KATOK FUCHSIAN GROUPS PDF

Although discontinuity and discreteness are equivalent in this case, this is not generally true for the case of an arbitrary group of conformal homeomorphisms acting on the full Riemann sphere as opposed to H. This motivates the following definition of a Fuchsian group. That is, any one of these three can serve as a definition of a Fuchsian group, the others following as theorems. Indeed, even the modular group PSL 2,Z , which is a Fuchsian group, does not act discontinuously on the real number line; it has accumulation points at the rational numbers. There may, however, be limit points on the real axis.

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Although discontinuity and discreteness are equivalent in this case, this is not generally true for the case of an arbitrary group of conformal homeomorphisms acting on the full Riemann sphere as opposed to H.

This motivates the following definition of a Fuchsian group. That is, any one of these three can serve as a definition of a Fuchsian group, the others following as theorems.

Indeed, even the modular group PSL 2,Z , which is a Fuchsian group, does not act discontinuously on the real number line; it has accumulation points at the rational numbers. There may, however, be limit points on the real axis. The limit set may be empty, or may contain one or two points, or may contain an infinite number. Otherwise, a Fuchsian group is said to be of the second type. Since it is nowhere dense, this implies that any limit point is arbitrarily close to an open set that is not in the limit set.

In other words, the limit set is a Cantor set. The type of a Fuchsian group need not be the same as its type when considered as a Kleinian group: in fact, all Fuchsian groups are Kleinian groups of type 2, as their limit sets as Kleinian groups are proper subsets of the Riemann sphere, contained in some circle.

This is the subgroup of PSL 2,R consisting of linear fractional transformations.

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Fuchsian Groups

Open to the public. The hyperbolic metric 1. You may purchase this title at these fine bookstores. Some Escher graphics gropus based on them for the disc model of hyperbolic geometry. You also may like to try some of these bookshopswhich may or may not sell this item. In this case, the group may be called the Fuchsian group of the surface. The Dirichlet region 3.

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