Quasifuchs group

Limes set of a quasi-fox group

In the mathematical theory of the Klein group , a quasifuchs group is a Klein group whose limit set is a Jordan curve . The term generalizes the concept of Fuchs groups .

definition

A Klein group is a discrete group of isometries of hyperbolic space . Your limit set is the set of accumulation points of any orbit in the edge at infinity . The group is a quasi-fox group when there is a Jordan curve. ${\displaystyle \Gamma }$ ${\displaystyle \Lambda (\Gamma )}$ ${\displaystyle S_{\infty }^{2}=\partial _{\infty }H^{3}}$ ${\displaystyle \Lambda (\Gamma )}$

While the Limes set of a Fuchs group is a circle , the Limes sets of general quasifuchs groups are often complicated fractal curves. Because the limit set is -invariant, these curves have a high degree of symmetry. ${\displaystyle \Gamma }$

Each quasi- Fuchs group can be conjugated into a Fuchs group using a quasi-conformal homeomorphism .

Quasi-Fuchs groups as generalizations of Fuchs groups

Occasionally there is also the more general definition in the literature that the Limes quantity should be contained in a Jordan curve. Quasifuchs groups of the first kind are then those for which the Limes set is a Jordan curve (i.e. they meet the above definition), while Klein groups, whose Limes set is a real subset of a Jordan curve, are then called quasifuchs groups of the second kind.

By embedding , Fuchs groups can be understood as Klein groups. The Limes set of a Fuchs group is contained in a circle. Quasi- Fuchs groups of the first and second kind thus generalize Fuchs groups of the first and second kind . ${\displaystyle PSL(2,\mathbb {R} )\to PSL(2,\mathbb {C} )=Isom^{+}(H^{3})}$

Fractal structure of the Limes set

For Fuchsian groups of the first kind, the Hausdorff dimension of the Limes set is always . ${\displaystyle 1}$

If a quasi- Fuchs group is not a Fuchs group, then the Hausdorff dimension of the Limes set is strictly larger than , so one obtains an -invariant fractal. ${\displaystyle \Gamma }$ ${\displaystyle 1}$ ${\displaystyle \Gamma }$

Topology of the quotient manifold

A torsion-free quasi - Fuchs group is isomorphic to the fundamental group of a surface . The Klein manifold is then homôomorphic to . ${\displaystyle \Gamma }$ ${\displaystyle S}$ ${\displaystyle \Gamma \backslash H^{3}}$ ${\displaystyle S\times \mathbb {R} }$

Finitely generated quasifuchs groups are geometrically finite , their convex core has finite volume.

Discontinuity area and conforming margin

According to the Jordan theorem of curves , the discontinuity area defined as the complement of the Limes set consists of two areas , both of which are homeomorphic to the circular disk. The group acts properly discontinuously in these areas, the quotient ${\displaystyle S_{\infty }^{2}\setminus \Lambda (\Gamma )}$ ${\displaystyle \Omega (\Gamma )}$ ${\displaystyle \Omega _{1},\Omega _{2}}$ ${\displaystyle \Gamma }$

${\displaystyle X=\Gamma \backslash \Omega _{1},Y=\Gamma \backslash \Omega _{2}}$

are therefore Riemann surfaces . They are called the conformal boundary of the Klein manifold . ${\displaystyle \Gamma \backslash H^{3}}$

Simultaneous uniformization

The uniformization of Lipman Bers says the moduli space of all quasi fox rule groups of given that isomorphism by the product of two copies of the Teichmüller space of parameterize can: ${\displaystyle \Gamma }$ ${\displaystyle \Gamma }$

${\displaystyle QF(\Gamma )\cong T(\Gamma )\times T(\Gamma )}$.

This bijection is obtained by assigning each quasi-fox group its conformal edge as an element in the product of two pond miller spaces.

In particular, for a surface group (i.e. the fundamental group of a closed orientable surface ) the module space of its quasi-Fuchsian deformations is -dimensional. ${\displaystyle (12g-12)}$

Simultaneous uniformization is used to construct Bers cuts and the skinning map .

AdS quasi-fox groups

In addition to the above quasi-fox groups acting on the hyperbolic space, there are also certain AdS quasi-fox groups acting in the anti-de-sitter space . These are the holonomy groups of GHMC manifolds (global hyperbolic maximally Cauchy compact manifolds). ${\displaystyle \mathbb {H} ^{3}=O(3,1)/O(3)}$ ${\displaystyle AdS^{3}=O(2,2)/O(2,1)}$

literature

• Matsuzaki, Katsuhiko; Taniguchi, Masahiko: Hyperbolic manifolds and Kleinian groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998. ISBN 0-19-850062-9

Individual evidence

1. ^ Rufus Bowen : Hausdorff dimension of quasicircles. Inst. Hautes Études Sci. Publ. Math. No. 1979, 50: 11-25. doi: 10.1007 / BF02684767? LI = true # page-1