Handlebody Groups Research Articles

Purely pseudo-Anosov subgroups of the genus 2 handlebody group

We prove that finitely generated, purely pseudo-Anosov subgroups of the genus 2 handlebody group are convex cocompact.

A finite presentation for the balanced superelliptic handlebody group

The balanced superelliptic handlebody group is the normalizer of the transformation group of the balanced superelliptic covering space in the handlebody group of the total space. We give a finite presentation for the balanced superelliptic handlebody group. To give this presentation, we construct a finite presentation for the liftable Hilden group.

Classification of Consistent Systems of Handlebody Group Representations

Abstract The classifying spaces of handlebody groups form a modular operad. Algebras over the handlebody operad yield systems of representations of handlebody groups that are compatible with gluing. We prove that algebras over the modular operad of handlebodies with values in an arbitrary symmetric monoidal bicategory $\mathcal{M}$ (we introduce for these the name ansular functor) are equivalent to self-dual balanced braided algebras in $\mathcal{M}$. After specialization to a linear framework, this proves that consistent systems of handlebody group representations on finite-dimensional vector spaces are equivalent to ribbon Grothendieck-Verdier categories in the sense of Boyarchenko-Drinfeld. Additionally, it produces a concrete formula for the vector space assigned to an arbitrary handlebody in terms of a generalization of Lyubashenko’s coend. Our main result can be used to obtain an ansular functor from vertex operator algebras subject to mild finiteness conditions. This includes examples of vertex operator algebras whose representation category has a non-exact monoidal product.

The handlebody group and the images of the second Johnson homomorphism

Given an oriented surface bounding a handlebody, we study the subgroup of its mapping class group defined as the intersection of the handlebody group and the second term of the Johnson filtration: $\mathcal{A} \cap J_2$. We introduce two trace-like operators, inspired by Morita's trace, and show that their kernels coincide with the images by the second Johnson homomorphism $\tau_2$ of $J_2$ and $\mathcal{A} \cap J_2$, respectively. In particular, we answer by the negative to a question asked by Levine about an algebraic description of $\tau_2(\mathcal{A} \cap J_2)$. By the same techniques, and for a Heegaard surface in $S^3$, we also compute the image by $\tau_2$ of the intersection of the Goeritz group $\mathcal{G}$ with $J_2$.

Lagrangian traces for the Johnson filtration of the handlebody group

In this paper, we define trace-like operators on a subspace of the space of derivations of the free Lie algebra generated by the first homology group H of a surface Σ. This definition depends on the choice of a Lagrangian of H, and we call these operators the Lagrangian traces. We suppose that Σ is the boundary of a handlebody with first homology group H′, and we show that, in degree greater than 2, the Lagrangian traces corresponding to the Lagrangian Ker(H→H′) vanish on the image by the Johnson homomorphisms of the elements of the Johnson filtration that extend to the handlebody.

A small generating set for the balanced superelliptic handlebody group

The balanced superelliptic handlebody group is the normalizer of the transformation group of the balanced superelliptic covering space in the handlebody group of the total space. We prove that the balanced superelliptic mapping class group is generated by four elements. To prove this, we also proved that the liftable Hilden group is generated by three elements. This generating set for the liftable Hilden group is minimal except for some hyperelliptic cases and the generating set for the balanced superelliptic mapping class group above is also minimal for several cases.

Stable subgroups of the genus 2 handlebody group

We show that a finitely generated subgroup of the genus two handlebody group is stable if and only if the orbit map to the disk graph is a quasi-isometric embedding. To this end, we prove that the genus two handlebody group is a hierarchically hyperbolic group, and that the maximal hyperbolic space in the hierarchy is quasi-isometric to the disk graph of a genus two handlebody by appealing to a construction of Hamenst\"adt-Hensel. We then utilize the characterization of stable subgroups of hierarchically hyperbolic groups provided by Abbott-Behrstock-Berlyne-Durham-Russell. We also present several applications of the main theorems, and show that the higher genus analogues of the genus two results do not hold.

Cyclic Framed Little Disks Algebras, Grothendieck–Verdier Duality And Handlebody Group Representations

AbstractWe characterize cyclic algebras over the associative and the framed little 2-disks operad in any symmetric monoidal bicategory. The cyclicity is appropriately treated in a coherent way, that is up to coherent isomorphism. When the symmetric monoidal bicategory is specified to be a certain symmetric monoidal bicategory of linear categories subject to finiteness conditions, we prove that cyclic associative and cyclic framed little 2-disks algebras, respectively, are equivalent to pivotal Grothendieck–Verdier categories and ribbon Grothendieck–Verdier categories, a type of category that was introduced by Boyarchenko–Drinfeld based on Barr’s notion of a $\star$-autonomous category. We use these results and Costello’s modular envelope construction to obtain two applications to quantum topology: I) We extract a consistent system of handlebody group representations from any ribbon Grothendieck–Verdier category inside a certain symmetric monoidal bicategory of linear categories and show that this generalizes the handlebody part of Lyubashenko’s mapping class group representations. II) We establish a Grothendieck–Verdier duality for the category extracted from a modular functor by evaluation on the circle (without any assumption on semisimplicity), thereby generalizing results of Tillmann and Bakalov–Kirillov.

(Un)distorted stabilisers in the handlebody group

We study geometric properties of stabilisers in the handlebody group. We find that stabilisers of meridians are undistorted, while stabilisers of primitive curves or annuli are exponentially distorted for large enough genus.

The Meyer function on the handlebody group

We give an explicit formula for the signature of handlebody bundles over the circle in terms of the homological monodromy. This gives a cobounding function of Meyer's signature cocycle on the mapping class group of a $3$-dimensional handlebody, i.e., the handlebody group. As an application, we give a topological interpretation for the generator of the first cohomology group of the hyperelliptic handlebody group.

The Geometry of the Handlebody Groups II: Dehn Functions

We show that the Dehn function of the handlebody group is exponential in any genus g≥3. On the other hand, we show that the handlebody group of genus 2 is cubical and biautomatic and therefore has a quadratic Dehn function.

Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex

Let M be a hyperbolic fibered 3-manifold whose first Betti number is greater than 1 and let S be a fiber with pseudo-Anosov monodromy \psi . We show that there exists a sequence (R_n, \psi_n) of fibers and monodromies contained in the fibered cone of (S,\psi) such that the asymptotic translation length of \psi_n on the curve complex \mathcal C(R_n) behaves asymptotically like 1=j .Rn/j2. As applications, we can reprove the previous result by Gadre–Tsai that the minimal asymptotic translation length of a closed surface of genus g asymptotically behaves like 1/|\chi(R_n)|^2 . We also show that this holds for the cases of hyperelliptic mapping class group and hyperelliptic handlebody group.

Rigidity and flexibility for handlebody groups

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 handlebody group into the genus g mapping class group is conjugation by a mapping class group element. On the other hand, we construct an injection of the genus g handlebody group into a genus h > g mapping class group which is not conjugate into a handlebody group.

The asymptotic behavior of the minimal pseudo-ANOSOV dilatations in the hyperelliptic 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. We prove that the logarithm of the minimal dilatation (i.e, the minimal entropy) of all pseudo-Anosov elements in the hyperelliptic handlebody group of genus $g$ is comparable to $1/g$. This means that the asymptotic behavior of the minimal pseudo-Anosov dilatation of the subgroup of genus $g$ in question is the same as that of the ambient mapping class group of genus $g$. We also determine finite presentations of the hyperelliptic handlebody groups.

CANNON–THURSTON MAPS FOR KLEINIAN GROUPS

We show that Cannon–Thurston maps exist for degenerate free groups without parabolics, that is, for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon–Thurston maps for surface groups, we show that Cannon–Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon–Thurston maps for degenerate free groups without parabolics correspond to endpoints of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon–Thurston maps for arbitrary finitely generated Kleinian groups without parabolics.

Right-angled Artin groups on finite subgraphs of disk graphs

Koberda proved that if a graph Γ is a full subgraph of a curve graph C(S) of an orientable surface S, then the right-angled Artin group A(Γ) on Γ is a subgroup of the mapping class group Mod(S) of S. On the other hand, for a sufficiently complicated surface S, Kim–Koberda gave a graph Γ which is not contained in C(S), but A(Γ) is a subgroup of Mod(S). In this paper, we prove that if Γ is a full subgraph of a disk graph D(H) of a handlebody H, then A(Γ) is a subgroup of the handlebody group Mod(H) of H. Further, we show that there is a graph Γ which is not contained in some disk graphs, but A(Γ) is a subgroup of the corresponding handlebody groups.

Realisation and dismantlability

We study dismantling properties of the arc, disc and sphere graphs. We prove that any finite subgroup H of the mapping class group of a surface with punctures, the handlebody group, or Out(F_n) fixes a filling (resp. simple) clique in the appropriate graph. We deduce realisation theorems, in particular the Nielsen Realisation Problem in the case of a nonempty set of punctures. We also prove that infinite H have either empty or contractible fixed point sets in the corresponding complexes. Furthermore, we show that their spines are classifying spaces for proper actions for mapping class groups and Out(F_n).

Isoperimetric inequalities for the handlebody groups

We show that the mapping class group of a handlebody of genus at least 2 has a Dehn function of at most exponential growth type.

Separable subgroups of mapping class groups

We investigate separability questions for the mapping class group of a surface. While this group is not subgroup separable in general, we prove a large family of interesting subgroups are separable. This includes many classically studied subgroups such as solvable subgroups, Heegaard and Handlebody groups, geometric subgroups, and all the terms in the Johnson filtration.

Heegaard diagrams and handlebody groups

Heegaard diagrams on the boundary of a handlebody are studied from the dynamics systems point of view. A relationship between the strongly irreducible condition of Casson–Gordan and the Masur's domain of discontinuity for the action of the handlebody group is established.