The structure of certain subgroups of the Picard group

1978 ◽  
Vol 84 (3) ◽  
pp. 427-436 ◽  
Author(s):  
Norbert Wielenberg
Keyword(s):  
Discrete Subgroup ◽  
Universal Covering ◽  
Picard Group ◽  
Hyperbolic Volume ◽  
Universal Covering Space ◽  
Hnn Extension ◽  
Free Product With Amalgamation ◽  
Quotient Manifold ◽  
Group Of Isometries ◽  
Torsion Free

A torsion-free discrete subgroup G of PSL(2, C) acts as a group of isometries of hyperbolic 3-space H3. The resulting quotient manifold M has H3 as its universal covering space with G as the group of cover transformations. We shall give examples where M has finite hyperbolic volume and is a link complement in S3. In these examples, G is a subgroup of the Picard group and in most cases is given as an HNN extension or a free product with amalgamation of kleinian groups with fuchsian groups as amalgamated or conjugated subgroups.

A quadratic parabolic group

1975 ◽  
Vol 77 (2) ◽  
pp. 281-288 ◽  
Author(s):  
Robert Riley
Keyword(s):  
Normal Form ◽  
Half Space ◽  
Transformation Group ◽  
Universal Covering ◽  
Covering Space ◽  
Fundamental Domains ◽  
Universal Covering Space ◽  
Other Information ◽  
Figure Eight Knot

When k is a 2-bridge knot with group πK, there are parabolic representations (p-reps) θ: πK → PSL(): = PSL(2, ). The most obvious problem that this suggests is the determination of a presentation for an image group πKθ. We shall settle the easiest outstanding case in section 2 below, viz. k the figure-eight knot 41, which has the 2-bridge normal form (5, 3). We shall prove that the (two equivalent) p-reps θ for this knot are isomorphisms of πK on πKθ. Furthermore, the universal covering space of S3\k can be realized as Poincaré's upper half space 3, and πKθ is a group of hyperbolic isometries of 3 which is also the deck transformation group of the covering 3 → S3\k. The group πKθ is a subgroup of two closely related groups that we study in section 3. We shall give fundamental domains, presentations, and other information for all these groups.

Floer type cohomology on cosymplectic manifolds

2015 ◽  
Vol 12 (07) ◽  
pp. 1550082 ◽  
Author(s):  
Yong Seung Cho ◽  
Young Do Chai
Keyword(s):  
Moduli Space ◽  
Loop Space ◽  
Type Theorem ◽  
Gradient Flow ◽  
Universal Covering ◽  
Cochain Complex ◽  
Action Functional ◽  
Flow Lines ◽  
Universal Covering Space ◽  
Cosymplectic Manifolds

We investigate a Floer type cohomology on cosymplectic manifolds M. To do this, we study a symplectic type action functional on the universal covering space of the loop space of contractible loops in M and the moduli space of gradient flow lines of the functional. The cochain complex induced by the critical points of the functional produces Floer type cohomology of M which is naturally isomorphic to a quantum type cohomology of M. We have an Arnold type theorem for Hamiltonian cosymplectomorphisms on compact semipositive cosymplectic manifolds. As an example, we consider the product of a Calabi–Yau 3-fold and the unit circle.

On a hyperbolic 3-manifold with some special properties

1993 ◽  
Vol 113 (1) ◽  
pp. 87-90 ◽  
Author(s):  
B. Zimmermann
Keyword(s):  
Coxeter Groups ◽  
Universal Covering ◽  
Finite Subgroup ◽  
Heegaard Splitting ◽  
Heegaard Genus ◽  
Normal Torsion ◽  
Covering Group ◽  
Minimal Index ◽  
Torsion Free Subgroup ◽  
Torsion Free

We present a closed hyperbolic 3-manifold M with some surprising properties. The universal covering group of M is a normal torsion-free subgroup of minimal index in one of the nine Coxeter groups G, generated by the reflections in the faces of one of the nine Lannér-tetrahedra (bounded tetrahedra in hyperbolic 3-space all of whose dihedral angles are of the form π/n with n ∈ ℕ see [1] or [3]). The corresponding Coxeter group G splits as a semidirect product G = π1M⋉A, where A is a finite subgroup of G, and G is the only one of the nine Coxeter groups associated to the Lannér-tetrahedra which admits such a splitting (this follows using results in [4]). We derive a presentation of π1M and show that the first homology group H1(M) of M is isomorphic to ℚ11. This is in sharp contrast to other torsion-free (non-normal) subgroups of finite index in Coxeter groups constructed in [1] which all have finite first homology (though it is known that they are all virtually ℚ-representable (see [5], p. 434). It follows from our computations that the Heegaard genus of M is 11, and that there exists a Heegaard splitting of M of genus 11 invariant under the action of the group I+(M) ≌ S5 ⊕ ℚ2 of orientation-preserving isometries of M (we compute this group in [4]), so that the Heegaard genus of M is equal to the equivariant Heegaard genus of the action of I+(M) on M. Moreover M is maximally symmetric in the sense of [4, 6]: the order 120 of the subgroup of index 2 in I+(M) which preserves both handle-bodies of the Heegaard splitting is the maximal possible order of a group of orientation-preserving diffeomorphisms of a handle-body of genus 11. (This maximal order is 12(g—1) for a handle-body of genus g; see [7].) By taking the coverings Mq of M corresponding to the surjections π1M→H1(M) ≌ ℚ11→(ℚq)11 for q ∈ ℕ, we obtain explicitly an infinite series of maximally symmetric hyperbolic 3-manifolds.

Rigidities of asymptotically Euclidean manifolds

1999 ◽  
Vol 60 (3) ◽  
pp. 521-528 ◽  
Author(s):  
Seong-Hun Paeng
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Fundamental Group ◽  
Compact Riemannian Manifold ◽  
Universal Covering ◽  
Covering Space ◽  
Universal Covering Space

Let M be an n-dimensional compact Riemannian manifold. We study the fundamental group of M when the universal covering space of M, M is close to some Euclidean space ℝs asymptotically.

scholarly journals Quaternionic arithmetic lattices of rank 2 and a fake quadric in characteristic 2

2018 ◽  
Vol 28 (06) ◽  
pp. 1049-1090 ◽  
Author(s):  
Nithi Rungtanapirom
Keyword(s):  
Fundamental Group ◽  
Euler Characteristic ◽  
Quaternion Algebra ◽  
Universal Covering ◽  
Arithmetic Lattice ◽  
Rank 2 ◽  
Characteristic 2 ◽  
Square Complex ◽  
Torsion Free

We construct a torsion-free arithmetic lattice in [Formula: see text] arising from a quaternion algebra over [Formula: see text]. It is the fundamental group of a square complex with universal covering [Formula: see text], a product of trees with constant valency [Formula: see text], which has minimal Euler characteristic. Furthermore, our lattice gives rise to a fake quadric over [Formula: see text] by means of non-archimedean uniformization.

scholarly journals A presentation for a group of automorphisms of a simplicial complex

1988 ◽  
Vol 30 (3) ◽  
pp. 331-337 ◽  
Author(s):  
M. A. Armstrong
Keyword(s):  
Simplicial Complex ◽  
Elementary Proof ◽  
Universal Covering ◽  
Covering Space ◽  
Covering Spaces ◽  
Group Of Automorphisms ◽  
Generators And Relations ◽  
Universal Covering Space ◽  
Simply Connected ◽  
Special Case

The Bass–Serre theorem supplies generators and relations for a group of automorphisms of a tree. Recently K. S. Brown [2] has extended the result to produce a presentation for a group of automorphisms of a simply connected complex, the extra ingredient being relations which come from the 2-cells of the complex. Suppose G is the group, K the complex and L the 1-skeleton of K. Then an extension of π1(L) by G acts on the universal covering space of L (which is of course a tree) and Brown's technique is to apply the work of Bass and Serre to this action. Our aim is to give a direct elementary proof of Brown's theorem which makes no use of covering spaces, deals with the Bass–Serre theorem as a special case and clarifies the roles played by the various generators and relations.

scholarly journals Geometric Approach to Optimal Path Problem with Uncertain Arc Lengths

2020 ◽  
Vol 9 (1) ◽  
pp. 11
Author(s):  
Atefeh Hasan-Zadeh
Keyword(s):  
Homotopy Class ◽  
Algebraic Topology ◽  
Optimal Path ◽  
Shortest Paths ◽  
Geometric Approach ◽  
Universal Coverage ◽  
Universal Covering ◽  
Universal Covering Space ◽  
Coverage Space ◽  
Arc Lengths

In this paper, the problem of finding the shortest paths, one of the most important problems in science and technology has been geometrically studied. Shortest path algorithm has been generalized to the shortest cycles in each homotopy class on a surface with arbitrary topology, using the universal covering space notion in the algebraic topology. Then, a general algorithm has been presented to compute the shortest cycles (geometrically rather than combinatorial) in each homotopy class. The algorithm can handle surface meshes with the desired topology, with or without boundary. It also provides a fundamental framework for other algorithms based on universal coverage space due to the capacity and flexibility of the framework. 

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