Proves certain subgroups of genus 2 handlebody group are convex cocompact.
problem Characterizing subgroups of genus 2 handlebody group.
method Proving convex cocompactness of purely pseudo-Anosov subgroups.
result Finitely generated, purely pseudo-Anosov subgroups are convex cocompact.
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 H1(Γ;Q) and showed that H1(Γ;Q)=0 for specific subgroups. result No finite-index subgroups of the handlebody group contain nontrivial rational abelianization.
Study Torelli subgroups of handlebody groups and their cohomology.
problem Understanding the cohomology of handlebody Torelli groups.
method Introduce Torelli subgroups, use Johnson homomorphisms, and symplectic representations.
result Describe cup products in the first rational cohomology groups of handlebody Torelli groups.
Proves a small set generates a subgroup of surface mapping classes.
problem Generating a subgroup of Torelli group efficiently.
method Proves genus one BP-maps generate handlebody subgroup.
result Proves a normal generating set for handlebody subgroup.
Stable subgroups identified in genus two handlebody group.
problem Characterizing stable subgroups in genus two handlebody group.
method Proving genus two handlebody group is hierarchically hyperbolic, using quasi-isometric embedding properties and Hamenstädt-Hensel construction.
result Stable subgroups identified and characterized.
Handlebody groups are rigid but not flexible in mapping class groups.
problem Understanding rigidity and flexibility of handlebody groups within mapping class groups.
method Analyzing finite index subgroups and constructing specific embeddings.
result Finite index subgroups of handlebody groups are rigid but not all embeddings are conjugate into handlebody groups.
Suppose subgroups A,B<MCG(S) in the mapping class group of a closed orientable surface S are given and let ⟨A,B⟩ be the subgroup they generate. We discuss a question by Minsky asking when ⟨A,B⟩≃A∗A∩BB for handlebody subgroups A,B.
The paper extends Koberda's theorem to handlebody groups.
problem Understanding subgroups of mapping class groups and handlebody groups.
method Analyzing right-angled Artin groups on disk graphs of handlebodies.
result Right-angled Artin groups on disk graphs of handlebodies are subgroups of handlebody groups.
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.
We consider the hyperelliptic handlebody group on a closed surface of genus g. This is the subgroup of the mapping class group on a closed surface of genus g consisting of isotopy classes of homeomorphisms on the surface that commute with some fixed hyperelliptic involution and that extend to homeomorphisms on the …
Handlebody groups are virtual duality groups in positive genus.
problem Understanding the structure of handlebody groups.
method Analyzing mapping class groups and their dualizing modules.
result Description of dualizing modules for handlebody groups.
New parametrization of 3-spheres using Johnson subgroups.
problem Constructing integral homology 3-spheres.
method Intrinsic description of equivalence relation on fourth Johnson subgroup.
result Intrinsic description of equivalence relation using fourth Johnson handlebody subgroups.
The main theorem of this paper generalizes recent results in Dehn surgery to the case of handlebody attachment. We consider attaching handlebodies and solid tori to the boundary of an irreducible, boundary-irreducible, atoroidal and acylindrical 3-manifold. We show that for a large class of homeomorphisms attaching the…
We show that if M is a closed three manifold with a Heegaard splitting with sufficiently big "handlebody distance" then the subgroup of the mapping class group of the Heegaard surface, which extend to both handlebodies is finite. As a corollary, this implies that under the same hypothesis, the mapping class group of …
Study subgroup of mapping class group related to handlebody, answering a question about Johnson homomorphism.
problem Understanding the image of the second Johnson homomorphism for a specific subgroup of mapping class groups.
method Introduced trace-like operators and used them to compute images of Johnson homomorphisms.
result Answered a question about algebraic description of the image of the second Johnson homomorphism.
New invariant for links in handlebodies defined using braid groups and Soergel bimodules.
problem Defining invariants for links in handlebodies.
method Using braid groups and complexes of Soergel bimodules.
result Generalized HOMFLYPT homology for links in handlebodies.
Study of handlebody group intersections with Torelli and Johnson kernels.
problem Intersection of handlebody group with Torelli and Johnson kernels.
method Use Birman--Craggs--Johnson (BCJ) homomorphism to study intersections and compute cup products.
result Determine images of BCJ homomorphism restricted to handlebody group intersections and compute cup products.
New quasi-homomorphisms on surface groups vanish on handlebody groups.
problem Understanding quasi-homomorphisms on mapping class groups and their behavior on handlebody subgroups.
method Constructing infinitely many linearly independent quasi-homomorphisms that vanish on a handlebody subgroup.
result Disproved a conjecture about bounded width in Heegaard splittings and found infinitely many quasi-invariants on Heegaard splittings.
Proves knots in handlebodies can be represented as plats of braids.
problem Representing knots in handlebodies as plats of braids.
method Established Hilden braid group and mixed braid group, proving algebraic equivalence.
result Plat closure representation for knots in handlebodies.
Loxodromic elements are pseudo-Anosov on specific graphs.
problem Characterizing loxodromic elements in specific groups.
method Analyzing subgroups acting on multiarc and curve graphs, and the handlebody group on disk graphs.
result Loxodromic elements are pseudo-Anosov on witness graphs.
Let F a closed connected orientable surface bounding a genus g handlebody H. In this paper we find a finite set of generators for the subgroup E(2,g) of the pure mapping class group of the twice punctured torus PMCG(2,g), consisting of the isotopy classes of homeomorphisms of F which admit an extension to H keeping a p…
Develops Johnson-Morita theory for 3D handlebody groups.
problem Algebraic structure of 3D handlebody groups.
method Analogue of Johnson-Morita theory for handlebody groups.
result Associated graded embeds into a Lie algebra of special derivations.
Let Hg be a genus g handlebody and MCG2n(Tg) be the group of the isotopy classes of orientation preserving homeomorphisms of Tg=∂Hg, fixing a given set of 2n points. In this paper we find a finite set of generators for E2ng, the subgro…
We consider braids on m+n strands, such that the first m strands are trivially fixed. We denote the set of all such braids by Bm,n. Via concatenation Bm,n acquires a group structure. The objective of this paper is to find a presentation for Bm,n using the structure of its corresponding pure braid sub…
We show that if the Hempel distance of a Heegaard splitting is larger than three then the mapping class group of the Heegaard splitting is isomorphic to a subgroup of the mapping class group of the ambient 3-manifold. This implies that given two handlebody sets in the curve complex for a surface that are distance at le…
Handlebody groups reduced to 3 or 4 generators for g ≥ 5 and 3 or 4 for g ≥ 3.
problem Finding minimal generating sets for handlebody groups.
method Using relations in Wajnryb's presentation to reduce the number of generators.
result Handlebody groups M(Vg) are generated by 3 or 4 elements for g≥5 and 3 or 4 for g≥3. We prove the existence of homeomorphisms of a closed, orientable surface of genus 3 or greater that do not extend to any handlebody bounded by the surface. We show that such homeomorphisms exist arbitrarily deep in the Johnson filtration of the mapping class group. The second and third terms of the Johnson filtration a…
Using certain Thom spectra appearing in the study of cobordism categories, we show that the odd half of the Miller-Morita-Mumford classes on the mappping class group of a surface with negative Euler characteristic vanish in integral cohomology when restricted to the handlebody subgroup. This is a special case of a more…
Let $\H_g$ be a genus g handlebody and $\MCG_{2n}(\T_g)$ be the 2n-punctured mapping class group of $\T_g=\partial\H_g$. In this paper we study two particular subgroups of $\MCG_{2n}(\T_g)$ which generalize Hilden groups. As well as Hilden groups are related to plate closures of braids, these generalizations are re…
Study of large group actions on surfaces, focusing on Hurwitz and handlebody groups.
problem Characterizing and understanding group actions on surfaces, especially maximal handlebody and Hurwitz groups.
method Analyzing various group actions, comparing Hurwitz and handlebody groups, and examining bounding actions.
result Relationship between Hurwitz groups and maximal handlebody groups, and insights into geometric bounding actions.
The study characterizes group actions on 3-manifolds achieving maximal order.
problem Understanding maximal group actions on 3-manifolds.
method Analyzing finite group-actions on handlebodies and 3-manifolds.
result Characterization of 3-manifolds and groups achieving maximal order.
The paper studies the topology and geometry of simple orbifolds, generalizing concepts from simple polytopes.
problem Understanding the topology and geometry of simple orbifolds.
method Generalizing concepts from simple polytopes to simple orbifolds, focusing on simple handlebodies.
result Characterization of orbifold-aspherical properties and the existence of rank-two free abelian subgroups in terms of combinatorics.
The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable 3-dimensional handlebody of genus g can be enumerated in terms of sets of generators of G. They correspond to the equivalence classes of generating n-vectors of elements of G, where n=1+(g-1)/|G|, u…
The study of periodic subgroups in homeomorphism groups of manifolds.
problem Burnside problem for homeomorphism groups of manifolds.
method Analyzing surface and circle homeomorphism groups, extending Tits alternative.
result Every finitely generated periodic subgroup is finite for most manifolds.
This paper gives an exposition of relative weight filtrations on completions of mapping class groups associated to a stable degeneration of marked genus g curves. These relative weight filtrations have been constructed using Galois theory (with Matsumoto) and Hodge theory (with Pearlstein and Terasoma). It is shown tha…
Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus g with one boundary component to ∧3H, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping cla…
We find polynomial-time solutions to the word problem for free-by-cyclic groups, the word problem for automorphism groups of free groups, and the membership problem for the handlebody subgroup of the mapping class group. All of these results follow from observing that automorphisms of the free group strongly resemble s…
The paper proves unique factorization of knotted handlebodies and examines handlebody-knot symmetry.
problem Uniqueness of factorization of knotted handlebodies along decomposing 2-spheres.
method Analyzes factorization of knotted handlebodies in the 3-sphere, proving uniqueness for specific cases.
result Determines chirality of 6_{10} handlebody-knot and constructs an infinite family of hyperbolic handlebody-knots.
Complete classification of handlebody links up to six crossings.
problem Classifying handlebody links with specific properties.
method Complete enumeration and ambient isotopy consideration.
result Complete classification of handlebody links with six crossings or less.
New invariants for handlebody-links using Milnor's invariants.
problem Classifying HL-homotopy classes of handlebody-links.
method Constructing invariants using Milnor's link-homotopy invariants.
result A bijection between HL-homotopy classes and tensor product space.
New algebraic structures help color handlebody-knots for invariants.
problem Coloring handlebody-knots for invariants.
method Introducing G-families of biquandles, partially multiplicative biquandles, and group decomposable biquandles. result Enhanced polynomial invariant of handlebody-knots using group G. Paper defines numerical criteria to test handlebody link irreducibility.
problem Determining the irreducibility of handlebody links.
method A set of numerical criteria to test handlebody links for irreducibility.
result Effective method to recognize irreducibility of handlebody knots and most handlebody links.
New algebraic structure for distinguishing twisted virtual handlebody-links.
problem Distinguishing twisted virtual handlebody-links.
method Introducing twisted virtual bikeigebras and using them to define invariants.
result New invariants distinguish some twisted virtual handlebody-links.
Formula for signature of handlebody bundles, interpreting cohomology.
problem Signature of handlebody bundles and their monodromy.
method Explicit formula using homological monodromy.
result Cobounding function of Meyer's signature cocycle on handlebody group.
High-dimensional handlebodies are shown to be products of simpler shapes.
problem Understanding the structure of high-dimensional handlebodies.
method Introducing Kirby diagrams to simplify the structure of k-handlebodies. result High-dimensional handlebodies are products of simpler shapes.
The JSJ decomposition helps classify genus two handlebody-knots.
problem Classifying genus two handlebody-knots.
method JSJ decomposition and annulus classification applied to handlebody-knot exteriors.
result Genus two handlebody-knots classified using annulus diagrams.
We give examples of knots in a genus 2 handlebody which have nontrivial Dehn surgeries yielding handlebodies and show that these knots are not 1--bridge.
Finite presentation for a specific group in 3D handlebody topology.
problem Finding a finite presentation for a specific group in 3D handlebody topology.
method Constructed a finite presentation for the liftable Hilden group to derive the presentation for the balanced superelliptic handlebody group.
result A finite presentation was given for the balanced superelliptic handlebody group.