Joseph Maher
We show that for certain hyperbolic 3-manifolds, all boundary slopes are slopes of immersed incompressible surfaces, covered by incompressible embeddings in some finite cover. The manifolds include hyperbolic punctured torus bundles and hyperbolic two-bridge knots.
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The paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3-manifold, which generalizes an invariant due to Rubinstein and Wang. The subsurface is virtually embedded if and only if the almost fiber part is aspiral, and in this case, the subsurface is virtually a leaf of a taut foliation. Besides other consequences, examples are exhibited that non-geometric 3-manifolds with no Seifert fibered pieces may contain essentially immersed but not virtually embedded closed subsurfaces.
We show that many 3-manifold groups have no nonabelian surface subgroups. For example, any link of an isolated complex surface singularity has this property. In fact, we determine the exact class of closed graph-manifolds which have no immersed pi_1-injective surface of negative Euler characteristic. We also determine the class of closed graph manifolds which have no finite cover containing an embedded such surface. This is a larger class. Thus, manifolds M^3 exist which have immersed pi_1-injective surfaces of negative Euler characteristic, but no such surface is virtually embedded (finitely …
We show that if a group is not virtually cyclic and is hyperbolic relative to a family of proper subgroups, then it has a hyperbolically embedded subgroup which contains a finitely generated non-abelian free group as a finite index subgroup.
Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group $G$ (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, ${\rm Out}(F_n)$, \dots) embeds via the natural restriction maps into the inverse limit of the second bounded cohomologies of its virtually free subgroups, and in fact even into the inverse limit of the second bounded cohomologies of its hyperbolically embedded virtually free subgroups. This result is new and non-trivial even in the case where $G$ is a (non-free) hyperbolic gro…
We consider the following properties of compact oriented irreducible graph-manifolds: to contain a $π_1$-injective surface (immersed, virtually embedded or embedded), be (virtually) fibered over $S^1$, and to carry a metric of nonpositive sectional curvature. It turns out that all these properties can be described from a unified point of view.
Let S be an immersed horizontal surface in a 3-dimensional graph manifold. We show that the fundamental group of the surface S is quadratically distorted whenever the surface is virtually embedded (i.e., separable) and is exponentially distorted when the surface is not virtually embedded.
It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. If a slope on the boundary of the link-manifold is given, there is an algorithm to determine if the slope bounds an embedded punctured-disk; if a meridian slope is given, then it can be determined if a longitude bounds an embedded punctured-disk. The methods use normal surface theory but do not follow the classical approach. Properties of minimal vertex triangulations, layered-triangulations, 0--efficient triangu…
Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in the fundamental group of M, and that the double cosets for crossing surfaces are also separable. We deduce that if there is a "sufficient" collection of surfaces in M, then the fundamental group of M is virtually the fundamental group of a special nonpositively curved cube complex. We provide a sufficient collection for graph manifolds with boundary thus proving that their fundamental groups are virtually special, and hence linear.
The A-polynomial is a knot invariant related to the space of $SL_2(\mathbb{C})$ representations of the knot group. In this paper our interests lies in the logarithmic Gauss map of the A-polynomial. We develop a homological point of view on this slope by extending the constructions of Degtyarev, the second author and Lecuona to the setting of non-abelian representations. It defines a rational function on the character variety, which unifies various known invariants such as the change of curves in the Reidemeister function, the modulus of boundary-parabolic representations, the boundary slope of…
Degeneracy slopes, boundary slopes and exceptional surgery slopes
本文研究了3流形中退化斜率、边界斜率和例外手术斜率的上界，通过三个应用展示了对素交替结和超曲面的贡献。
A diagonal surface in a link exterior M is a properly embedded, incompressible, boundary incompressible surface which furthermore has the same number of boundary components and same slope on each component of the boundary of M. We derive a formula for the boundary slope of a diagonal surface in the exterior of a 2-bridge link which is analogous to the formula for the boundary slope of a 2-bridge knot found by Hatcher and Thurston. Using this formula we show that the diameter of a 2-bridge link, that is, the difference between the smallest and largest finite slopes of diagonal surfaces, is equa…
Slope norm and an algorithm to compute the crosscap number
基于0-高效三角剖分与正常曲面理论，提出三种计算纽结crosscap number的算法，并推广至一般不可约三维流形上的slope norm计算，通过程序实现确定了196个此前未知的纽结crosscap number。
Let K be a knot embedded in a Heegaard surface S for a closed orientable 3-manifold M. We define K-stable equivalence between pairs (S, K) and (S', K) in M, and we prove that any two pairs are K-stably equivalent in M if they have the same surface slope.
A 3-manifold is foliar if it supports a codimension-one co-oriented taut foliation. Suppose $M$ is an oriented 3-manifold with connected boundary a torus, and suppose $M$ contains a properly embedded, compact, oriented, surface $R$ with a single boundary component that is Thurston norm minimizing in $H_2(M, \partial M)$. We define a readily recognizable type of sutured manifold decomposition, which for notational reasons we call double-diamond taut, and show that if $R$ admits a double-diamond taut sutured manifold decomposition, then every boundary slope except one is strongly realized by a c…
Let M be an orientable and irreducible 3-manifold whose boundary is an incompressible torus. Suppose that M does not contain any closed nonperipheral embedded incompressible surfaces. We will show in this paper that the immersed surfaces in M with the 4-plane property can realize only finitely many boundary slopes. Moreover, we will show that only finitely many Dehn fillings of M can yield 3-manifolds with nonpositive cubings. This gives the first examples of hyperbolic 3-manifolds that cannot admit any nonpositive cubings.
Polynomial bounds for surfaces in cusped 3-manifolds
本文为尖点三维流形中的嵌入本质曲面（闭或任意边界斜率，可定向或不可定向）建立了显式多项式数量界限，覆盖交错链环补空间及更广流形族，推广了先前指数界与交错链环特例结果。
In this paper, we will compute the dimension of the space of spun and ordinary normal surfaces in an ideal triangulation of the interior of a compact 3-manifold with incompressible tori or Klein bottle components. Spun normal surfaces have been described in unpublished work of Thurston. We also define a boundary map from spun normal surface theory to the homology classes of boundary loops of the 3-manifold and prove the boundary map has image of finite index. Spun normal surfaces give a natural way of representing properly embedded and immersed essential surfaces in a 3-manifold with tori and …
We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic n-space, satisfying a natural condition on their parabolic subgroups, there are finite index subgroups which generate a subgroup that is an amalgamated free product. Constructions of infinite volume hyperbolic n-manifolds are described by gluing lower dimensional manifolds. It is shown that every slope on a cusp of a hyperbolic 3-manifold is a multi…
We determine all hyperbolic 3-manifolds $M$ admitting two toroidal Dehn fillings at distance 4 or 5. We show that if $M$ is a hyperbolic 3-manifold with a torus boundary component $T_0$, and $r,s$ are two slopes on $T_0$ with $Δ(r,s) = 4$ or 5 such that $M(r)$ and $M(s)$ both contain an essential torus, then $M$ is either one of 14 specific manifolds $M_i$, or obtained from $M_1, M_2, M_3$ or $M_{14}$ by attaching a solid torus to $\partial M_i - T_0$. All the manifolds $M_i$ are hyperbolic, and we show that only the first three can be embedded into $S^3$. As a consequence, this leads to a com…
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