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 these handlebodies, the fundamental group of the resulting manifold contains the fundamental group of a closed surface of genus at least two.
We prove that the handlebody subgroup of the Torelli group of an orientable surface is generated by genus one BP-maps. As an application, we give a normal generating set for the handlebody subgroup of the level $d$ mapping class group of an orientable surface.
Suppose subgroups $A,B < MCG(S)$ in the mapping class group of a closed orientable surface $S$ are given and let $\langle A, B \rangle$ be the subgroup they generate. We discuss a question by Minsky asking when $\langle A, B \rangle \simeq A*_{A \cap B} B$ for handlebody subgroups $A,B$.
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.
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…
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 $M$ is finite.
We construct infinitely many linearly independent quasi-homomorphisms on the mapping class group of a Riemann surface with genus at least two which vanish on a handlebody subgroup. As a corollary, we disprove a conjecture of Reznikov on bounded width in Heegaard splittings. Another corollary is that there are infinitely many linearly independent quasi-invariants on the Heegaard splittings of three-manifolds.
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 are the well-known Torelli group and Johnson subgroup, respectively. Richard Hain has obtained very similar results by different methods.
Koberda proved that if a graph $Γ$ is a full subgraph of a curve graph $\mathcal{C}(S)$ of an orientable surface $S$, then the right-angled Artin group $A(Γ)$ on $Γ$ is a subgroup of the mapping class group ${\rm Mod}(S)$ of $S$. On the other hand, for a sufficiently complicated surface $S$, Kim-Koberda gave a graph $Γ$ which is not contained in $\mathcal{C}(S)$, but $A(Γ)$ is a subgroup of ${\rm Mod}(S)$. In this paper, we prove that if $Γ$ is a full subgraph of a disk graph $\mathcal{D}(H)$ of a handlebody $H$, then $A(Γ)$ is a subgroup of the handlebody group ${\rm Mod}(H)$ of $H$. Further,…
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 general theorem valid in all dimensions: universal characteristic classes made from monomials in the Pontrjagin classes (and even powers of the Euler class) vanish when pulled back from BDiff(\partial W) to BDiff(W).
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 least four apart, the group of automorphisms of the curve complex that preserve both handlebody sets is finite.
Let $\textup{H}_g$ be a genus $g$ handlebody and $\textup{MCG}_{2n}(\textup{T}_g)$ be the group of the isotopy classes of orientation preserving homeomorphisms of $\textup{T}_g=\partial\textup{H}_g$, fixing a given set of $2n$ points. In this paper we find a finite set of generators for $\mathcal{E}_{2n}^g$, the subgroup of $\textup{MCG}_{2n}(\textup{T}_g)$ consisting of the isotopy classes of homeomorphisms of $\textup{T}_g$ admitting an extension to the handlebody and keeping fixed the union of $n$ disjoint properly embedded trivial arcs. This result generalizes a previous one obtained by th…
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 related to Heegaard splittings of manifolds and to bridge decompositions of links. Connections between these subgroups and motion groups of links in closed 3-manifolds are also provided.
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 properly embedded trivial arc fixed. This subgroup turns out to be important for the study of knots in closed 3-manifolds via (g,1)-decomposition. In fact, the knots represented by the isotopy classes belonging to the same left cosets of E(2,g) in PMCG(2,g) are equivalent.
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|, under Nielsen equivalence (or weak Nielsen equivalence). For abelian and dihedral G, this allows a complete determination of the equivalence and weak equivalence classes of actions for all genera. Additional information is obtained for solvable groups and for the groups PSL(2,3^p)…
We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g. The maximal possible order of a finite group-action of an orientable or nonorientable handlebody of genus g > 1 is 24(g-1), and in the present paper we characterize the 3-manifolds M and groups G for which the maximal possible order |G| = 24(g-1) is obtained, for some G-invariant Heegaard splitting of genus g > 1. If M is reducible then it is obtained by doubling an action of maximal possible order 2…
Recall that a group is called large if it has a finite index subgroup which surjects onto a non-abelian free group. By work of Agol and Cooper-Long-Reid, most 3-manifold groups are large; in particular, the fundamental groups of hyperbolic 3-manifolds are large. In previous work, the first author gave examples of closed, hyperbolic 3-manifolds with arbitrarily large first homology rank but whose fundamental groups do not surject onto a non-abelian free group. We call a group very large if it surjects onto a non-abelian free group. In this paper, we consider the question of whether the groups o…
