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In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces. There are some variations of the definition: sometimes the Fuchsian group is assumed to be finitely generated, sometimes it is allowed to be a subgroup of PGL(2,R) (so that it contains orientation-reversing elements), and sometimes it is allowed to be a Kleinian group (a discrete subgroup of PSL(2,C)) which is conjugate to a subgroup of PSL(2,R).

Fuchsian groups are used to create Fuchsian models of Riemann surfaces. In this case, the group may be called the Fuchsian group of the surface. In some sense, Fuchsian groups do for non-Euclidean geometry what crystallographic groups do for Euclidean geometry. Some Escher graphics are based on them (for the disc model of hyperbolic geometry).

General Fuchsian groups were first studied by Henri Poincaré (1882), who was motivated by the paper (Fuchs 1880), and therefore named them after Lazarus Fuchs.

Fuchsian groups on the upper half-plane

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Let be the upper half-plane. Then is a model of the hyperbolic plane when endowed with the metric

The group PSL(2,R) acts on by linear fractional transformations (also known as Möbius transformations):

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

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

  • For every in , the orbit has no accumulation point in .

An equivalent definition for to be Fuchsian is that be a discrete group, which means that:

  • Every sequence of elements of converging to the identity in the usual topology of point-wise convergence is eventually constant, i.e. there exists an integer such that for all , , where 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 full Riemann sphere (as opposed to ). Indeed, the Fuchsian group PSL(2,Z) is discrete but has accumulation points on the real number line : elements of PSL(2,Z) will carry to every rational number, and the rationals Q are dense in R.

General definition

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A linear fractional transformation defined by a matrix from PSL(2,C) will preserve the Riemann sphere P1(C) = C ∪ ∞, but will send the upper-half plane H to some open disk Δ. Conjugating by such a transformation will send a discrete subgroup of PSL(2,R) to a discrete subgroup of PSL(2,C) preserving Δ.

This motivates the following definition of a Fuchsian group. Let Γ ⊂ PSL(2,C) act invariantly on a proper, open disk Δ ⊂ C ∪ ∞, that is, Γ(Δ) = Δ. Then Γ is Fuchsian if and only if any of the following three equivalent properties hold:

  1. Γ is a discrete group (with respect to the standard topology on PSL(2,C)).
  2. Γ acts properly discontinuously at each point z ∈ Δ.
  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 so-called Picard group PSL(2,Z[i]) is discrete but does not preserve any disk in 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.

It is most usual to take the invariant domain Δ to be either the open unit disk or the upper half-plane.

Limit sets

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Because of the discrete action, the orbit Γz 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 Λ(Γ) be the limit set of Γ, that is, the set of limit points of Γz for z ∈ H. Then Λ(Γ) ⊆ R ∪ ∞. The limit set may be empty, or may contain one or two points, or may contain an infinite number. In the latter case, there are two types:

A Fuchsian group of the first type is a group for which the limit set is the closed real line R ∪ ∞. This happens if the quotient space H/Γ has finite volume, but there are Fuchsian groups of the first kind of infinite covolume.

Otherwise, a Fuchsian group is said to be of the second type. Equivalently, this is a group for which the limit set is a perfect set that is nowhere dense on R ∪ ∞. 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.

Examples

[edit]

An 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

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

Other Fuchsian groups include the groups Γ(n) for each integer n > 0. Here Γ(n) consists of linear fractional transformations of the above form where the entries of the matrix

are congruent to those of the identity matrix modulo n.

A co-compact example is the (ordinary, rotational) (2,3,7) triangle group, containing the Fuchsian groups of the Klein quartic and of the Macbeath surface, as well as other Hurwitz groups. More generally, any hyperbolic von Dyck group (the index 2 subgroup of a triangle group, corresponding to orientation-preserving isometries) is a Fuchsian group.

All these are Fuchsian groups of the first kind.

  • All hyperbolic and parabolic cyclic subgroups of PSL(2,R) are Fuchsian.
  • Any elliptic cyclic subgroup is Fuchsian if and only if it is finite.
  • Every abelian Fuchsian group is cyclic.
  • No Fuchsian group is isomorphic to Z × Z.
  • Let Γ be a non-abelian Fuchsian group. Then the normalizer of Γ in PSL(2,R) is Fuchsian.

Metric properties

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If h is a hyperbolic element, the translation length L of its action in the upper half-plane is related to the trace of h as a 2×2 matrix by the relation

A similar relation holds for the systole of the corresponding Riemann surface, if the Fuchsian group is torsion-free and co-compact.

See also

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References

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Fuchsian group

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A Fuchsian group is a discrete of the projective PSL(2, ℝ), consisting of 2×2 real matrices with 1, up to scalar multiples, and acting via Möbius transformations on the upper half-plane model of the hyperbolic plane. This action is faithful, orientation-preserving, and properly discontinuous, meaning that for any compact set in the upper half-plane, only finitely many group elements map it to overlapping regions, ensuring the quotient space is a manifold. The term "Fuchsian" honors the German mathematician Lazarus Fuchs, whose work on linear differential equations with regular singular points inspired the concept, particularly through connections to hypergeometric functions. systematically developed the theory in his 1882 paper "Théorie des groupes fuchsiens," where he explored these groups as tools for constructing automorphic functions and addressing the uniformization of Riemann surfaces. Poincaré's insights revealed that such groups generate discrete actions that tile the hyperbolic plane via fundamental domains, like polygons with paired sides satisfying angle conditions. Fuchsian groups play a central role in hyperbolic geometry and topology, as their quotients by torsion-free subgroups yield Riemann surfaces of genus g≥2g \geq 2, endowing them with a natural hyperbolic metric via the uniformization theorem. Examples include the modular group PSL(2, ℤ), which is Fuchsian and classifies elliptic curves over the complex numbers, and translation subgroups like {z↦z+n∣n∈Z}\{ z \mapsto z + n \mid n \in \mathbb{Z} \}, generating infinite-area quotients with cusps. Elements are classified by trace: elliptic (∣tr⁡∣<2|\operatorname{tr}| < 2), parabolic (∣tr⁡∣=2|\operatorname{tr}| = 2), or hyperbolic (∣tr⁡∣>2|\operatorname{tr}| > 2), determining fixed points and dynamics on the boundary circle. Their study extends to arithmetic cases, Shimura curves, and representations in number theory, while geometrically finite subgroups admit fundamental polygons whose areas relate to the Euler characteristic by Gauss-Bonnet: μ(H2/Γ)=2π(2g−2+∑(1−1/mi))\mu(\mathbb{H}^2 / \Gamma) = 2\pi (2g - 2 + \sum (1 - 1/m_i)), where gg is the genus and mim_i are elliptic orders.

Definitions

On the Upper Half-Plane

The upper half-plane H\mathbb{H} is the open set {z=x+iy∈C∣y>0}\{ z = x + iy \in \mathbb{C} \mid y > 0 \} in the complex plane, endowed with the Riemannian metric ds2=dx2+dy2y2ds^2 = \frac{dx^2 + dy^2}{y^2}, which induces a geometry of constant curvature −1-1. This metric defines the hyperbolic distance between points, ensuring that geodesics are either vertical lines or semicircles orthogonal to the real axis, and it models the hyperbolic plane H2\mathbb{H}^2. The group of orientation-preserving isometries of H\mathbb{H} under this metric is the projective special linear group PSL(2,R)\mathrm{PSL}(2, \mathbb{R}), consisting of 2×22 \times 2 real matrices with determinant 1, modulo {±I}\{\pm I\}. Elements of PSL(2,R)\mathrm{PSL}(2, \mathbb{R}) act on H\mathbb{H} via Möbius transformations of the form z↦az+bcz+dz \mapsto \frac{az + b}{cz + d}, where a,b,c,d∈Ra, b, c, d \in \mathbb{R} and ad−bc=1ad - bc = 1. These transformations preserve the hyperbolic metric, mapping geodesics to geodesics and fixing the real line as the boundary at infinity. A Fuchsian group Γ\Gamma is a discrete subgroup of PSL(2,R)\mathrm{PSL}(2, \mathbb{R}), where discreteness means that Γ\Gamma has no accumulation points in the topological space PSL(2,R)\mathrm{PSL}(2, \mathbb{R}) equipped with the subspace topology from GL(2,R)\mathrm{GL}(2, \mathbb{R}). Equivalently, the action of Γ\Gamma on H\mathbb{H} is properly discontinuous: for every z∈Hz \in \mathbb{H}, there exists a neighborhood UU of zz such that γU∩U=∅\gamma U \cap U = \emptyset for all γ∈Γ∖{e}\gamma \in \Gamma \setminus \{e\}, ensuring that orbits Γz\Gamma z are discrete subsets of H\mathbb{H} with no limit points in H\mathbb{H}. This condition excludes elliptic elements of infinite order, as such rotations around a fixed point in H\mathbb{H} would generate accumulating orbits. Fuchsian groups were introduced by in 1882 as discrete groups of Möbius transformations generating motions in non-Euclidean (, motivated by the uniformization of Riemann surfaces. They represent the two-dimensional analogue of the more general Kleinian groups, which act on hyperbolic 3-space.

General Formulation

A Fuchsian group is defined abstractly as a discrete of the group of orientation-preserving isometries of the hyperbolic plane H2\mathbb{H}^2. Proper discontinuity of the action means that for every compact subset K⊂H2K \subset \mathbb{H}^2, the set {g∈G∣gK∩K≠∅}\{ g \in G \mid gK \cap K \neq \emptyset \} is finite. This ensures that the quotient space H2/G\mathbb{H}^2 / G is a hyperbolic 2-orbifold. Every Fuchsian group GG is conjugate in the full of H2\mathbb{H}^2 to a of PSL(2,R)\mathrm{PSL}(2, \mathbb{R}), the group of [2×](/page/2Times)2[2 \times](/page/2_Times) 2 real matrices of 1 acting by Möbius transformations on the upper half-plane model of H2\mathbb{H}^2. Fuchsian groups form a special case of Kleinian groups, which are discrete s of PSL(2,C)\mathrm{PSL}(2, \mathbb{C}) acting on hyperbolic 3-space; specifically, a Kleinian group is Fuchsian if its lies on a in the at infinity. The fundamental groups of closed orientable hyperbolic surfaces provide canonical examples of Fuchsian groups, as each such surface is isometric to H2/π1(S)\mathbb{H}^2 / \pi_1(S) for a surface SS of g≥2g \geq 2. The Ahlfors–Bers theorem establishes a parametrization of the of Fuchsian groups of fixed topological type via quasiconformal extensions: for a finitely generated Fuchsian group Γ\Gamma of the first kind, the T(Γ)T(\Gamma) is realized as an open domain in the complex of holomorphic quadratic differentials on the quotient surface, where points correspond to equivalence classes of quasiconformal deformations of Γ\Gamma with Beltrami coefficients bounded by the Bers constant.

Examples

Elementary Fuchsian Groups

Elementary Fuchsian groups are discrete subgroups of PSL(2,R)\mathrm{PSL}(2,\mathbb{R}) whose limit sets contain at most two points on the boundary ∂H\partial \mathbb{H} of the upper half-plane H\mathbb{H}. These groups are characterized by having a finite orbit under the group action on ∂H\partial \mathbb{H}, making their dynamics simple compared to non-elementary cases. Up to conjugacy in PSL(2,R)\mathrm{PSL}(2,\mathbb{R}), elementary Fuchsian groups are either cyclic or isomorphic to the infinite dihedral group D∞D_\infty. Cyclic elementary Fuchsian groups are generated by a single non-identity element g∈PSL(2,[R](/page/R))g \in \mathrm{PSL}(2,\mathbb{[R](/page/R)}), whose type is determined by the of the trace of a lift to SL(2,R)\mathrm{SL}(2,\mathbb{R}): elliptic if ∣tr(g)∣<2|\mathrm{tr}(g)| < 2, parabolic if ∣tr(g)∣=2|\mathrm{tr}(g)| = 2, or hyperbolic if ∣tr(g)∣>2|\mathrm{tr}(g)| > 2. Elliptic cyclic groups are finite, isomorphic to Zn\mathbb{Z}_n for some n≥2n \geq 2, and conjugate to subgroups of the rotation group SO(2)\mathrm{SO}(2) acting by finite-order rotations around a fixed point in H\mathbb{H}. For example, the group generated by an elliptic element of order nn fixes a point in H\mathbb{H} and has an empty . Parabolic cyclic groups are infinite, isomorphic to Z\mathbb{Z}, and fix exactly one point on ∂H\partial \mathbb{H}; a standard example is the subgroup ⟨z↦z+1⟩\langle z \mapsto z + 1 \rangle, which acts by horizontal translations and produces a horocyclic consisting of points at constant imaginary part approaching the cusp at . Hyperbolic cyclic groups are also infinite, isomorphic to Z\mathbb{Z}, and fix two distinct points on ∂H\partial \mathbb{H}, with the expanding distances along the axis connecting these points. Non-cyclic elementary Fuchsian groups are conjugate to D∞D_\infty, generated by two elements fixing the same point on ∂H\partial \mathbb{H} (e.g., two parabolic elements) or by a hyperbolic element and an order-2 elliptic element interchanging the fixed points of the hyperbolic one. These groups have limit sets consisting of one or two points. All elementary Fuchsian groups are virtually abelian, with rank at most 2, and their up to conjugacy is fully determined by the traces and fixed points of generators.

Non-Elementary Fuchsian Groups

Non-elementary Fuchsian groups are discrete subgroups of PSL(2,R)\mathrm{PSL}(2, \mathbb{R}) whose limit sets consist of infinitely many points on the boundary R∪{∞}\mathbb{R} \cup \{\infty\} of the upper half-plane H\mathbb{H}. This distinguishes them from elementary Fuchsian groups, which have limit sets containing at most two points and are either finite or virtually cyclic. Non-elementary groups exhibit rich dynamics, often being free or surface groups, and play a central role in the uniformization of Riemann surfaces of genus greater than 1. A canonical example is Γ=PSL(2,[Z](/page/Z))\Gamma = \mathrm{PSL}(2, \mathbb{[Z](/page/Z)}), which acts on H\mathbb{H} via Möbius transformations with integer coefficients. It is generated by the elliptic element S:z↦−1/zS: z \mapsto -1/z of order 2 and the parabolic element T:z↦z+1T: z \mapsto z + 1. The fundamental domain for this action is the defined by ∣Re(z)∣≤1/2|\mathrm{Re}(z)| \leq 1/2 and ∣z∣≥1|z| \geq 1, which tiles H\mathbb{H} under the , yielding the modular surface as the . This group, first studied by Poincaré in connection with automorphic functions, has finite covolume and limit set the entire real line, showcasing Cantor-like structure in its dynamics. Triangle groups provide a broad family of non-elementary Fuchsian groups associated with hyperbolic tessellations. The (p,q,r)(p, q, r)- is generated by rotations of orders pp, qq, and rr about the vertices of an ideal or finite with corresponding angles π/p\pi/p, π/q\pi/q, and π/r\pi/r, where 1/p+1/q+1/r<11/p + 1/q + 1/r < 1. These groups are cocompact when all orders are finite and act by tessellating H\mathbb{H} with congruent triangles, producing quotients that are hyperbolic surfaces. For instance, the (2,3,7)- uniformizes the Bolza surface, a genus-2 Riemann surface with high symmetry. Hecke groups extend the modular group to a parameterized family. The Hecke group HqH_q for integer q≥3q \geq 3 is generated by S:z↦−1/zS: z \mapsto -1/z and Uq:z↦z+2cos⁡(π/q)U_q: z \mapsto z + 2\cos(\pi/q), reducing to the modular group when q=3q=3. These are Fuchsian groups of the first kind with a fundamental domain that is an infinite hyperbolic triangle with vertices at cusps and angles 00, π/2\pi/2, π/q\pi/q. They have finite covolume and are used in studying modular forms and cusp forms on non-compact surfaces. Fuchsian Schottky groups offer freely generated examples of non-elementary Fuchsian groups, constructed via pairing disjoint semicircles (or intervals) on the boundary. A classical Fuchsian Schottky group of rank g≥2g \geq 2 is freely generated by 2g2g hyperbolic elements pairing gg pairs of disjoint boundary arcs, with the region between each pair serving as a generator of the fundamental domain. Every free Fuchsian group of finite rank at least 2 arises this way, and their quotients are compact of genus gg. The limit set is a Cantor set, and these groups were shown by to uniformize punctured spheres before Maskit's generalization to higher genera. Infinite dihedral groups refer to non-cyclic virtually cyclic infinite Fuchsian subgroups isomorphic to D∞D_\infty, which stabilize a geodesic in H\mathbb{H} and are generated by a hyperbolic element (acting by translations along the geodesic) and an order-2 elliptic element (interchanging the fixed points on the boundary). Groups generated by a single hyperbolic element are instead infinite cyclic. However, these are all elementary, and non-elementary examples like the above exhibit more complex orbital dynamics on infinite limit sets.

Limit Sets

Construction and Basic Properties

The limit set Λ(Γ)\Lambda(\Gamma) of a Fuchsian group Γ≤PSL(2,R)\Gamma \leq \mathrm{PSL}(2,\mathbb{R}) acting on the upper half-plane H\mathbb{H} is defined as the closure of the accumulation points of the orbit Γ⋅z0\Gamma \cdot z_0 for any fixed z0∈Hz_0 \in \mathbb{H}, and this set is contained in the boundary ∂H=R∪{∞}\partial \mathbb{H} = \mathbb{R} \cup \{\infty\}. Equivalently, Λ(Γ)=⋃z∈HAP∂H(Γz)\Lambda(\Gamma) = \bigcup_{z \in \mathbb{H}} \mathrm{AP}_{\partial \mathbb{H}}(\Gamma z), where AP∂H(Γz)\mathrm{AP}_{\partial \mathbb{H}}(\Gamma z) denotes the set of accumulation points of Γz\Gamma z in ∂H\partial \mathbb{H}. This definition is independent of the choice of z0z_0, as all orbits accumulate in the same closed subset of the boundary. The limit set can be constructed explicitly in certain cases, such as through sequences of nested intervals in the classical Schottky construction for free , where pairs of disjoint intervals are mapped to each other by group generators, and the limit set emerges as the intersection of shrinking nested components. More generally, points in Λ(Γ)\Lambda(\Gamma) arise as radial limits: for a sequence {γn}⊂Γ\{\gamma_n\} \subset \Gamma with γnz→ξ∈∂H\gamma_n z \to \xi \in \partial \mathbb{H} along geodesics from z∈Hz \in \mathbb{H} to ξ\xi, where the hyperbolic distance d(z,γnz)→∞d(z, \gamma_n z) \to \infty. For non-elementary , Λ(Γ)\Lambda(\Gamma) is perfect—closed and containing no isolated points—and compact as a closed subset of the compactified real line RP1\mathbb{RP}^1. Basic properties of Λ(Γ)\Lambda(\Gamma) include invariance under the action of Γ\Gamma, meaning γΛ(Γ)=Λ(Γ)\gamma \Lambda(\Gamma) = \Lambda(\Gamma) for all γ∈Γ\gamma \in \Gamma, which follows directly from the group action preserving accumulation points of orbits. For Schottky-type Fuchsian groups, generated freely by hyperbolic elements pairing disjoint intervals, Λ(Γ)\Lambda(\Gamma) is nowhere dense in RP1\mathbb{RP}^1, forming a Cantor-like subset with empty interior. The Hausdorff dimension dim⁡HΛ(Γ)\dim_H \Lambda(\Gamma) satisfies 0<dim⁡HΛ(Γ)≤10 < \dim_H \Lambda(\Gamma) \leq 1, with the upper bound reflecting the one-dimensional nature of the boundary and equality holding for groups of the first kind whose limit sets fill R∪{∞}\mathbb{R} \cup \{\infty\}. Points in Λ(Γ)\Lambda(\Gamma) are classified as ordinary or exceptional based on the nature of radial limits. An ordinary point ξ∈Λ(Γ)\xi \in \Lambda(\Gamma) admits radial limits of holomorphic functions or geodesics approaching ξ\xi, while exceptional points lack such uniform approximation; however, the set of radial limit points is dense in Λ(Γ)\Lambda(\Gamma) for non-elementary groups. Fuchsian groups, as a subclass of Kleinian groups, inherit the property that their limit sets have Lebesgue measure zero on R∪{∞}\mathbb{R} \cup \{\infty\} when the group is of the second kind (i.e., the limit set is not the entire boundary), as established in Ahlfors' finiteness theorem and subsequent resolutions of the measure conjecture for finitely generated cases. By the dichotomy theorem, the limit set of a Fuchsian group has Lebesgue measure zero or full measure on the boundary, with the latter occurring precisely for groups of the first kind.

Classification of Limit Sets

Fuchsian groups are classified as elementary or non-elementary based on the cardinality and structure of their limit sets. Elementary Fuchsian groups have limit sets consisting of at most two points on the real projective line RP1\mathbb{RP}^1. The trivial group has an empty limit set (0 points). Cyclic groups generated by a parabolic element have a limit set consisting of a single point, corresponding to the cusp at infinity. Cyclic groups generated by a hyperbolic element have a limit set of exactly two points, which are the fixed points of the generator and serve as the endpoints of its geodesic axis in the hyperbolic plane. Non-elementary Fuchsian groups have perfect limit sets, meaning they are closed, nonempty, and every point is a limit point. These limit sets exhibit diverse structures: for Schottky groups, which are free groups generated by hyperbolic elements satisfying a ping-pong condition, the limit set is a totally disconnected Cantor set of measure zero. In contrast, for lattices such as the modular group PSL(2,Z)\mathrm{PSL}(2,\mathbb{Z}), the limit set is the entire RP1\mathbb{RP}^1, filling the boundary completely and having positive (full) measure. Fuchsian groups of the first kind, which include such lattices, always have the full RP1\mathbb{RP}^1 as their limit set, while those of the second kind have proper subsets. Sullivan developed a classification of measures on limit sets via conformal densities, extending Patterson's construction for Fuchsian groups. A conformal density of dimension δ(Γ)\delta(\Gamma) is a family of measures {μx}x∈H2\{\mu_x\}_{x \in \mathbb{H}^2} on the limit set Λ(Γ)\Lambda(\Gamma) satisfying dμxdμy(ξ)=eδ(Γ)bξ(x,y)\frac{d\mu_x}{d\mu_y}(\xi) = e^{\delta(\Gamma) b_\xi(x,y)} for ξ∈Λ(Γ)\xi \in \Lambda(\Gamma), where bξb_\xi is the Busemann function; these densities are invariant under the group action and unique up to scalar if the action is ergodic. For Fuchsian groups, such densities support the Patterson-Sullivan measures when the Poincaré series diverges at the critical exponent. The Hausdorff dimension of the limit set Λ(Γ)\Lambda(\Gamma) equals the critical exponent δ(Γ)\delta(\Gamma) of the Poincaré series, defined as the infimum of s>0s > 0 such that ∑γ∈Γe−sd(o,γo)<∞\sum_{\gamma \in \Gamma} e^{-s d(o, \gamma o)} < \infty for a basepoint o∈H2o \in \mathbb{H}^2: dim⁡HΛ(Γ)=δ(Γ).\dim_H \Lambda(\Gamma) = \delta(\Gamma). This equality holds for non-elementary Fuchsian groups, with δ(Γ)≤1\delta(\Gamma) \leq 1; equality to 1 occurs precisely for groups of the first kind. Patterson-Sullivan measures, constructed as weak limits of measures from the Poincaré series at the critical exponent, are finite, nonatomic, and supported on Λ(Γ)\Lambda(\Gamma). For Fuchsian groups without parabolic elements, these measures are mutually absolutely continuous with respect to Hausdorff measure of dimension δ(Γ)\delta(\Gamma) and are δ(Γ)\delta(\Gamma)-conformal, satisfying d(μ∘γ−1)dμ(ξ)=∥γ′(ξ)∥δ(Γ)\frac{d(\mu \circ \gamma^{-1})}{d\mu}(\xi) = \| \gamma'(\xi) \|^{\delta(\Gamma)} for ξ∈Λ(Γ)\xi \in \Lambda(\Gamma). In the context of hyperbolic groups, they provide a geometric measure of maximal entropy on the boundary, facilitating the study of thermodynamic formalism and equidistribution of orbits.

Geometric Properties

Fundamental Domains and Tessellations

A fundamental domain for a Fuchsian group Γ\Gamma acting on the upper half-plane H\mathbb{H} is a closed region D⊂HD \subset \mathbb{H} such that the union Γ⋅D=H\Gamma \cdot D = \mathbb{H} and the interiors of distinct translates gD∘gD^\circ for g∈Γg \in \Gamma are disjoint, ensuring DD contains exactly one representative from each orbit under the group action. This construction allows Γ\Gamma to tile H\mathbb{H} via the translates Γ⋅D\Gamma \cdot D, providing a fundamental region whose boundary identifications yield the quotient space. The Dirichlet fundamental domain centered at a point p∈Hp \in \mathbb{H} with trivial stabilizer is defined as D(p)={z∈H:d(z,p)≤d(gz,p) ∀g∈Γ}D(p) = \{ z \in \mathbb{H} : d(z, p) \leq d(gz, p) \ \forall g \in \Gamma \}, where dd denotes the hyperbolic distance, forming the convex hull of points in H\mathbb{H} closer to pp than to any other group translate g(p)g(p). Its boundary consists of geodesic segments (sides) perpendicularly bisecting the geodesics joining pp to g(p)g(p), with each side paired to another via a unique group element g≠1g \neq 1, generating Γ\Gamma through these side-pairings. For finitely generated Γ\Gamma of cofinite area, D(p)D(p) is a hyperbolic polygon with finitely many sides. For infinite-volume Fuchsian groups, the Ford domain provides an alternative construction, defined in the unit disk model D\mathbb{D} as the closure of the intersection over g∈Γ∖{1}g \in \Gamma \setminus \{1\} of the exteriors of the isometric circles of gg, intersected with a fundamental domain for any parabolic subgroup stabilizing infinity, such as a vertical strip. This yields a fundamental domain that is both Dirichlet and Ford (DF domain) precisely when Γ\Gamma is an index-2 subgroup of a reflection group generated by reflections in the sides of a hyperbolic polygon. Fuchsian triangle groups, generated by rotations around the vertices of a hyperbolic triangle with angles π/p\pi/p, π/q\pi/q, π/r\pi/r where 1/p+1/q+1/r<11/p + 1/q + 1/r < 1, produce regular tessellations of H\mathbb{H} by congruent copies of that triangle, with vertices of orders pp, qq, rr. For the (2,3,7)(2,3,7) triangle group, this tessellation underlies the Hurwitz surface of genus 3, the compact Riemann surface of maximal symmetry, obtained as a torsion-free subgroup of index 168. The quotient H/Γ\mathbb{H}/\Gamma forms a hyperbolic surface, a non-compact Riemann surface of finite area if Γ\Gamma has cofinite area, or an orbifold if finite stabilizers (elliptic points) are present, with the topology determined by the side-pairings of the fundamental domain. On this quotient, closed geodesics correspond to conjugacy classes of hyperbolic elements in Γ\Gamma, and the Selberg zeta function, defined as Z(s)=∏{γ}∏k=0∞(1−e−(s+k)ℓ(γ))Z(s) = \prod_{\{\gamma\}} \prod_{k=0}^\infty (1 - e^{-(s+k) \ell(\gamma)}) over primitive closed geodesics of length ℓ(γ)\ell(\gamma), encodes their lengths and relates to the spectrum of the Laplacian on L2(H/Γ)L^2(\mathbb{H}/\Gamma).

Metric Structures

The upper half-plane model of the hyperbolic plane H2\mathbb{H}^2 carries the Riemannian metric ds2=dx2+dy2y2ds^2 = \frac{dx^2 + dy^2}{y^2}, where points are represented as z=x+iyz = x + iy with y>0y > 0. This metric induces the hyperbolic distance between two points z,w∈H2z, w \in \mathbb{H}^2 given by d(z,w)=\arcosh(1+∣z−w∣22ℑ(z)ℑ(w)).d(z, w) = \arcosh\left(1 + \frac{|z - w|^2}{2 \Im(z) \Im(w)}\right). The geodesics in this model, which are the shortest paths minimizing this distance, consist uniquely of semicircles centered on the real axis and orthogonal to it, or vertical rays extending to . These geodesics are invariant under the isometric action of Fuchsian groups, preserving the hyperbolic structure on the . For a hyperbolic element γ\gamma in a Fuchsian group Γ⊂PSL(2,R)\Gamma \subset \mathrm{PSL}(2, \mathbb{R}), the translation length ℓ(γ)\ell(\gamma) measures the minimal displacement along the unique geodesic axis fixed by γ\gamma, computed as ℓ(γ)=2\arcosh(∣tr⁡(γ)∣2),\ell(\gamma) = 2 \arcosh\left(\frac{|\operatorname{tr}(\gamma)|}{2}\right), where tr⁡(γ)\operatorname{tr}(\gamma) is the trace of a matrix representative in SL(2,R)\mathrm{SL}(2, \mathbb{R}). This length quantifies the hyperbolic isometry's effect and remains invariant under conjugation within the group. On the quotient surface X=Γ\H2X = \Gamma \backslash \mathbb{H}^2, such lengths correspond to closed geodesics, influencing the surface's geometry. The relates the geometry of XX to its topology: for a complete hyperbolic surface of constant −1-1, the area satisfies area⁡(X)=−2πχ(X)\operatorname{area}(X) = -2\pi \chi(X), where the is χ(X)=2−2g−n\chi(X) = 2 - 2g - n for a surface of gg with nn punctures. This formula arises from integrating the form over a fundamental domain and applying the theorem's general statement for surfaces. For non-compact surfaces arising from Fuchsian groups, the metric completion includes infinite-area ends: cusps, corresponding to parabolic fixed points, form horocyclic neighborhoods with exponentially decaying widths, while funnels, associated with hyperbolic elements bounding the group, exhibit flaring hyperbolic funnels of infinite area. A key metric decomposition of hyperbolic surfaces is the thick-thin partition, defined using the Margulis constant ε>0\varepsilon > 0, a (approximately 0.263 in dimension 2) such that the ε\varepsilon-thin part consists of embedded tubes around geodesics shorter than ε\varepsilon and neighborhoods of cusps, while the thick part has injectivity radius at least ε/2\varepsilon/2. This decomposition highlights regions of high distortion near short geodesics or cusps, with the thin parts having bounded diameter in the thick complement. Such structures can be analyzed via fundamental domains to compute volumes of these components.

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