self-similar ends, focusing on infinite and one maximal ends.
result For self-similar surfaces with infinite maximal ends, their mapping class groups are generated by involutions and are uniformly perfect.
Study shows pants graph automorphisms match mapping class groups of nonorientable surfaces.
problem Understanding automorphisms of pants graphs on nonorientable surfaces.
method Analyzing mapping class groups and proving isomorphism.
result Automorphism group of pants graphs isomorphic to mapping class groups.
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 H1(N;Z) for a non-orientable surface. result Determined the first homology group for the mapping class group of a specific surface.
Study homology groups for non-orientable surfaces with boundary.
problem Determine the first homology group for mapping class groups of non-orientable surfaces.
method Calculate the first homology group with twisted coefficients.
result Explicitly determined the first homology group for various mapping class groups.
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. 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.
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.
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.
Multitwists cannot generate all compactly supported mapping class groups on infinite-type surfaces.
problem Generating compactly supported mapping class groups on infinite-type surfaces.
method Analyzing closure of compactly supported mapping class groups and their relation to multitwists.
result Closure of compactly supported mapping class groups is not generated by multitwists.
We construct new monomorphisms between mapping class groups of surfaces. The first family of examples injects the mapping class group of a closed surface into that of a different closed surface. The second family of examples are defined on mapping class groups of once-punctured surfaces and have quite curious behaviour…
It is a classical result of Powell that pure mapping class groups of connected, orientable surfaces of finite type and genus at least three are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology g…
Study mapping class group actions on surface fundamental groups.
problem Understanding group actions on surface fundamental groups.
method Computing image of mapping class group generators as outer automorphisms.
result Computed images of mapping class group generators on fundamental groups.
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.
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.
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…
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. Involutions generate mapping class groups of infinite surfaces.
problem Generating involutions for mapping class groups of infinite surfaces.
method Analyzing infinite surfaces with n ends, showing involutions generate groups for n ≥ 6 and n ≥ 3.
result Involutions generate mapping class groups for n ≥ 6 and n ≥ 3.
Abstract: Mapping class groups act on cohomology of surfaces via Hochschild cohomology.
problem Understanding the action of mapping class groups on cohomology of surfaces.
method Associate cochain complexes to surfaces, with mapping class groups acting projectively on cohomology.
result Projective action of mapping class groups on Hochschild cohomology of Hopf algebras.
Study shows conjugacy of torsion in genus 2 surfaces.
problem Torsion elements in mapping class groups of surfaces.
method Proved congruence subgroup property for centralizers of finite subgroups.
result Torsion elements in surfaces of genus ≤ 2 are conjugacy distinguished.