Jörgensen’s inequality and purely loxodromic two-generator free Kleinian groups

The principal character of a representation of the free group of rank two into [Formula: see text] is a triple of complex numbers that determines an irreducible representation uniquely up to conjugacy. It is a central problem in the geometry of discrete groups and low dimensional topology to determine when such a triple represents a discrete group which is not virtually abelian, that is, a Kleinian group. A classical necessary condition is Jørgensen’s inequality. Here, we use certain shifted Chebyshev polynomials and trace identities to determine new families of such inequalities, some of which are best possible. The use of these polynomials also shows how we can identify the principal character of some important subgroups from that of the group itself.

Download Full-text

Spaces of Kleinian Groups

10.1017/cbo9781139106993 ◽

2006 ◽

Keyword(s):

Kleinian Groups

Download Full-text

Iteration theory and inequalities for Kleinian groups

Bulletin of the American Mathematical Society ◽

10.1090/s0273-0979-1989-15761-3 ◽

1989 ◽

Vol 21 (1) ◽

pp. 57-64 ◽

Cited By ~ 17

Author(s):

F. W. Gehring ◽

G. J. Martin

Keyword(s):

Kleinian Groups ◽

Iteration Theory

Download Full-text

Iterated inversion system: an algorithm for efficiently visualizing Kleinian groups and extending the possibilities of fractal art

Journal of Mathematics and the Arts ◽

10.1080/17513472.2021.1943998 ◽

2021 ◽

pp. 1-31

Author(s):

Kento Nakamura

Keyword(s):

Kleinian Groups

Download Full-text

Free vs. locally free Kleinian groups

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2016-0005 ◽

2019 ◽

Vol 2019 (746) ◽

pp. 149-170

Author(s):

Pekka Pankka ◽

Juan Souto

Keyword(s):

Hausdorff Dimension ◽

Kleinian Group ◽

Cantor Set ◽

Cantor Sets ◽

Kleinian Groups ◽

The Other ◽

Limit Set ◽

Limit Sets ◽

Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

Download Full-text