Fuchsian group

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In mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane. A Fuchsian group is always a discrete group, and thus is a lattice in one of the two semisimple Lie groups PSL(2,R) or PSL(2,C).

Fuchsian groups are used to create Fuchsian models of Riemann surfaces. In some sense, Fuchsian groups do for non-Euclidean geometry what crystallographic groups do for Euclidean geometry, but the theory is much richer. Some Escher graphics are based on them (for the disc model of hyperbolic geometry).

Overview

There are several equivalent definitions of a Fuchsian group that can be made, defining it to be a discrete group or as group that acts properly discontinuously; several different notions of discreteness may also apply. Furthermore, a Fuchsian group can be defined to be a subgroup of either PSL(2,R) or PSL(2,C), depending on whether one chooses to work with hyperbolic geometry on the upper half-plane or the open unit disk. In either case, a subgroup that is conjugate (in PSL(2,C)) to a Fuchsian group is also Fuchsian; although in this case the invariant domain need not be the upper half-plane or the disk, but rather a subset of the Riemann sphere Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbb {C}}=\mathbb{C}\cup\infty} that is isometric to the half-plane or disk.

The various different definitions can be seen to stem from a single, formal, abstract definition, which is thus stated first. This formal definition is then elucidated with simpler definitions below.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma \subset PSL(2,\mathbb{C})} act invariantly on a proper, open disk Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta\subset \hat{\mathbb {C}}} , that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(\Delta)=\Delta} . Then Γ is Fuchsian if and only if any of the following three properties hold:

  1. Γ is a discrete group (with respect to the standard topology on PSL(2,C)).
  2. Γ acts properly discontinuously on some point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\in\Delta} .
  3. The set Δ is a subset of the region of discontinuity Ω(Γ) of Γ.

That is, any one of these three can serve as a definition of a Fuchsian group, the others following as theorems. The notion of an invariant proper subset Δ is important; the Picard group PSL(2,Z[i]) on the Gaussian integers Z[i] is discrete but does not act discontinuously anywhere on the Riemann sphere. 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. Similarly, the idea that Δ is a proper subset of the region of discontinuity is important; when it is not, the subgroup is called a Kleinian group.

When the invariant domain Δ is the upper half-plane, then every Fuchsian group leaving the upper half-plane invariant is a subgroup of PSL(2,R). Every Fuchsian group is conjugate (in PSL(2,C)) to a discrete subgroup of PSL(2,R), which leads to the most common and simplest definition, given below. Alternately, any Fuchsian group can be made conjugate to one which leaves the open unit disk invariant.

Fuchsian groups on the upper half-plane

Let H = {z in C : Im(z) > 0} be the upper half-plane. Then H is a model for hyperbolic plane geometry, when given the element of arc length

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The group PSL(2,R) acts on H by linear fractional transformations:

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This action is faithful, and in fact PSL(2,R) is isomorphic to the group of all orientation-preserving isometries of H.

A Fuchsian group Γ may be defined to be a subgroup of PSL(2,R), which acts discontinuously on H. That is,

An equivalent definition for Γ to be Fuchsian is that Γ be discrete, in the following sense:

  • Every sequence {γn} of elements of Γ converging to the identity in the usual topology of point-wise convergence is eventually constant, i.e. there exists an integer N such that for all n > N, γn = I, where I is the identity matrix.

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 Riemann sphere. Indeed, the Fuchsian group PSL(2,Z) has accumulation points on the real number line Im z = 0; elements of PSL(2,Z) will carry z = 0 to every rational number; the rationals Q are dense in R.

Limit sets

Because of the discrete action, the orbit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma z} of a point z in the upper half-plane under the action of Γ has no accumulation points in the upper half-plane. There may, however, be limit points on the real axis. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda(\Gamma)} be the limit set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} , that is, the set of limit points of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma z} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \in \mathbb{H}} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda(\Gamma) \subseteq \mathbb{R} \cup \infty} . The limit set may be empty, or may contain one or two points, or may contain an infinite number. In the later case, there are two types:

A Fuchsian group of the first kind is a group for which the limit set is the closed real line Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R} \cup \infty} .

A Fuchsian group of the second kind is a group for which the limit set is a perfect set that is nowhere dense on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R} \cup \infty} . This implies that the limit set is a closed set, as perfect sets are closed. Since it is nowhere dense, this implies that every 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.

Examples

By far the most prominent example of a Fuchsian group is the modular group, PSL(2,Z). This is the subgroup of PSL(2,R) consisting of linear fractional transformations

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where a,b,c,d are integers. The quotient space H/PSL(2,Z) is the moduli space of elliptic curves.

Other famous Fuchsian groups include the groups Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} (n) for each integer n > 0. Here Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} (n) consists of linear fractional transformations of the above form where the entries of the matrix

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are congruent to those of the identity matrix modulo n.

All these are Fuchsian groups of the first kind, meaning that the quotient of H by these groups has finite volume.

  • Any elliptic cyclic subgroup is Fuchsian if and only if it is finite.
  • Every abelian Fuchsian group is finite cyclic.
  • No Fuchsian group is isomorphic to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z} \times \mathbb{Z}} .
  • Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} be a non-abelian Fuchsian group. Then the normalizer of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} in PSL(2,R) is Fuchsian.

References

  • Svetlana Katok, Fuchsian Groups (1992), University of Chicago Press, Chicago ISBN 0-226-42583-5
  • Hershel M. Farkas, Irwin Kra, Theta Constants, Riemann Surfaces and the Modular Group, American Mathematical Society, Providence RI, ISBN 0-8218-1392-7 (See section 1.6)
  • Peter J. Nicholls, The Ergodic Theory of Discrete Groups, (1989) London Mathematical Society Lecture Note Series 143, Cambridge University Press, Cambridge ISBN 0-521-37674-2
  • Henryk Iwaniec, Spectral Methods of Automorphic Forms, Second Edition, (2002) (Volume 53 in Graduate Studies in Mathematics), America Mathematical Society, Providence, RI ISBN 0-8218-3160-7 (See Chapter 2).