Classification of epimorphisms of fundamental groups of surfaces onto free groups

1990 ◽  
Vol 48 (2) ◽  
pp. 736-742
Author(s):  
R. I. Grigorchuk ◽  
P. F. Kurchanov
Keyword(s):  
Free Groups ◽  
Fundamental Groups

scholarly journals On the classification of rank-two representations of quasiprojective fundamental groups

2008 ◽  
Vol 144 (5) ◽  
pp. 1271-1331 ◽  
Author(s):  
Kevin Corlette ◽  
Carlos Simpson
Keyword(s):  
Fundamental Groups ◽  
Monodromy At Infinity ◽  
Dense Representation

AbstractSuppose that X is a smooth quasiprojective variety over ℂ and ρ:π1(X,x)→SL(2,ℂ) is a Zariski-dense representation with quasiunipotent monodromy at infinity. Then ρ factors through a map X→Y with Y either a Deligne–Mumford (DM) curve or a Shimura modular stack.

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 Classification of negatively pinched manifolds with amenable fundamental groups

2006 ◽  
Vol 196 (2) ◽  
pp. 229-260 ◽  
Author(s):  
Igor Belegradek ◽  
Vitali Kapovitch
Keyword(s):  
Fundamental Groups

Some Hyperbolic 3-Manifolds with Totally Geodesic Boundary

2010 ◽  
Vol 17 (03) ◽  
pp. 457-468 ◽  
Author(s):  
Agnese Ilaria Telloni
Keyword(s):  
Fundamental Groups ◽  
Isometry Groups ◽  
Geodesic Boundary ◽  
Totally Geodesic ◽  
Cyclic Coverings

We construct a family of compact hyperbolic 3-manifolds with totally geodesic boundary, depending on three integer parameters. Then we determine geometric presentations of the fundamental groups of these manifolds and prove that they are cyclic coverings of the 3-ball branched along a specified tangle with two components. Finally, we give a classification of these manifolds up to homeomorphism (resp., isometry), and determine their isometry groups.

scholarly journals All Finite Generalized Tetrahedron Groups

2008 ◽  
Vol 15 (04) ◽  
pp. 555-580 ◽  
Author(s):  
Benjamin Fine ◽  
Miriam Hahn ◽  
Alexander Hulpke ◽  
Volkmar große Rebel ◽  
Gerhard Rosenberger ◽  
...  
Keyword(s):  
Complete Classification ◽  
Fundamental Groups ◽  
Triangle Groups ◽  
Reduced Word ◽  
Generalized Tetrahedron Group

A generalized tetrahedron group is defined to be a group admitting a presentation [Formula: see text] where l, m, n, p, q, r ≥ 2, each Wi(a,b) is a cyclically reduced word involving both a and b. These groups appear in many contexts, not least as fundamental groups of certain hyperbolic orbifolds or as subgroups of generalized triangle groups. In this paper, we build on previous work to give a complete classification of all finite generalized tetrahedron groups.

scholarly journals Generalized Fibonacci groups H(r,n,s) that are connected labelled oriented graph groups

2019 ◽  
Vol 22 (1) ◽  
pp. 23-39 ◽  
Author(s):  
Gerald Williams
Keyword(s):  
Cyclic Group ◽  
Betti Numbers ◽  
Oriented Graph ◽  
Complete Classification ◽  
Circulant Matrices ◽  
Fundamental Groups ◽  
Infinite Cyclic Group ◽  
Torus Knot ◽  
Graph Groups

Abstract The class of connected Labelled Oriented Graph (LOG) groups coincides with the class of fundamental groups of complements of closed, orientable 2-manifolds embedded in {S^{4}} , and so contains all knot groups. We investigate when Campbell and Robertson’s generalized Fibonacci groups {H(r,n,s)} are connected LOG groups. In doing so, we use the theory of circulant matrices to calculate the Betti numbers of their abelianizations. We give an almost complete classification of the groups {H(r,n,s)} that are connected LOG groups. All torus knot groups and the infinite cyclic group arise and we conjecture that these are the only possibilities. As a corollary we show that {H(r,n,s)} is a 2-generator knot group if and only if it is a torus knot group.

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