D. B. McReynolds
In this article we prove a generalization of Selberg's lemma on the existence of torsion free, finite index subgroups of arithmetic groups. Some of the geometric applications are the resolution a conjecture of Nimershiem and answers to questions of Long-Reid and the author.
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Let G be a finite group and let M be a G-manifold. We introduce the concept of generalized orbifold invariants of M/G associated to an arbitrary group Gamma, an arbitrary Gamma-set, and an arbitrary covering space of a connected manifold Sigma whose fundamental group is Gamma. Our orbifold invariants have a natural and simple geometric origin in the context of locally constant G-equivariant maps from G-principal bundles over covering spaces of Sigma to the G-manifold M. We calculate generating functions of orbifold Euler characteristic of symmetric products of orbifolds associated to arbitrary…
In this article we examine the conjecture of Neumann and Reid that the only hyperbolic knots in the $3$-sphere which admit hidden symmetries are the figure-eight knot and the two dodecahedral knots. Knots whose complements cover hyperbolic reflection orbifolds admit hidden symmetries, and we verify the Neumann-Reid conjecture for knots which cover small hyperbolic reflection orbifolds. We also show that a reflection orbifold covered by the complement of an AP knot is necessarily small. Thus when $K$ is an AP knot, the complement of $K$ covers a reflection orbifold exactly when $K$ is either th…
We discuss properties of complex algebraic orbifold groups, their characteristic varieties, and their abelian covers. In particular, we deal with the question of (quasi)-projectivity of orbifold groups. We also prove a structure theorem for the variety of characters of normal-crossing quasi-projective orbifold groups. Finally, we extend Sakuma's formula for the first Betti number of abelian covers of orbifold fundamental groups. Several examples are presented, including a compact orbifold group which is not projective and a Zariski pair of plane projective curves that can be told by considerin…
Let O be a compact orientable 3-orbifold with non-empty singular locus and a finite volume hyperbolic structure. (Equivalently, O is the quotient of hyperbolic 3-space by a lattice in PSL(2,C) with torsion.) Then we prove that O has a tower of finite-sheeted covers {O_i} with linear growth of p-homology, for some prime p. This means that the dimension of the first homology, with mod p coefficients, of the fundamental group of O_i grows linearly in the covering degree. The proof combines techniques from 3-manifold theory with group-theoretic methods, including the Golod-Shafarevich inequality a…
We prove that any generating tuple of the fundamental group of a sufficiently large 2-dimensional orbifold is represented by an almost orbifold covering. As a corollary we obtain a generalization of Louder's Theorem which asserts that any two generating tuples of the fundamental group of a closed surface are Nielsen equivalent.
Given a rational homology sphere M, whose splice diagram satisfy the semigroup condition, Neumann and Wahl were able to define a complete intersection surface singularity called splice diagram singularity from the splice diagram of M. They were also able to show that under an additional hypothesis on M called the congruence condition, the link of the splice diagram singularity is the universal abelian cover of M. In this article we generalize the congruence condition to the class of orbifolds called graph orbifold. We show that under a small additional hypothesis, this orbifold congruence cond…
A three-dimensional orbifold $(Σ, γ_i, n_i)$, where $Σ$ is a rational homology sphere, has a universal abelian orbifold covering, whose covering group is the first orbifold homology. A singular pair $(X,C)$, where $X$ is a normal surface singularity with $\mathbb Q$HS link and $C$ is a Weil divisor, gives rise on its boundary to an orbifold. One studies the preceding orbifold notions in the algebro-geometric setting, in particular defining the universal abelian log cover of a pair. A first key theorem computes the orbifold homology from an appropriate resolution of the pair. In analogy with th…
For a branched cover between two closed orientable surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces, the total degree of the cover, and the total length of the partitions of the degree given by the local degrees at the preimages of the branching points. A very old problem asks whether a collection of partitions of an integer having the appropriate total length (that we call a candidate cover) always comes from some branched cover. The answer is known to be in the affirmative whenever the candidate base surface is not the 2-sphere, while for the 2-sphere e…
Good, but not very good orbifolds
本文构造了被流形覆盖但不被有限覆盖的有效闭orbifolds示例。
We give a complete proof of Thurston's Orbifold Theorem for very good 3-orbifolds of cyclic type. An orbifold is said to be very good when it has a finite cover which is a manifold. A 3-orbifold is of cyclic type if the singular set is a non-empty 1-manifold transverse to the boundary.
We prove that an irreducible 3-manifold whose fundamental group satisfies a certain group-theoretic property called RFRS is virtually fibered. As a corollary, we show that 3-dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic form virtually fiber. These include the Seifert Weber dodecahedral space and the Bianchi orbifolds. Moreover, we show that a taut sutured compression body has a finite-sheeted cover with a depth one taut-oriented foliation.
In this paper, we prove several structure theorems for locally conformally flat, positive Yamabe orbifolds and nonnegative scalar curvature, ALE manifolds. These two kinds of spaces can be related by conformal blow-up and conformal compactification. For the orbifolds, we prove that such orbifolds admit a manifold cover. For the ALE manifolds, the homomorphism of the fundamental group for the ALE space induced by the embedding of the ALE end is always injective. Using these properties, several classifications of such ALE manifolds and orbifolds are given in low dimensions. As an application to …
We analyze the orbifolds that can be obtained as quotients of hyperbolic 3-manifolds admitting a Heegaard splitting of genus two by their orientation preserving isometry groups. The genus two hyperbolic 3-manifolds are exactly the hyperbolic 2-fold branched coverings of 3-bridge links. If the 3-bridge link is a knot, we prove that the underlying topological space of the quotient orbifold is either the 3-sphere or a lens space and we describe the combinatorial setting of the singular set for each possible isometry group. In the case of 3-bridge links with two or three components, the situation …
We show that if $g$ is a Riemannian metric on a closed piecewise locally symmetric manifold $M$, then the lift of $g$ to the universal cover $\widetilde{M}$ has a discrete isometry group. We also show that the index $[\Isom(\widetilde{M}): π_1(M)]$ is bounded by a constant independent of $g$.
Symplectic four-manifolds and hyperbolic $π$-orbifolds
该研究构造了具有双曲π-orbifold结构的闭辛四维流形，解决LeBrun猜想，其最简例欧拉示性数为304，可作为双曲120胞腔拼接而成，存在4次分支覆盖到亏格4的两曲面乘积。
We study the size of the isometry group Isom(M, g) of Riemannian manifolds (M, g) as g varies. For M not admitting a circle action, we show that the order of Isom(M, g) can be universally bounded in terms of the bounds on Ricci curvature, diameter, and injectivity radius of M. This generalizes results known for negative Ricci curvature to all manifolds. More generally we establish a similar universal bound on the index of the deck group pi_1(M) in the isometry group Isom(X, g) of the universal cover X in the absence of suitable actions by connected groups. We apply this to characterize locally…
Let M be a closed orientable Seifert fibered 3-manifold with a hyperbolic base 2-orbifold, or equivalently, admitting a geometry modeled on H^2 \times R or the universal cover of SL(2,R). Our main result is that the connected component of the identity map in the diffeomorphism group Diff(M) is either contractible or homotopy equivalent to the circle, according as the center of the fundamental group of M is trivial or infinite cyclic. Apart from the remaining case of non-Haken infranilmanifolds, this completes the homeomorphism classifications of Diff(M) and of the space of Seifert fiberings of…
We provide two new proofs of a theorem of Cooper, Long and Reid which asserts that, apart from an explicit finite list of exceptional manifolds, any compact orientable irreducible 3-manifold with non-empty boundary has large fundamental group. The first proof is direct and topological; the second is group-theoretic. These techniques are then applied to prove a string of results about (possibly closed) 3-orbifolds, which culminate in the following theorem. If K is a knot in a compact orientable 3-manifold M, such that the complement of K admits a finite-volume hyperbolic structure, then the orb…
To a rational homology sphere graph manifold one can associate a weighted tree invariant called splice diagram. In this article we prove a sufficient numerical condition on the splice diagram for a graph manifold to be a singularity link. We also show that if two manifolds have the same splice diagram, then their universal abelian covers are homeomorphic. To prove the last theorem we have to generalize our notions to orbifolds.
We investigate a question of Cooper adjacent to the Virtual Haken Conjecture. Assuming certain conjectures in number theory, we show that there exist hyperbolic rational homology 3-spheres with arbitrarily large injectivity radius. These examples come from a tower of abelian covers of an explicit arithmetic 3-manifold. The conjectures we must assume are the Generalized Riemann Hypothesis and a mild strengthening of results of Taylor et al on part of the Langlands Program for GL_2 of an imaginary quadratic field. The proof of this theorem involves ruling out the existence of an irreducible two …
Topology and geometry of flagness and beltness of simple orbifolds
本文将简单多面体的旗与带概念推广到简单轨道流形，证明简单手柄体的轨道非球面性等价于旗性、秩二自由阿贝尔子群存在性等价于□-带存在性，并给出三维直角双曲结构的纯组合刻画，推广了Pogorelov定理。
Subgroups of bounded rank in hyperbolic 3-manifold groups
本文证明双曲3-流形群中有界秩子群的有限性定理，并由此推出闭双曲3-流形的有界秩覆盖塔必为与1-轨道形上纤维化相关的有限覆盖塔。
A construction of representations of 3-manifold groups into PU(2,1) through Lefschetz fibrations
本文通过 Lefschetz 纤维化构造了 3-流形基本群在复双曲空间全纯等距群格点中的无穷多个非共轭表示，该格点对应于 Hirzebruch 超平面布置的分支覆盖配边基本群。
We prove the quasimodularity of generating functions for counting pillowcase covers, with and without Siegel-Veech weight. Similar to prior work on torus covers, the proof is based on analyzing decompositions of half-translation surfaces into horizontal cylinders. It provides an alternative proof of the quasimodularity results of Eskin-Okounkov and a practical method to compute area Siegel-Veech constants. A main new technical tool is a quasi-polynomiality result for 2-orbifold Hurwitz numbers with completed cycles.
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