An effective bound for the Huber constant for cofinite Fuchsian groups

Let $\Gamma$ be a cofinite Fuchsian group acting on hyperbolic two-space $\HH.$ Let $M=\Gamma \setminus \HH $ be the corresponding quotient space. For $\gamma,$ a closed geodesic of $M$, let $l(\gamma)$ denote its length. The prime geodesic counting function $\pi_{M}(u)$ is defined as the number of $\Gamma$-inconjugate, primitive, closed geodesics $\gamma $ such that $e^{l(\gamma)} \leq u.$ The \emph{prime geodesic theorem} implies: $$\pi_{M}(u)=\sum_{0 \leq \lambda_{M,j} \leq 1/4} \text{li}(u^{s_{M,j}}) + O_{M}(\frac{u^{3/4}}{\log{u}}), $$ where $0=\lambda_{M,0}<\lambda_{M,1}<...$ are the eigenvalues of the hyperbolic Laplacian acting on the space of smooth functions on $M$ and $s_{M,j} = \frac{1}{2}+\sqrt{\frac{1}{4} - \lambda_{M,j}}. $ Let $C_{M}$ be smallest implied constant so that $$|\pi_{M}(u)-\sum_{0 \leq \lambda_{M,j} \leq 1/4} \text{li}(u^{s_{M,j}})|\leq C_{M}\frac{u^{3/4}}{\log{u}} \quad \text{\text{for all} $u>1.$}$$ We call the (absolute) constant $C_{M}$ the Huber constant. The objective of this paper is to give an effectively computable upper bound of $C_{M}$ for an arbitrary cofinite Fuchsian group. As a corollary we estimate the Huber constant for $\PSL(2,\ZZ),$ we obtain $C_{M} \leq 16,607,349,020,658 \approx \exp(30.44086643)$.


INTRODUCTION
Let Γ be a cofinite Fuchsian group 1 , and let M = Γ \ H be the corresponding hyperbolic orbifold.Let C(M ) denote the set of closed geodesics of M, and let P(M ) denote the set of prime (or primitive) closed geodesics (see [Bus92,page 245]).For each γ ∈ C(M ) there exists a unique prime geodesic γ 0 and a unique exponent m ≥ 1 so that γ = γ m 0 .Let l(γ) denote the length of γ.Associated to γ is a unique hyperbolic conjugacy class {P γ } Γ with norm N γ ≡ N (P γ ) = e l(γ) .
The prime geodesic counting function π M (u) is defined to be the number of Γinconjugate, primitive, hyperbolic elements γ ∈ Γ such that e lγ < u.
Selberg [Sel56] and Huber [Hub59,Hub61a,Hub61b] independently proved the prime geodesic theorem, which is the asymptotic formula Later, Huber proved a stronger version of the prime geodesic theorem, with error terms 2 : li(u sM,j ) + O M u 3/4 log u , where 0 = λ M,0 < λ M,1 < • • • are the eigenvalues of the hyperbolic Laplacian acting on the space of smooth functions on M and For a proof of this theorem, via the Selberg zeta function, see [Hej76,Hej83] and [Ran78].Randol [Ran84] and Sarnak [Sar80] also gave of proof using the Selberg trace formula.We call the implied constant inherent in O M ( u 3/4 log u ) the Huber constant, denoted by C M .The Huber constant is an absolute constant, not an asymptotic constant 3 .The Huber constant is the minimal constant that satisfies π M (u) − 0≤λM,j ≤.1/4 li(u sM,j ) ≤ C M u 3/4 log u for all u > 1.
The main goal of this paper is to estimate C M for an arbitrary cofinite Fuchsian group.We determine an effective algorithm by which one can obtain explicit bounds for C M using elementary geometric and spectral theoretic data associated to M .In particular, we apply the algorithm in the case of the full modular group PSL(2, Z) and obtain a precise, numerical bound for the Huber constant.
In [JK02], the authors studied the Huber constant in the following situation: Let Γ 0 be a cofinite Fuchsian group, and let Γ be a finite-index subgroup of Γ 0 .They showed that C M ≤ [Γ 0 : Γ] • C M0 .Hence if one could estimate this constant for Γ 0 = PSL(2, Z), one would have an estimate for any congruence subgroup.For more applications of these ideas, see [JK09,JK06,JK01].
In the articles [JK09], [JK06] and [JK01], the authors studied analytic aspects of Arakelov theory, ultimately developing bounds for special values of Selberg's zeta functions, Green's functions and Faltings's delta function.The bounds for these analytic functions involved many explicitly computable analytic and geometric quantity, together with the Huber constant.As such, specific, effective bounds for the Huber constant then could be used to make the bounds derived in [JK09], [JK06] and [JK01] effective.With this said, the results in the present paper complete the analysis in the aforementioned articles, thus providing effective, numerically computable, bounds for various analytic quantities which appear in the Arakelov theory of algebraic curves.
Main result.Our main result is Theorem 4.15 (see §4.8), where we determine an upper bound for the Huber constant C M in terms of various well studied invariants of Γ and M such as: the number of small eigenvalues A, the number of exceptional poles of the scattering matrix N , the smallest positive eigenvalue λ 1 , the length of the smallest closed geodesic of M, the decomposition of the fundamental For cocompact Γ d is the (hyperbolic) diameter of F A number of eigenvalues λ of ∆, such that λ ≤ 1 4 , including λ 0 = 0 here λ 1 is the smallest non-zero eigenvalue are all the poles of φ(s) in the interval ( 1 2 , 1] c 1 minimum of the lower left hand matrix entry of non-parabolic elements c chosen to satisfy 1 < c < N (Γ) for all hyperbolic γ ∈ Γ µ log c, a positive lower bound for the length of all closed geodesics domain of Γ into cusp sectors, the area of M, and a few other simple, easily computable, invariants4 .We could give the main result by stating our upper bound for C M .However the result is so complicated that it would take us two pages just to list out the equation.So, we present our results as an algorithm, rather than a formula.To help the reader appreciate the complexity (for the case of a general cofinite Fuchsian group), we state, as a corollary, the simple case of a cocompact, torsion-free Fuchsian group.
In the process of proving Theorem 4.15, we give an upper bound for the implied (absolute) constant of the spectral counting µ(r).See Theorem 3.3.In other words, for an arbitrary cofinite Fuchsian group, we find an explicit constant C, depending on Γ, so that Next, consider the following table of constants, which are valid only for Γ cocompact, torsion-free.
Theorem.Let Γ be a cocompact, torsion-free Fuchsian group.Then the constant C u is an upper bound for the Huber constant C M , and the constant C is an upper bound for the implied constant of the spectral counting function of ∆ .
In order to estimate C M for general cofinite Fuchsian group we need the following modified table.Some constants from the cocompact table are redefined.
Theorem.Let Γ be a cofinite Fuchsian group.Then the constant C u is an upper bound for the Huber constant C M , and the constant C is an upper bound for the implied constant of the spectral counting function of ∆ .
Using the theory laid out in [Sar82], and the excellent open-source computational program PARI/GP [The05], we can explicitly calculate (and list out) the length spectrum for PSL(2, Z).Our computations suggest that So it seems that our result is not sharp.
Why is our result for PSL(2, Z) so much larger than the experimental result?There are many reasons.Our bound for the spectral counting function is probably one million times too big (when compared to Weyl's law, an asymptotic result).We are mostly interested in how the Huber constant grows as a function of the various fundamental constants.Estimating the spectral counting function required absolute estimates of various infinite sums over primitive hyperbolic classes; and we used trivial bounds for the hyperbolic counting function.The non-trivial absolute bound being the sole objective of this paper.However, using PSL(2, Z) specific bounds on the spectral counting function, the result could be markedly improved.
Connection to existing literature.As stated above, the bound for the Huber constant obtained in the present paper allows one to complete the analysis in [JK09], [JK06] and [JK01], resulting in explicit and computable bounds for certain analytic quantities in Arakelov theory.We refer the interested reader to these articles for the precise statements of the main results, from which one can immediately employ the bounds obtained here in order to further strengthen the analysis presented in [JK09], [JK06] and [JK01].We will discuss further applicability of the present paper to other computational problems.
In [ECJMB], the authors used ideas from Arakelov theory and developed a strategy to compute Galois representations modulo ℓ associated to a fixed modular form of arbitrary weight.Ultimately, the goal of [ECJMB] is to devise an algorithm, which has complexity that is polynomial in ℓ.A summary of the ideas from [ECJMB] is given in [Edi06], where the purpose is to focus attention to the weight twelve modular form associated to PSL(2, Z).The ideas from [ECJMB] have been used to achieve advances in many other computation problems; see, for example, [Bo07], [CL09], [Cou09], and [La06] for specific results, as well as the survey article [Chl08] for more general comments.
As stated in [ECJMB], a key component of the algorithm was to determine bounds for Arakelov Green's functions.F. Merkl provided sufficient bounds to complete the algorithm; however, the bounds from [JK06] are sharper and, as a result, improve the efficiency of the algorithm from [ECJMB].Theorem 4.8 of [JK06] summarizes a bound for the Arakelov Green's function in terms of hyperbolic data, including a special value of the Selberg zeta function.This special value was studied in [JK09] and now can be bounded explicitly using the bounds for the Huber constant developed in the present paper.Thus, the bounds obtained for the Huber constant proved here ultimately can be used in the algorithms constructed in [ECJMB] and subsequent articles.
Going forward, it may be possible that the algorithms of [ECJMB], or similarly developed ideas, could yield further applications, in which case effective bounds for the analytic aspects of Arakelov theory would play a role in the implementation of the resulting algorithms.Indeed, there has been recent progress in expanding Arakelov theory to allow for singular, or non-compact metrics (see [BKK05], [BKK07], [FG09], and [Ha09]).As a result, one can anticipate further computational algorithms which utilize these recent extensions of Arakelov theory, in which case the bounds from [JK09], [JK06] and [JK01] and the present paper may play a role.
Outline.In brief, the present paper amounts to a careful analysis of the proof of the prime geodesic theorem, making sure that each step in the proof yields explicitly computable constants.However, straightforward the methodology may sound, there are many details which involve considerable analytic difficulties.
In §1 we prove some useful lemmas and develop necessary notation.Our argument consists of many estimates, each with an implicit constant, and our notation helps to keep track of these constants.At the heart of our argument is a variant of Karamata's Tauberian theorem (Lemma 1.1 in §1.2) that gives an absolute, instead of an asymptotic, result.
In §2 we study the heat kernel e − ∆ t of the Laplace-Beltrami operator for the hyperbolic orbifold M = Γ \ H.Our main tool is the Selberg trace formula.We specifically show that e − ∆ t = O(1/t) for t ∈ (0, 5], and find the corresponding explicit O-constant.Note that we focus on the interval (0,5] because small eigenvalues (λ < 1/4) are treated differently in many applications of the Selberg trace formula.We will estimate the spectral counting function µ(x) for λ ≥ 1/5 giving us a small margin of error from the 1/4 cutoff.
Next, in §3, using results from §2 and our variant of Karamata's Tauberian theorem (Lemma 1.1), we prove Theorem 3.3, which provides an effective bound for the spectral counting function.In §3.2 we specialize Theorem 3.3 to the case of Γ = PSL(2, Z).
Finally, in §4, using Theorem 3.3, we obtain an upper bound for the Huber constant (Theorem 4.15) and in §4.9, we specialize to the case of Γ = PSL(2, Z).

Concluding remarks.
We conclude the introduction by addressing a number of points which naturally arise when studying the computations presented in this article.
First, it is possible to further contract the algorithm for bounding the Huber constant and obtain a bound which can be related to the geometry of M .For example, the constant c 1 is the radius of the largest isometric circle in a certain fundamental domain for M ; see page 38 of [Iwa97].Also, the number of small eigenvalues can be bounded by the topology of M ; see [Zo82].However, to employ general results would only further worsen the bound for C M for special important cases.For example, for the congruence subgroups, the Selberg 1/4 conjecture asserts there is only one small eigenvalue, whereas the bound from [Zo82] implies that the number of small eigenvalues grows at most like the genus of the surface, which may be true in general cases but most likely is not true in special cases of particular importance.
Second, the heat kernel style of test functions which we analyze could be replaced by studying a different approach to the prime geodesic theorem, namely by finite difference methods applied to the test functions denoted by ψ k ; see [Hej83].
In our investigation, we saw no significant difference when looking ahead to the problem of bounding C M , so we simply selected one approach.We will leave the problem of studying other ways of proving the prime geodesic theorem to see if the methodologies lead to improved bounds for the Huber constant.
Finally, it is somewhat striking to see that in the case of the full modular group the bound we prove is approximately exp(30.45)whereas the numerical investigations lead one to believe that the true bound is log 2, which differ by an order of 10 14 .In Fall 2003, the second named author (J.J.) was giving a talk at the Courant Institute at NYU on preliminary results associated to [JK01].(As described in [ECJMB], it was then that Peter Sarnak communicated to the second named author (J.J.) the problem of bounding Arakelov Green's functions.)During the lecture, when the results of [JK09] were described, Sarnak stated that he felt it would be difficult to obtain a precise, numerical bound for the Huber constant for any group, including the full modular group.With this stated, the present paper successfully addresses the problem of determining precise, effective bounds for C M , and it remains to be seen how, or rather if, methods can be developed to refine the bound for the full modular group.

PRELIMINARY LEMMAS
In this section we group together certain lemmas that will be used throughout this paper.Our key lemma is a variant of Karamata's Tauberian Theorem.
1.1.Big 'O' notation.Let i ∈ N, and let f, g be functions with a common domain If the domain in question is not clear, we will explicitly state it.We call the constant C i absolute5 .1.2.Karamata's Tauberian theorem.In this section we prove a variation Karamata's Tauberian Theorem [Wid41,.Our version keeps track of the implied constants in the original theorem.
We will apply the lemma below with the parameter d = 5.Small eigenvalues (λ < 1/4) are treated differently in many applications of the Selberg trace formula.
We will estimate the spectral counting function µ(x) for λ ≥ 1/5 giving us a small margin of error from the 1/4 cutoff.Lemma 1.1.Let α(t) be a non-negative, non-decreasing function, continuous on all but a possibly countable subset of (0, ∞).Suppose that α(0) = 0 and that the integral converges for all s > 0. Suppose further that for C > 0, d > 0 Then Next, observing that g(e −w ) = e w 0 ≤ w ≤ 1 0 1 < w, an elementary calculation using the fact that α(0) = 0 shows that Hence we have Setting t = 1/s implies that Later on, we will let α(t) be the spectral counting function for the Laplacian ∆ .

Trivial counting lemma.
Let Γ be a cofinite Fuchsian group.Let π(x) denote the number of primitive conjugacy classes of hyperbolic elements (of Γ) of norm not exceeding x (the details are given below).In this section (following [Fis87, pages 47-49]) we give an explicit constant B so that for all x ≥ 1, π(x) ≤ Bx.This result is called the trivial counting lemma. 6et ρ(z, w) denote the hyperbolic distance on H.For T hyperbolic, define the norm of T, N (T ), by Note that the norm is constant within a fixed conjugacy class  Note that π(x) = 0 for x ≤ 1.
Here λ n = 1 4 + r 2 n are the eigenvalues of the hyperbolic Laplacian ∆ and {γ} Γ are the conjugacy classes for the hyperbolic elements of Γ with primitive generator γ 0 ; φ(s) is the determinant of the scattering matrix S(s); |F | is the hyperbolic area of the fundamental domain F of Γ; g(r) is the Fourier transform of h(z); {R} Γ are conjugacy classes for the elliptic elements of Γ with m R = |C(R)| (the order of the centralizer of R in Γ) and tr(R) = 2 cos(θ R ), 0 < θ R < π; τ is the number of cusps of Γ; and ψ(s) is the logarithmic derivative of the Gamma function.See see [Hej76,Hej83], [Iwa02], [Ven82], for more details.

HEAT KERNEL FOR THE HYPERBOLIC LAPLACIAN
Let Γ be a cofinite Fuchsian group, and let H(Γ) denote the Hilbert space of measurable functions f : Let ∆ : H(Γ) → H(Γ) be the self-adjoint extension of the (two-dimensional hyperbolic) Laplacian.In this section we will estimate (for Plugging in the pair h, g into the Selberg trace formula gives an explicit formula for the (regularized) trace of the heat kernel e − ∆ t : (2.1) where λ n = 1 4 + r 2 n are the eigenvalues of ∆ .We will now estimate each of the right-hand-side terms of equation (2.1) for 0 < t ≤ 5.
Elliptic term.Next we estimate the elliptic term 1 + e −2πr e −tr 2 dr, where m R are integers and 0 < θ R < π.
In particular, if .
The lemma now follows from the equation .

Identity term.
Lemma 2.3.For all t > 0 Hyperbolic term.Since Γ is discrete, there exists a hyperbolic conjugacy class {P s } with minimal norm N (P s ) > 1. Choose c > 0 so that Proof.First note that for a hyperbolic element P, N (P ) = N (P 0 ) j for some integer j ≥ 0. Since N (P ) > 1, N (P 0 ) ≤ N (P ).Noting that Rewriting the last sum as a Stieltjes integral Integrating by parts, and the trivial counting lemma (Lemma 1.3) yield The last estimate is a trivial estimate.

Spectral terms.
We have now bounded all the terms on the right side of the trace formula.By Lemmas 2.1, 2.2, 2.3, and 2.4; we have Lemma 2.5.Let

EXPLICIT BOUND FOR THE SPECTRAL COUNTING FUNCTION
We would like to apply Lemma 1.1 to the equation but a minor problem stops us: The real valued function −φ ′ φ ( 1 2 + ir) may be negative for some values of r.However, at the cost of a small error term, we will (following Selberg) replace the signed function −φ ′ φ ( 1 2 + ir) with a non-negative function.
(2) follows from elementary estimates: Taking the logarithmic derivative of both sides of the equation below plugging in s = 1 2 + ir, and recalling that Then the following hold: (2) Proof.(1) A simple calculation using Note that both sides of the above inequality are positive.Let Then rewriting (3.1) gives us But Lemma 3.1, and a crude estimate (see equation (3.6) for the idea of the estimate), show that and since (1) is now proved.
Lemma 3.2(3) immediately implies that We summarize our result:

Explicit estimate for the modular group.
Let Γ = PSL(2, Z).In this section we will give an explicit upper bound for the number C which depends on B, C 3 , C 4 , C 5 , C 6 , and C 7 .
The constant B. From the trivial counting lemma ( §1.3) we have where τ = 1.Let F be the standard fundamental domain of Γ The euclidian distance of F , to the x-axis, is ǫ = √ 3 2 .We can decompose F = F 0 ∪ F (Y ), where Im z Im w , where ρ(z, w) is the hyperbolic distance between z and w.An elementary calculation shows that d 1 < 2.3.Hence B < 753.The estimate for C is probably not sharp.However, we did not use any of the number theoretic properties of PSL(2, Z) to arrive at the estimate; we only assumed that Γ was a cofinite Fuchsian group.

BOUND FOR THE HUBER CONSTANT
In this section, we adapt Randol's proof ([Ran84]) of the prime geodesic theorem to an arbitrary cofinite Fuchsian group.Along the way, using Theorem 3.3, we will obtain an upper bound for the Huber constant C M .Our main references are [Ran84, pages 295-300] and [Bus92, Section 9.6].
4.1.Explicit bump function.For T > 0 set For a functions f, g on R, we set and Lemma 4.1.There exists a nonnegative function φ(x) with the following properties: and let (1) and ( 2) now follow from the definition of φ.

Explicit lemmas.
In this section we prove some lemmas which allow us to obtain explicit constants from Randol's proof of the prime geodesic theorem.Recall ( §4.1), and the definitions of I T (x) and φ.For ǫ > 0, define and define where Note that φ ǫ (w) = φ(ǫw).
The main idea behind the proof is to apply the Selberg trace formula to the pair h ǫ T (r), g ǫ T (x); and to extract the leading and error term from the trace formula.Much of our effort will be spent on giving explicit estimates to each term in the trace formula.
We next set up some important notation.For 0 ≤ λ k ≤ 1 4 , a small eigenvalue, define r k , and s k , by ).
Let A denote the number small eigenvalues: Proof.We prove (1); (2) and (3) will then follow from elementary estimates.
Recall that for r ≥ 0, where and note that φ ǫ (w) = φ(ǫw).By Part 4 of Lemma 4.1, with b = 1/2, it follows that where Hence, The next lemma will be used repeatedly.Integrating by parts, yields A similar argument shows that 4.3.The leading term of the prime geodesic theorem.In this section we show that the sum gives us the leading term of the prime geodesic theorem.In the next section, we will see that the above sum is a term in the Selberg trace formula.
Note that H ǫ (T ) is an approximation of H(T ), and that for any ǫ > 0, ([Ran84, page 298]) An application of the Selberg trace formula yields (4.3) )) Our next goal is to let ǫ = e −T , and estimate each term, in terms of the variable T.
Recall from Theorem 3.3 that Then ). Proof.
Lemma 4.11.Let Proof.By Lemma 4.10, ). Since we can rewrite H(T ) as Both of the above terms are positive for all T. Thus, from Lemma 4.10 and a crude estimate (for each small eigenvalue, 1 2 ≤ s k ≤ 1, hence e s k t ≤ e t .), it follows that The last inequality is obtained by integrating by parts and using (4.6).Next, using the crude bound (1 + T ) ≤ 2e 3T /4 , we obtain Choose µ > 0 so that µ < l(γ) for all γ ∈ C(M ).Lemma 4.12. where By (4.6), The lemma now follows.
Hence, putting everything together, for x > 1, And using the trivial inequality 1 < x 3/4 log x , we obtain where 4.9.The Huber constant for PSL(2, Z).For Γ = PSL(2, Z), we can take c = 6.85.Also, Γ has no small eigenvalues (except for the trivial one), so A = 1.Using the computations in §3.2, an explicit value for C u can be calculated: Theorem 4.16.Let Γ = PSL(2, Z).Then C u = 16,607,349,020,658 is an upper bound for the Huber constant.
Using the theory laid out in [Sar82], and the excellent open-source computational program PARI/GP [The05], we can explicitly calculate (and list out) the length spectrum for PSL(2, Z).Our computations suggest that C M < 2.
So it seems that our result is probably not sharp!COMPUTER EXPERIMENTS FOR PSL(2, Z) Let Γ 0 = PSL(2, Z), and Γ be a finite index subgroup of Γ 0 .Let {P 0 } denote the set of primitive hyperbolic conjugacy classes of Γ.We compute length spectrum (norms of primitive hyperbolic conjugacy classes) using the ideas of [Sar82].Using the above ideas, one can compute the primitive conjugacy classes for Γ 0 = PSL(2, Z).We next explain how to compute the length spectrum for Γ, a finiteindex normal subgroup of Γ 0 .
Let Γ ⊳ Γ 0 be a finite-index normal subgroup of Γ 0 = PSL(2, Z), G = Γ 0 /Γ, and n = |G|.For each primitive conjugacy class {P γ } of Γ 0 let m represent the order of P γ ∈ G.It follows that {P m γ } is a subset of Γ that splits into conjugacy classes with respect to Γ.

4. 4 .
Application of the trace formula.Let Γ be a cofinite Fuchsian group, and let M = Γ \ H be the corresponding hyperbolic-orbifold.Let C(M ) denote the set of closed geodesics of M, and let P(M ) denote the set of prime (or primitive) closed geodesics (see[Bus92, page 245]).For each γ ∈ C(M ) there exists a unique prime geodesic γ 0 and a unique exponent m ≥ 1 so that γ = γ m 0 .Let l(γ) denote the length of γ.Associated to γ is a unique hyperbolic conjugacy class {P γ } Γ with norm
Let Q(x, y) = ax 2 + bxy + cy 2 be a primitive, indefinite binary quadratic form of discriminant d = b 2 −4ac > 0. Two forms [a, b, c] and [a ′ , b ′ , c ′ ] are called equivalent (in the narrow sense) if they are related by a unimodular transformation.Consider the map φ which sends the form [a, b, c] to the matrixt d −bu d 2 −cu d au d t d +bu d 2where t d , u d > 0 is the fundamental solutions of Pell's equation t 2 − du 2 = 4. Setǫ d = t d + √ d u d 2 .Let D be the set of fundamental discriminants {d > 0, d ≡ 0, 1 mod 4} and let h(d) denote the number of inequivalent primitive forms of discriminant d.For eachd ∈ D let Q 1 (d), . . .Q h(d) (d)be a complete set of inequivalent forms of discriminate d.Finally, set QF = d∈D {Q 1 (d), . . .Q h(d) (d)} Lemma 4.17.([Sar82]) The set φ(QF ) is the set of conjugacy classes of primitive hyperbolic elements of PSL(2, Z).The norm of each hyperbolic element φ(Q i (d)) is ǫ 2 d .

TABLE 3 .
Algorithm for the general cofinite Fuchsian case a fundamental domain for Γ, where each F Y j is a cusp sector and F Y is compact.Let ζ 1 , . . ., ζ τ be a complete list of representatives of the Γ equivalence classes of cusps, and let A 1 , . . ., A τ be elements of PSL(2, R)