scholarly journals Nilpotent quotients of fundamental groups of special 3-manifolds with boundary

1998 ◽  
Vol 58 (2) ◽  
pp. 233-237
Author(s):  
Gabriela Putinar
Keyword(s):  
Fundamental Group ◽  
Betti Number ◽  
Fundamental Groups ◽  
Manifolds With Boundary ◽  
Manifold With Boundary ◽  
Maximal Torsion ◽  
Torsion Free

We use a Betti number estimate of Freedman-Hain-Teichner to show that the maximal torsion-free nilpotent quotient of the fundamental group of a 3-manifold with boundary is either Z or Z ⊕ Z. In particular we reobtain the Evans-Moser classification of 3-manifolds with boundary which have nilpotent fundamental groups.

Elementary amenable groups and 4-manifolds with Euler characteristic 0

1991 ◽  
Vol 50 (1) ◽  
pp. 160-170 ◽  
Author(s):  
Jonathan A. Hillman
Keyword(s):  
Normal Subgroup ◽  
Fundamental Group ◽  
Euler Characteristic ◽  
Fundamental Groups ◽  
Normal Subgroups ◽  
Euler Characteristics ◽  
Amenable Groups ◽  
Amenable Subgroup ◽  
Finite Subgroups ◽  
Torsion Free

AbstractWe extend earlier work relating asphericity and Euler characteristics for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups. In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical. Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends or is virtually an extension of Z by a subgroup of Q, or the manifold is asphencal and the group is virtually poly- Z of Hirsch length 4.

scholarly journals GROUPS OF LOCALLY-FLAT DISK KNOTS AND NON-LOCALLY-FLAT SPHERE KNOTS

2005 ◽  
Vol 14 (02) ◽  
pp. 189-215 ◽  
Author(s):  
GREG FRIEDMAN
Keyword(s):  
Fundamental Group ◽  
Piecewise Linear ◽  
Complete Classification ◽  
Fundamental Groups ◽  
Singular Set ◽  
Flat Disk ◽  
Set Of Points

The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of the knot complement and that of the complement of the "boundary knot" that occurs around the singular set, the set of points at which the embedding is not locally-flat. If a knot has only point singularities, this is equivalent to studying the groups of a PL locally-flat disk knot and its boundary sphere knot; in this case, we obtain a complete classification of all such group pairs in dimension ≥6. For more general knots, we also obtain complete classifications of these group pairs under certain restrictions on the singularities. Finally, we use spinning constructions to realize further examples of boundary knot groups.

scholarly journals Coarse fundamental groups and box spaces

2019 ◽  
Vol 150 (3) ◽  
pp. 1139-1154
Author(s):  
Thiebout Delabie ◽  
Ana Khukhro
Keyword(s):  
Fundamental Group ◽  
Betti Number ◽  
Equivalence Classes ◽  
Fundamental Groups ◽  
Normal Subgroups ◽  
Finitely Presented Groups ◽  
Rank Gradient ◽  
Box Space ◽  
Finitely Presented ◽  
Uniformly Bounded

AbstractWe use a coarse version of the fundamental group first introduced by Barcelo, Kramer, Laubenbacher and Weaver to show that box spaces of finitely presented groups detect the normal subgroups used to construct the box space, up to isomorphism. As a consequence, we have that two finitely presented groups admit coarsely equivalent box spaces if and only if they are commensurable via normal subgroups. We also provide an example of two filtrations (Ni) and (Mi) of a free group F such that Mi > Ni for all i with [Mi:Ni] uniformly bounded, but with $\squ _{(N_i)}F$ not coarsely equivalent to $\squ _{(M_i)}F$. Finally, we give some applications of the main theorem for rank gradient and the first ℓ2 Betti number, and show that the main theorem can be used to construct infinitely many coarse equivalence classes of box spaces with various properties.

scholarly journals Hyperbolic groups with boundary an n-dimensional Sierpinski space

2019 ◽  
Vol 11 (01) ◽  
pp. 233-247
Author(s):  
Jean-François Lafont ◽  
Bena Tshishiku
Keyword(s):  
Fundamental Group ◽  
Negative Curvature ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Manifolds With Boundary ◽  
Geodesic Boundary ◽  
Totally Geodesic ◽  
Visual Boundary ◽  
Aspherical Manifolds ◽  
Torsion Free

For [Formula: see text], we show that if [Formula: see text] is a torsion-free hyperbolic group whose visual boundary [Formula: see text] is an [Formula: see text]-dimensional Sierpinski space, then [Formula: see text] for some aspherical [Formula: see text]-manifold [Formula: see text] with non-empty boundary. Concerning the converse, we construct, for each [Formula: see text], examples of aspherical manifolds with boundary, whose fundamental group [Formula: see text] is hyperbolic, but with visual boundary [Formula: see text] not homeomorphic to [Formula: see text]. Our examples even support (metric) negative curvature, and have totally geodesic boundary.

scholarly journals Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Biplab Basak ◽  
Manisha Binjola
Keyword(s):  
Fundamental Group ◽  
Large Class ◽  
Lower Bounds ◽  
Manifolds With Boundary ◽  
Manifold With Boundary ◽  
Regular Genus

AbstractLet 𝑀 be a connected compact PL 4-manifold with boundary. In this article, we give several lower bounds for regular genus and gem-complexity of the manifold 𝑀. In particular, we prove that if 𝑀 is a connected compact 4-manifold with ℎ boundary components, then its gem-complexity k(M) satisfies the inequalities k(M)\geq 3\chi(M)+7m+7h-10 and k(M)\geq k(\partial M)+3\chi(M)+4m+6h-9, and its regular genus \mathcal{G}(M) satisfies the inequalities \mathcal{G}(M)\geq 2\chi(M)+3m+2h-4 and \mathcal{G}(M)\geq\mathcal{G}(\partial M)+2\chi(M)+2m+2h-4, where 𝑚 is the rank of the fundamental group of the manifold 𝑀. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of a PL 4-manifold with boundary. Further, the sharpness of these bounds is also shown for a large class of PL 4-manifolds with boundary.

scholarly journals Topological 4-manifolds with right-angled Artin fundamental groups

2019 ◽  
Vol 11 (04) ◽  
pp. 777-821
Author(s):  
Ian Hambleton ◽  
Alyson Hildum
Keyword(s):  
Fundamental Group ◽  
Cohomological Dimension ◽  
Fundamental Groups ◽  
Connected Sum ◽  
Dimension Formula ◽  
Assembly Map ◽  
Torsion Free

We classify closed, spin[Formula: see text], topological [Formula: see text]-manifolds with fundamental group [Formula: see text] of cohomological dimension [Formula: see text] (up to [Formula: see text]-cobordism), after stabilization by connected sum with at most [Formula: see text] copies of [Formula: see text]. In general, we must also assume that [Formula: see text] satisfies certain [Formula: see text]-theory and assembly map conditions. Examples for which these conditions hold include the torsion-free fundamental groups of [Formula: see text]-manifolds and all right-angled Artin groupswhose defining graphs have no [Formula: see text]-cliques.

scholarly journals Classification of extremal elliptic K3 surfaces and fundamental groups of open K3 surfaces

2001 ◽  
Vol 161 ◽  
pp. 23-54 ◽  
Author(s):  
Ichiro Shimada ◽  
De-Qi Zhang
Keyword(s):  
Fundamental Group ◽  
K3 Surfaces ◽  
Rational Curves ◽  
Complete List ◽  
Fundamental Groups ◽  
Sufficient Condition

We present a complete list of extremal elliptic K3 surfaces (Theorem 1.1). As an application, we give a sufficient condition for the topological fundamental group of complement to an ADE-configuration of smooth rational curves on a K3 surface to be trivial (Proposition 4.1 and Theorem 4.3).

scholarly journals TOPOLOGICAL 4-MANIFOLDS WITH GEOMETRICALLY TWO-DIMENSIONAL FUNDAMENTAL GROUPS

2009 ◽  
Vol 01 (02) ◽  
pp. 123-151 ◽  
Author(s):  
IAN HAMBLETON ◽  
MATTHIAS KRECK ◽  
PETER TEICHNER
Keyword(s):  
Fundamental Group ◽  
Fundamental Groups ◽  
Two Dimensional ◽  
Intersection Form

Closed oriented 4-manifolds with the same geometrically two-dimensional fundamental group (satisfying certain properties) are classified up to s-cobordism by their w2-type, equivariant intersection form and the Kirby–Siebenmann invariant. As an application, we obtain a complete homeomorphism classification of closed oriented 4-manifolds with solvable Baumslag–Solitar fundamental groups, including a precise realization result.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

Sign in / Sign up

Export Citation Format

Share Document