1 Introduction

Let B be an indefinite quaternion algebra over \(\mathbb {Q}\) of reduced discriminantFootnote 1D and \(R \subset B\) be an Eichler order of level Q, that is an order R such that locally for each prime and for \(p \div D\) \(R_p\) is the unique maximal order in \(B_p\). Let \(\Gamma \) be the subset of proper units (elements of norm one) of R. Further, fix an isomorphism \(B(\mathbb {R})\) with the matrix algebra \(\mathop {\text {Mat}}_{2 \times 2}(\mathbb {R})\).Footnote 2 In the case where B is already split over \(\mathbb {Q}\) (\(D=1\)), we may choose this identification such that

is the familiar congruence lattice. Under the identification of \(B(\mathbb {R})\), \(\Gamma \) is a lattice in \(\mathop {\text {SL}}_{2}(\mathbb {R})\) of co-volume \(V_{\Gamma } = (DQ)^{1+o(1)}\). Furthermore, \(\Gamma \) is co-compact if and only if \(B(\mathbb {Q})\) is a division algebra \((D \ne 1)\). We refer to [4] for further details on the above.

A typical example of a fundamental domain for the action of \(\Gamma \) on the homogeneous space\({\mathop {\text {SL}}_{2}(\mathbb {R})}/{\mathop {\text {SO}}_{2}(\mathbb {R})} \cong \mathbb {H}\), the upper half-plane, is a normal polygon, also known as a Dirichlet domain, which is given as follows. Let \(w \in \mathbb {H}\) be a point whose stabiliser \(\Gamma _w\) in \(\Gamma \) consists only of plus and minus the identity element. Then, the normal polygon with centre w is given by

$$\begin{aligned} {\mathcal {F}}_{\Gamma ,w} = \{ z \in \mathbb {H} \, | \, d(z,w) < d(\gamma z, w), \ \forall \gamma \in \Gamma , \gamma \ne \pm I \}, \end{aligned}$$

where d denotes the hyperbolic distance. If B is split and \(\Gamma = \Gamma _0(Q)\), then another typical example of a fundamental domain is given by the standard polygon:

$$\begin{aligned} {\mathcal {F}}_{\Gamma ,\infty } = \Bigg \{ z \in \mathbb {H} \, | \, |\Re (z)| < \tfrac{1}{2}, \, \Im (z) > \Im (\gamma z), \ \forall \gamma \in \Gamma -\Gamma _{\infty }\Bigg \}, \end{aligned}$$

also referred to as a Ford domain. The sides of these polygons may be paired up such that the corresponding side-pairing motions together with \(-I\) generate the group \(\Gamma \) [12, Chap. 2]. Thus, it comes as no surprise that the ‘size’ of these fundamental domains, e.g. the diameter of \({\Gamma }\backslash {\mathbb {H}}\) if the latter is compact, are related to the size of generators of \(\Gamma \). Algorithms to compute fundamental domains and, subsequently, a set of generators have been devised by Johansson [14], Voight [32], and subsequently improved by Rickards [26] if \(\Gamma \) is co-compact, and Kurth–Long [18] if \(\Gamma \) is a finite index subgroup of \(\mathop {\text {SL}}_{2}(\mathbb {Z})\) through the use of Farey symbols [17]. In the latter case, further algorithms based on the Reidemeister–Schreier process [25, 30] are available to determine an independent set of generators for \(\Gamma (p)\), where p is a prime, by Frasch [8], for \(\Gamma _0(p)\) by Rademacher [23], and for \(\Gamma _0(Q)\), for general \(Q\in \mathbb {N}\), by Chuman [3]. Albeit the former algorithms due Johansson, Voight, Rickards, and Kurth–Long work well in practise, they don’t give any answer to the question regarding the asymptotic size of the (produced) generators and, thus, an upper bound on their time complexity. The algorithms based on the Reidemeister–Schreier process do give an explicit set of generators whose elements are of polynomial size in the co-volume, but they are far from the generating set whose elements are of least size. Stronger results in that direction were given by Khoai [16], who showed that \(\Gamma _0(Q)\) is generated by elements of Frobenius norm bounded by \(O(Q^2)\), respectively O(Q) if Q is a prime power, and Chu–Li [2] who managed to show that \(\Gamma \) co-compact is generated by its elements of Frobenius norm bounded by \(O_{\epsilon }(V_{\Gamma }^{2.56+\epsilon })\).Footnote 3 In this paper, we shall prove the following theorems.

Theorem 1

\(\Gamma _0(Q)\) is generated by its elements of Frobenius norm \(O_{\epsilon }(Q^{1+\epsilon })\).

Theorem 2

Let \(\Gamma \subset \mathop {\text {SL}}_{2}(\mathbb {R})\) be a co-compact arithmetic lattice of co-volume \(V_{\Gamma }\) stemming from the proper units of an Eichler order R of level Q in a quaternion algebra B over \(\mathbb {Q}\) of reduced discriminant D. Then, for almost every \(\sigma \in {\Gamma }\backslash {\mathop {\text {SL}}_{2}(\mathbb {R})}\), \(\sigma ^{-1}\Gamma \sigma \) is generated by its elements of Frobenius norm \(O_{\epsilon }(V_{\Gamma }^{2+\epsilon })\). In other words, almost every embedding \(\Gamma \) of the proper units of the order R into \(\mathop {\text {SL}}_{2}(\mathbb {R})\) is generated by its elements of Frobenius norm \(O_{\epsilon }(V_{\Gamma }^{2+\epsilon })\). If one assumes either of the following conditions:

  • Q is square-free,

  • Selberg’s eigenvalue conjecture for \(\Gamma _0(DQ)\),

  • the sup-norm conjecture in the level aspect for exceptional eigenforms on \({\Gamma }\backslash {\mathop {\text {SL}}_{2}(\mathbb {R})}/{\mathop {\text {SO}}_{2}(\mathbb {R})}\),

then, \(\Gamma \) is in fact generated by its elements of Frobenius norm \(O_{\epsilon }(V_{\Gamma }^{2+\epsilon })\) regardless of the embedding.

Theorem 1 follows from carefully bounding the standard polygon (1.2) by isometric circles of large radius. This is layed out in Sect. 3. Theorem 2 follows from showing that the normal polygon (1.1) is contained in a ball of small radius.

Theorem 3

Let \(\Gamma \subset \mathop {\text {SL}}_{2}(\mathbb {R})\) be a co-compact arithmetic lattice of co-volume \(V_{\Gamma }\) stemming from the proper units of an Eichler order R of level Q in a quaternion algebra B over \(\mathbb {Q}\) of reduced discriminant D. Then, for every \(\eta >0\), satisfies

$$\begin{aligned} \sup _{z \in \mathbb {H}} \min _{\gamma \in \Gamma } d(\gamma z, w) \le (2+\eta ) \log 3V_{\Gamma } \end{aligned}$$

with probability \(1-o(1)\) as \(V_{\Gamma } \rightarrow \infty \). Suppose either of the following statements is true:

  • Q is square-free,

  • Selberg’s eigenvalue conjecture for \(\Gamma _0(DQ)\), or

  • the sup-norm conjecture in the level aspect for exceptional eigenforms on


Then, for every \(\eta > 0\) and \(w \in \mathbb {H}\), satisfies

$$\begin{aligned} \min _{\gamma \in \Gamma } d(\gamma z, w) \le (1+\eta ) \log 3V_{\Gamma } \end{aligned}$$

with probability \(1-o(1)\) as \(V_{\Gamma } \rightarrow \infty \). In particular, the diameter of the hyperbolic surface is bounded by \((2+o(1)) \log 3 V_{\Gamma }\) as \(V_{\Gamma } \rightarrow \infty \).

The bound (1.4) for the almost diameter is sharp and one may speculate whether the actual diameter is around the same length. The latter would imply that the co-compact lattices under consideration are generated by its elements of norm \(O_{\epsilon }(V_{\Gamma }^{1+\epsilon })\) and thus would bring it onto equal footing with Theorem 1. We should further remark that the bound depends on some Siegel-zero estimates and is, thus, not effective, respectively, can be made effective with at most one exception.

In order to prove Theorem 2, one could make use of Ratner’s exponential mixing for \(\mathop {\text {SL}}_{2}(\mathbb {R})\) [24]. This approach was taken by Chu–Li [2], who showed (1.3) for every \(w \in \mathbb {H}\) with 2 replaced by 2.56. Instead, we simplify the argument a bit by using the operator which averages over a sphere of a given radius. The latter operator was employed by Golubev–Kamber [10] who showed, amongst many results of related nature, that the almost diameter under the Selberg eigenvalue conjecture is bounded by \(\log (3V_{\Gamma })+(2+o(1))\log \log (9V_{\Gamma })\) under the mild assumption that \({\Gamma }\backslash {\mathbb {H}}\) has not too many points of small injectivity radius.

Our improvement compared to previous results comes from the incorporation of a density estimate for exceptional eigenvalues as well as the newly available fourth moment bound for Maass forms in the level aspect by Khayutin–Nelson–Steiner [15].

At last, we would also like to touch on the closely related problem of bounding the diameter of expander graphs. A rich family of expander graphs, so-called Ramanujan graphs, may be constructed from arithmetic data associated to definite quaternion algebras [20, 22]. These Ramanujan graphs (by definition) enjoy a large spectral gap, which in turn yields a small upper bound on the diameter. The incorporation of a density estimate for large eigenvalues for (homogeneous) expander graphs has proved valuable in showing that they admit a smaller diameter than what could be directly inferred from their spectral gap [9]. In Sect. 4, we demonstrate the usefulness of the fourth moment bound of Khayutin–Nelson–Steiner [15] also in the context of expander graphs, by proving upper bounds on the diameter of certain Ramanujan graphs without the use of the Ramanujan bound which are of equal strength.

2 Co-compact lattices

Write \(G=\mathop {\text {SL}}_{2}(\mathbb {R})\) and \(K=\mathop {\text {SO}}_{2}(\mathbb {R})\) for short. Let \(\mu \) denote the Haar measure on G normalised such that \(\text {d}\mu (n(x)a(y)k(\theta ))=\frac{1}{y^2}\text {d}x\text {d}y\text {d}\theta \), where

$$\begin{aligned} n(x) = \begin{pmatrix} 1 &{} x \\ 0 &{} 1\end{pmatrix}, \quad a(y) = \begin{pmatrix} y^{\frac{1}{2}} &{} 0 \\ 0 &{} y^{-\frac{1}{2}} \end{pmatrix}, \quad k(\theta ) = \begin{pmatrix} \cos \theta &{} -\sin \theta \\ \sin \theta &{} \cos \theta \end{pmatrix}. \end{aligned}$$

Let \(\Gamma \) be a co-compact lattice as in Sect. 1. Denote by \(V_{\Gamma }\) the co-volume of \(\Gamma \) with respect to \(\mu \). Further, \(\mu \) descends to a finite measure \(\nu \) on \({\Gamma }\backslash {G}\), which we normalise to a probability measure. With \(\nu _{\star }\) we denote the push forward of \(\nu \) to \({\Gamma }\backslash {G}/{K}\). Let \(\{u_j\}_j\) be an orthonormal basis of Hecke–Maass forms on \(L^2({\Gamma }\backslash {G}/{K},\nu _{\star })\), which we may also regard as an orthonormal basis of the K-invariant subspace of \(L^2({\Gamma }\backslash {G},\nu )\). We denote the Laplace eigenvalue of \(u_j\) with \(-\lambda _j=-(\frac{1}{4}+t_j^2)\), where \(t_j \in \mathbb {R} \cup i[0,\frac{1}{2}]\). We let d denote the hyperbolic distance on the upper half-plane \(\mathbb {H} \cong {G}/{K}\) and u the related quantity

$$\begin{aligned} u(z,w)= \tfrac{1}{2}(\cosh (d(z,w))-1) = \frac{|w-z|^2}{4 \mathop {\text { Im}}(z) \mathop {\text { Im}}(w)}. \end{aligned}$$

We note that

$$\begin{aligned} u(z,w) = \tfrac{1}{4} \mathop {\text {tr}}(gg^{t})-\tfrac{1}{2} =: u(g) \end{aligned}$$

where g is any matrix that takes z to w and \(u:G \rightarrow \mathbb {R}^+_0\) is left and right K-invariant. Let \(S: \mathbb {R}^{+}_0 \rightarrow [0,1]\) be a smooth bump function supported on \([0,\delta ]\) for some small \(\delta >0\), such that its Selberg/Harish-Chandra transform:

$$\begin{aligned} h(t) = 4\pi \int _{0}^{\infty } S(u) \cdot {}_2F_1(\tfrac{1}{2}+it,\tfrac{1}{2}-it;1;-u) \textrm{d}u \end{aligned}$$

is non-negative and

$$\begin{aligned} h(\pm \tfrac{i}{2}) = 4 \pi \int _0^{\infty } S(u) \textrm{d}u \asymp 1, \end{aligned}$$

where the implied constants may depend on \(\delta \). We note that

$$\begin{aligned} |h(t)| \le 4\pi \int _0^{\infty } |S(u)| \textrm{d}u = h( \pm \tfrac{i}{2} ) \ll 1. \end{aligned}$$


$$\begin{aligned} B(g_1,g_2) = \sum _{\gamma \in \Gamma } S( u(g_2^{-1}\gamma g_1)), \end{aligned}$$

where \(S(u(g_2^{-1}g_1))\) is a point-pair invariant, and thus we get the spectral expansion (cf. [12, Theorem 1.14])

$$\begin{aligned} B(g_1,g_2) = \frac{2 \pi }{ V_{\Gamma } } \sum _j h(t_j) u_j(g_1)\overline{u_j(g_2)}. \end{aligned}$$

\(B_g:=B(g,\cdot )\) will take the role of a smooth right K-invariant ball on \({\Gamma }\backslash {G}\). The following lemma is of crucial importance. Essentially, it says that \(B_g\) may be compared to an Euclidean ball.

Lemma 4

Suppose \(\delta \) is sufficiently small but (strictly) positive. Then, we have for any \(g_1 \in G\) that

$$\begin{aligned} \int _{{\Gamma }\backslash {G}} |B(g_1,g_2)|^2 \textrm{d}\nu (g_2) \ll \frac{1}{V_{\Gamma }}, \end{aligned}$$

where the implied constant may depend on \(\delta \).


We have

$$\begin{aligned} \begin{aligned} \int _{{\Gamma }\backslash {G}} |B(g_1,g_2)|^2 \textrm{d}\nu (g_2)&= \frac{4\pi ^2}{V_{\Gamma }^2} \sum _j |h(t_j)|^2 |u_j(g_1)|^2 \\&\quad = \frac{2\pi }{V_{\Gamma }} \sum _{\gamma \in \Gamma } ((S \circ u) \star (S \circ u))(g_1^{-1}\gamma g_1), \end{aligned} \end{aligned}$$

where the convolution is taken on G. We note that \((S \circ u) \star (S \circ u)\) is bounded for fixed \(\delta \). Moreover, it is supported on \(g \in G\) with \(u(g) \le 4\delta (1+\delta )\). We now note that \(u(g) \ge \frac{(\mathop {\text {tr}}g)^2}{4}-1\). Thus, if \(\delta \) is sufficiently small the sum over all hyperbolic \(\gamma \in \Gamma \) is zero since for those \(\mathop {\text {tr}}\gamma \) is at least 3. Now again for \(\delta \) sufficiently small, by the Margulis’ Lemma (cf. [7, Sect. 3.1]), the subgroup \({\widetilde{\Gamma }}\) generated by the remaining \(\gamma \) for which \(u(g_1^{-1}\gamma g_1) \le 4\delta (1+\delta )\) is virtually abelian and in particular of one of the following types:

  1. (i)

    An infinite cyclic group generated by an hyperbolic or parabolic isometry;

  2. (ii)

    A finite cyclic group generated by an elliptic isometry;

  3. (iii)

    An infinite dihedral group generated by two elliptic isometries of order 2.

The first case only contains the elliptic elements plus minus the identity. In the second case, \({\widetilde{\Gamma }}\) is finite and of order bounded by 6 as the characteristic polynomial is a cyclotomic polynomial of degree at most two over \(\mathbb {Q}\). In the third case, the subgroup \({\widetilde{\Gamma }}\) fixes a geodesic and all elliptic elements not equal to plus minus the identity fix a single point on this geodesic (and perform a rotation by \(\pi \) around the point). These fixpoints are evenly spread along the geodesic by the distance corresponding to the translation of the minimal hyperbolic element in \({\widetilde{\Gamma }}\). Thus, if \(\delta \) is small enough, there are at most four elliptic elements \(\gamma \in {\widetilde{\Gamma }}\) such \(u(g_1^{-1}\gamma g_1) \le 4\delta (1+\delta )\). We conclude the lemma. \(\square \)

We now get to the heart of the argument. We shall use an averaging operator which averages over a sphere of radius T:

where \(g_z\) is any matrix that takes i to z.

We shall prove the following proposition.

Proposition 5

For \(T\ge 1\), we have

$$\begin{aligned}{} & {} \int _{{\Gamma }\backslash {G}} \sup _{g_2 \in {\Gamma }\backslash {G}/{K}} \left| \langle A_T B_{g_1}, B_{g_2} \rangle -\langle B_{g_1}, u_0 \rangle \overline{\langle B_{g_2}, u_0 \rangle } \right| ^2 \textrm{d}\nu (g_1) \nonumber \\{} & {} \quad \ll _{\epsilon } T^2 V_{\Gamma }^{-2+\epsilon } \left( \textrm{e}^{-T} + \textrm{e}^{-\frac{T}{2}} V_{\Gamma }^{-1} \right) . \end{aligned}$$

Assume either of the following conditions:

  • Q is square-free,

  • Selberg’s eigenvalue conjecture for \(\Gamma _0(DQ)\),

  • the sup-norm conjecture in the level aspect for exceptional eigenforms on \({\Gamma }\backslash {\mathop {\text {SL}}_{2}(\mathbb {R})}/{\mathop {\text {SO}}_{2}(\mathbb {R})}\),

then we have the stronger bound

$$\begin{aligned} \Vert A_TB_g-\langle B_g,u_0 \rangle u_0\Vert _2^2 \ll _{\epsilon } T^2V_{\Gamma }^{-2+\epsilon } \left( \textrm{e}^{-T}V_{\Gamma }+\textrm{e}^{-\frac{T}{2}}V_{\Gamma }^{\frac{1}{2}} \right) . \end{aligned}$$

Before proceeding with the proof of the proposition, we shall show how Theorem 3 follows from it. The first half of the Theorem follows from choosing \(T=(2+o(1)) \log 3 V_{\Gamma }\) in (2.4). As the main term

$$\begin{aligned} \langle B_{g_1}, u_0 \rangle \overline{\langle B_{g_2}, u_0 \rangle } = \frac{4 \pi ^2}{V_{\Gamma }^2} |h(\tfrac{i}{2})|^2 \asymp V_{\Gamma }^{-2}, \end{aligned}$$

it follows that \(\langle A_T B_{g_1}, B_{g_2} \rangle >0\) for most \(g_1\). In order to prove the second half of the theorem, we shall use (2.5) with \(T_0=(1+o(1)) \log 3 V_{\Gamma }\). We then find

$$\begin{aligned} \nu _{\star }(\{z \in {\textstyle {\Gamma }\backslash {G}/{K}} \, | \, A_{T_0}B_g(z)=0 \}) \ll V_{\Gamma }^2 \Vert A_{T_0}B_g-\langle B_g,u_0 \rangle u_0\Vert _2^2 = o(1). \end{aligned}$$

Proof of Proposition 5

We first note that each \(u_i\) is an eigenfunction of \(A_T\) with eigenvalue \({}_2F_1(\frac{1}{2}-it_i,\frac{1}{2}+it_i;1;\frac{1}{2}-\frac{1}{2}\cosh (T))=P_{-\frac{1}{2}-it_i}(\cosh (T))\), cf. [12, Corollary 1.13]. We have that this eigenvalue is bounded by

$$\begin{aligned} {\left\{ \begin{array}{ll} (T+1)\textrm{e}^{-\frac{T}{2}(1-\sqrt{1-4\lambda _i})}, &{} 0\le \lambda _i \le \frac{1}{4}, \\ (T+1)\textrm{e}^{-\frac{T}{2}}, &{} \frac{1}{4} \le \lambda _i, \end{array}\right. } \end{aligned}$$

cf. [10, Props. 2.3 & 7.2]. Hence, after referring to the spectral expansion (2.3), we find that

$$\begin{aligned}{} & {} \langle A_{T} B_{g_1}, B_{g_2} \rangle - \langle B_{g_1}, u_0 \rangle \overline{\langle B_{g_2}, u_0 \rangle } \nonumber \\{} & {} \quad = O \left( \frac{1}{V_{\Gamma }^2} \sum _{0<\lambda _j < \frac{1}{4}} T\left( \textrm{e}^{-\frac{T}{2}} \right) ^{1-\sqrt{1-4\lambda _j}} |h(t_j)|^2 |u_j(g_1)u_j(g_2)| \right) \nonumber \\ {}{} & {} \qquad +O\left( T\textrm{e}^{-\frac{T}{2}} \Vert B_{g_1}\Vert _2 \Vert B_{g_2}\Vert _2 \right) \end{aligned}$$

for any two \(g_1,g_2 \in G\) and \(T \ge 1\). We note that the second smallest eigenvalue \(\lambda _1 \ge \frac{3}{16}\), due to Selberg [31] and the Jacquet–Langlands correspondence [13]. By Lemma 4, we find that the second error term is bounded by \(O(T\textrm{e}^{-\frac{T}{2}} V_{\Gamma }^{-1})\). It remains to deal with the first error term. By Cauchy–Schwarz, we may estimate

$$\begin{aligned}{} & {} \frac{1}{V_{\Gamma }^2} \sum _{0<\lambda _j< \frac{1}{4}} T\left( \textrm{e}^{-\frac{T}{2}} \right) ^{1-\sqrt{1-4\lambda _j}} |h(t_j)|^2 |u_j(g_1)u_j(g_2)| \\{} & {} \quad \le \left( \frac{1}{V_\Gamma ^2} \sum _{\frac{3}{16}\le \lambda _j< \frac{1}{4}} |h(t_j)|^2 |u_j(g_2)|^2 \right) ^{\frac{1}{2}}\\ {}{} & {} \qquad \times \left( \frac{1}{V_\Gamma ^2} \sum _{\frac{3}{16}\le \lambda _j< \frac{1}{4}} T^2\left( \textrm{e}^{-\frac{T}{2}}\right) ^{2(1-\sqrt{1-4\lambda _j})} |h(t_j)|^2 |u_j(g_1)|^2 \right) ^{\frac{1}{2}} \\{} & {} \quad \ll \Vert B_{g_2}\Vert _2 \left( \frac{1}{V_\Gamma ^2} \sum _{\frac{3}{16}\le \lambda _j< \frac{1}{4}} T^2\left( \textrm{e}^{-\frac{T}{2}}\right) ^{2(1-\sqrt{1-4\lambda _j})} |h(t_j)|^2 |u_j(g_1)|^2 \right) ^{\frac{1}{2}} \\{} & {} \quad \ll \frac{1}{V_{\Gamma }^{\frac{1}{2}}} \left( \frac{1}{V_\Gamma ^2} \sum _{\frac{3}{16}\le \lambda _j < \frac{1}{4}} T^2\left( \textrm{e}^{-\frac{T}{2}}\right) ^{2(1-\sqrt{1-4\lambda _j})} |h(t_j)|^2 |u_j(g_1)|^2 \right) ^{\frac{1}{2}}. \end{aligned}$$

Thus, we find

$$\begin{aligned}{} & {} \int _{{\Gamma }\backslash {G}} \sup _{g_2 \in {\Gamma }\backslash {G}/{K}} \left| \langle A_T B_{g_1}, B_{g_2} \rangle -\langle B_{g_1}, u_0 \rangle \overline{\langle B_{g_2}, u_0 \rangle } \right| ^2 \textrm{d}\nu (g_1) \\{} & {} \quad \ll \frac{1}{V_{\Gamma }^3} \left( \sum _{\frac{3}{16}\le \lambda _j< \frac{1}{4}} T^2\left( \textrm{e}^{-\frac{T}{2}}\right) ^{2(1-\sqrt{1-4\lambda _j})} |h(t_j)|^2 \int _{{\Gamma }\backslash {G}} |u_j(g_1)|^2 \textrm{d}\nu (g_1) \right) \\ {}{} & {} \qquad + T^2\textrm{e}^{-T} V_{\Gamma }^{-2} \\{} & {} \quad \ll \frac{1}{V_{\Gamma }^3} \left( \sum _{\frac{3}{16}\le \lambda _j < \frac{1}{4}} T^2\left( \textrm{e}^{-\frac{T}{2}}\right) ^{2(1-\sqrt{1-4\lambda _j})} \right) + T^2\textrm{e}^{-T} V_{\Gamma }^{-2}. \end{aligned}$$

The sum over the exceptional eigenvalues we may bound using the density estimate (cf. [12, Theorem 11.7])

$$\begin{aligned} \sharp \{j>0 \, | \, s_j > \sigma \} \ll _{\epsilon } (DQ)^{3-4\sigma +\epsilon }, \end{aligned}$$

for any \(\sigma \ge \frac{1}{2}\), where \(s_j=\frac{1}{2}-it_j\). We note that this density estimate, which a priori holds for \(\Gamma _0(DQ)\), also holds for the lattice \(\Gamma \) under consideration. This is the case since we may consider an explicit Jacquet–Langlands transfer sending a form on \(\Gamma \) to its corresponding newform form on \(\Gamma _0(N) \supseteq \Gamma _0(DQ)\) for some \(D \div N \div DQ\) which has the same eigenvalue. Under this map, at most \(Q^{o(1)}\) forms get mapped onto the same image. This follows from [1, Theorem 1 & its Corollary, Theorem 4] and noting that at the places dividing D the corresponding representations are one-dimensional. Using the estimate (2.7), we find

$$\begin{aligned} \sum _{\frac{3}{16}\le \lambda _j< \frac{1}{4}} T^2\left( \textrm{e}^{-\frac{T}{2}}\right) ^{2(1-\sqrt{1-4\lambda _j})}&= T^2 \sum _{\frac{1}{2} < s_j \le \frac{3}{4}} \left( \textrm{e}^{\frac{T}{2}}\right) ^{4(s_j-1)} \nonumber \\&\ll _{\epsilon } (DQ)^{\epsilon } T^2\left( \textrm{e}^{-T} (DQ) + \textrm{e}^{-\frac{T}{2}}\right) \nonumber \\&\ll _{\epsilon } V_{\Gamma }^{\epsilon } T^2 \left( \textrm{e}^{-T} V_{\Gamma }+\textrm{e}^{-\frac{T}{2}} \right) , \end{aligned}$$

Hence, we conclude the first part of the proposition. For the second part, we need to bound

$$\begin{aligned}{} & {} \Vert A_TB_g-\langle B_g,u_0 \rangle u_0\Vert _2^2 \\{} & {} \quad \ll \frac{1}{V_{\Gamma }^2} \sum _{\frac{3}{16} \le \lambda _j < \frac{1}{4}} T^2\left( \textrm{e}^{-\frac{T}{2}} \right) ^{2(1-\sqrt{1-4\lambda _j})} |h(t_j)|^2 |u_j(g)|^2 + T^2\textrm{e}^{-T} \Vert B_{g}\Vert _2^2. \end{aligned}$$

We note that the second summand is \(\ll T^2\textrm{e}^{-T}V_{\Gamma }^{-1}\), which is satisfactory. Furthermore, we find that the first summand is empty if we assume Selberg’s eigenvalue conjecture, which takes care of that case. In the other cases, we first recall (2.8), which implies there is a \(g_0 \in {\Gamma }\backslash {G}\) such that

$$\begin{aligned} \frac{1}{V_{\Gamma }^2} \sum _{\frac{3}{16} \le \lambda _j < \frac{1}{4}} T^2\left( \textrm{e}^{-\frac{T}{2}} \right) ^{2(1-\sqrt{1-4\lambda _j})} |h(t_j)|^2 |u_j(g_0)|^2 \ll _{\epsilon } T^2 V_{\Gamma }^{-2+\epsilon } \left( \textrm{e}^{-T} V_{\Gamma }+\textrm{e}^{-\frac{T}{2}} \right) . \end{aligned}$$

Thus, in order to prove (2.5), it is sufficient to bound

$$\begin{aligned}{} & {} \frac{1}{V_{\Gamma }^2} \sum _{\frac{3}{16} \le \lambda _j< \frac{1}{4}} T^2\left( \textrm{e}^{-\frac{T}{2}} \right) ^{2(1-\sqrt{1-4\lambda _j})} |h(t_j)|^2 \left( |u_j(g)|^2 -|u_j(g_0)|^2\right) \\{} & {} \quad \le \frac{1}{V_{\Gamma }^2} \left( \sum _{\frac{3}{16}\le \lambda _j< \frac{1}{4}} T^4\left( \textrm{e}^{-\frac{T}{2}}\right) ^{4(1-\sqrt{1-4\lambda _j})} \right) ^{\frac{1}{2}} \left( \sum _{\frac{3}{16}\le \lambda _j < \frac{1}{4}} \left( |u_j(g)|^2-|u_j(g_0)|^2 \right) ^2 \right) ^{\frac{1}{2}}. \end{aligned}$$

If we assume that the level Q of R is square free, then [15, Theorem 1.10] shows that

$$\begin{aligned} \sum _{\frac{3}{16}\le \lambda _j < \frac{1}{4}} \left( |u_j(g)|^2 -|u_j(g_0)|^2 \right) ^2 \ll _{\epsilon } V_{\Gamma }^{1+\epsilon }. \end{aligned}$$

The same conclusion also holds if we assume the sup-norm conjecture and referring to Weyl’s law. Finally, we may estimate the first factor by once more referring to the density estimate (2.7):

$$\begin{aligned} \sum _{\frac{3}{16}\le \lambda _j < \frac{1}{4}} T^4\left( \textrm{e}^{-\frac{T}{2}}\right) ^{4(1-\sqrt{1-4\lambda _j})} \ll _{\epsilon } V_{\Gamma }^{\epsilon }T^4 \left( \textrm{e}^{-2T} V_{\Gamma }+\textrm{e}^{-T} \right) . \end{aligned}$$

We conclude the proof. \(\square \)

We are left to infer Theorem 2 from Theorem 3. Suppose the stabiliser of the point \(i \in \mathbb {H}\) consists only of plus minus the identity and consider the normal polygon \({\mathcal {F}}_{\Gamma ,i}\). Suppose further that \({\mathcal {F}}_{\Gamma ,i}\) is contained in a ball of radius r. We shall now translate this picture from the upper half-plane to the Poincaré disk through the Cayley transformation \(\phi : \mathbb {H} \rightarrow \mathfrak {D}\), which maps \(z\mapsto (z-i)/(z+i)\). Under this map, \({\mathcal {F}}_{\Gamma ,i}\) gets mapped to a Ford domain \(\mathfrak {F}_{\Gamma }\). A motion \(\gamma = \left( {\begin{matrix} a &{} b \\ c &{} d \end{matrix}}\right) \in \mathop {\text {SL}}_{2}(\mathbb {R})\) gets transferred to the motion

$$\begin{aligned} \gamma ^{\phi } = \phi \gamma \phi ^{-1} = \begin{pmatrix} \frac{a+d}{2}+i\frac{b-c}{2} &{} \frac{a-d}{2}-i\frac{b+c}{2} \\ \frac{a-d}{2}+i\frac{b+c}{2} &{} \frac{a+d}{2}-i\frac{b-c}{2} \end{pmatrix} =: \begin{pmatrix} {\overline{F}} &{} {\overline{E}} \\ E &{} F\end{pmatrix} \end{aligned}$$

in \(\textrm{SU}(1,1)\). Now, Ford [6] proved that \(\Gamma ^{\phi }=\phi \Gamma \phi ^{-1}\) is generated by the motions \(\gamma ^{\phi } \in \Gamma ^{\phi }\) whose partial arc of its isometric circle forms part of the boundary of \(\mathfrak {F}_{\Gamma }\). We have that the isometric circle corresponding to the motion (2.9) is given by the equation \(|Ez+F|\), a circle with radius 1/|E| and centre \(-F/E\). Thus, in order for the isometric circle of \(\gamma ^{\phi }\) to intersect \(\mathfrak {F}_{\Gamma }\) one must have \((|F|-1)/|E| \le \mathop {\text {artanh}}(\frac{r}{2})\), which after a calculation yields \(|E| \le \sinh (r)\) and thus

$$\begin{aligned} \Vert \gamma \Vert _F^2 = 4|E|^2+2 \ll \textrm{e}^{2r}. \end{aligned}$$

Thus, we may conclude Theorem 2 from Theorem 3 after noting that the points \(z\in \mathbb {H}\) with non-trivial stabiliser are a null-set; hence, they may be excluded for the first part and for the second part one may conjugate the group by a tiny bit if i happens to be such a point.

3 Non-co-compact lattices

In this section, we shall bound the standard polygon (1.2). We shall also point out that a similar argument has been carried out in the appendix to [28].

We start with two preparatory Lemmata.

Lemma 6

For any \(\epsilon >0\), we may find a constant \(C_{\epsilon }>0\) with the following property. For two relatively prime integers ab and natural number D, we may find a natural number \(k \le C_{\epsilon } D^{\epsilon }\) such that \((a+kb,D)=1\).


This is [27, Lemma 2.1]. \(\square \)

Lemma 7

For any \(\epsilon >0\), we may find a constant \(C_{\epsilon }>0\) with the following property. For any real number x and natural number D, we either have that

  1. (i)

    there is an integer c such that \(|x-c| \le \frac{1}{2(1+C_{\epsilon }D^{\epsilon })}\), or that

  2. (ii)

    there is a natural number \(b \le 2(1+C_{\epsilon } D^{\epsilon })^2\) and an integer \((a,bD)=1\) such that

    $$\begin{aligned} |bx-a| \le \tfrac{1}{2}. \end{aligned}$$


By Dirichlet’s Approximation Theorem, we may find \(d \in \mathbb {N}\) with \(d \le K=2(1+C_{\epsilon }D^{\epsilon })\) and \(c \in \mathbb {Z}\) such that \(|\textrm{d}x-c|\le \frac{1}{K}\). Without loss of generality, we may assume \((c,d)=1\). Now, if \(d=1\), then we are done as the first condition is satisfied. Suppose now that \(d \ge 2\), then we may find a pair of integers ab such that \(ad-bc=\pm 1\), where the sign is chosen such that \(\frac{a}{b}\) and x lie on the same side of \(\frac{c}{d}\) on the number line. We note that the pair of integers \((a+kc,b+kd)\), where k is any integer, also satisfies the same equation. Hence, we may assume \(0< b \le d\). Note that we have \((a,c)=1\) and hence we may apply Lemma 6 to even require \((a,bD)=1\) at the cost of increasing the size of b to at most \((1+C_{\epsilon }D^{\epsilon })d\). We have

$$\begin{aligned} |x-\tfrac{a}{b}| \le \max \left\{ \frac{1}{dK}, \frac{1}{bd} \right\} \le \frac{1}{2b}. \end{aligned}$$

The conclusion follows. \(\square \)

We note that the standard polygon (1.2) agrees with the Ford domain

$$\begin{aligned} {\mathcal {F}}_{\Gamma ,\infty } = \{ z \in \mathbb {H} \,|\,|\Re (z)|\le \tfrac{1}{2} \text { and } |cQz+d|\ge 1, \, \forall (cQ,d)=1 \}. \end{aligned}$$

We proceed by showing that the isometric circles \(|cQz+d|=1\), corresponding to the motion \(\left( {\begin{matrix} a &{} b \\ cQ &{} d \end{matrix}}\right) \in \Gamma \), that form part of the boundary of \({\mathcal {F}}_{\Gamma ,\infty }\) must have their radius 1/|cQ| bounded below by \(Q^{-1+o(1)}\).

Let \(z=x+iy \in {\mathcal {F}}_{\Gamma ,\infty }\). We now apply Lemma 7 to xQ with \(D=Q\). Thus, we have either

  1. (i)

    \(|Qx-c| \le \frac{1}{2(1+C_{\epsilon }Q^{\epsilon })}\) for some integer c, or

  2. (ii)

    \(|bQx-a| \le \frac{1}{2}\) some natural number \(b \le 2(1+C_{\epsilon }Q^{\epsilon })^2\) and integer a with \((a,bQ)=1\).

Let us first deal with the second case, which corresponds to z being away from the cusps other than \(\infty \). Since z is in the Ford domain (3.1), we must have \(|bQz-a|\ge 1\) and thus \(y \ge \frac{1}{2} Q^{-1} (1+C_{\epsilon }Q^{\epsilon })^{-2}\). Returning to the first case, if \((c,Q)=1\) then once again we must have \(|Qz-c|\ge 1\) and hence \(y \ge \frac{1}{2} Q^{-1}\). Finally, if \((c,Q)>1\) then we may find natural numbers \(k_{\pm } \le C_{\epsilon }Q^{\epsilon }\) such that \((k_{\pm }c \pm 1,Q)=1\). Thus, we have

$$\begin{aligned} \frac{k_{-}c-1}{k_{-}Q}< x < \frac{k_{+}c+1}{k_{+}Q} \end{aligned}$$

and the isometric circles \(|(k_{\pm }Q)w+(k_{\pm }c\pm 1)|=1\) include the cusp \(\frac{c}{Q}\). Hence, we find that every point \(z \in {\mathcal {F}}_{\Gamma ,\infty }\) has either \(y \ge \frac{1}{2}(1+C_{\epsilon }Q^{\epsilon })^{-2}Q^{-1}\) or is in a cuspidal region in between to isometric circles of radius at least \(C_{\epsilon }Q^{-1-\epsilon }\). In particular, we find that every isometric circle, which is part of the boundary of \({\mathcal {F}}_{\Gamma ,\infty }\) must have radius at least \(\frac{1}{2}(1+C_{\epsilon }Q^{\epsilon })^{-2}Q^{-1}\). Thus, the side-pairing motions \(\left( {\begin{matrix} a &{} b \\ cQ &{} d \end{matrix}}\right) \) of \({\mathcal {F}}_{\Gamma ,\infty }\), which generate \(\Gamma \) must have \(|cQ| \ll _{\epsilon } Q^{1+2 \epsilon }\), \(|a|,|d| \ll _{\epsilon } Q^{1+2\epsilon }\) and consequently \(|b| \ll _{\epsilon } Q^{1+4 \epsilon }\) as \(ad-bcQ=1\). We conclude Theorem 1.

4 Graphs

We consider the Brandt–Ihara–Pizer “super singular isogeny graphs”, \(G(p, \ell )\), where \(p, \ell \) are primes with \(p \equiv 1 \mod (12)\). They are constructed by interpreting Brandt matrices \(B(\ell )\) associated to a maximal order R in the quaternion algebra \(B_{p,\infty }\) over \(\mathbb {Q}\) ramified at exactly \(p,\infty \) as adjacency matrices. They constitute a rich family of non-bipartite \((\ell +1)\)-regular Ramanujan graphs on \(n:=\frac{p-1}{12}+1\) vertices [22]. Let \(f_j \in L^2(G(p,\ell ))\), equipped with probability measure, be an orthonormal eigenbasis of the adjacency matrix \(B(\ell )\) with \(f_0 \equiv 1\). We may and shall also assume that they are eigenfunctions of all other Brandt matrices B(m) for \((m,p)=1\). We shall denote the eigenvalue of \(f_j\) with respect to B(m) by \(\lambda _j(m)\). By identifying the vertices of \(G(p,\ell )\) with the class set \(B_{p,\infty }^{\times }\backslash (B_{p,\infty } \otimes \mathbb {A}_f)^{\times } /(R \otimes \widehat{\mathbb {Z}})^{\times }\) of R, we may interpret the eigenfunctions \(f_j\) as automorphic forms in \(L^2({\text {PB}}_{p,\infty }^{\times }(\mathbb {Q}) \backslash {\text {PB}}_{p,\infty }^{\times }(\mathbb {A}) /K_{\infty }K_f)\), where \(K_{\infty }\) is a maximal torus and \(K_f\) the projective image of \((R \otimes \widehat{\mathbb {Z}})^{\times }\), constant on each connected component (as a real manifold). The automorphic forms \(f_j\) are in one-to-one correspondence with their theta lift, a modular form of weight 2, level p, and trivial character, which is cuspidal if and only if \(j \ne 0\). They form a basis of Hecke eigenforms of said space. The m-th Hecke eigenvalue of the theta lift of \(f_j\) is given by the eigenvalue \(\lambda _j(m)\) for \((m,p)=1\) [5, 11]. Thus, by the Petersson trace formula, one has the density estimate (cf. [29, Eq. (4)])

$$\begin{aligned} \sum _{j \ne 0} |\lambda _j(m)|^2 \ll _{\epsilon } (p \ell )^{\epsilon } \left( p+m^{\frac{1}{2}}\right) . \end{aligned}$$

Likewise, the fourth moment bound [15, Theorem 2.1] reads

$$\begin{aligned} \sup _{x,y \in G(p,\ell )}\sum _{j \ne 0}\left( f_j(x)^2-f_j(y)^2 \right) ^2 \ll _{\epsilon } p^{1+\epsilon }. \end{aligned}$$

It is known that the Ramanujan graphs \(G(p,\ell )\) have diameter bounded by \((2+o(1))\log _{\ell }(n)\).Footnote 4 Here, we shall give an alternative proof which avoids using the Ramanujan bound. Instead, we shall make use of the two inequalities (4.1) and (4.2). For \(x,y \in G(p,\ell )\), let \(K_t(x,y)\) denote the number of non-backtracking random walks of length t from x to y. We have the equality (see [19])

$$\begin{aligned} \sum _{0 \le i \le \frac{t}{2}} K_{t-2i}(x,y) = \frac{1}{n} \sum _j \lambda _j(\ell ^t) f_j(x)f_j(y). \end{aligned}$$

If xy are of distance larger than t, then the left-hand side of (4.3) is zero. We find that

$$\begin{aligned} \lambda _0(\ell ^t) f_0(x)f_0(y) \ll \sum _{j \ne 0} |\lambda _j(\ell ^t)| |f_j(x)f_j(y)|, \end{aligned}$$

from which we infer

$$\begin{aligned} \ell ^t \ll \sup _{x \in G(p,\ell )} \sum _{j \ne 0} |\lambda _j(\ell ^t)|f_j(x)^2, \end{aligned}$$

since \(\lambda _0(\ell ^t) = \frac{\ell ^{t+1}-1}{\ell -1}\). By orthonormality, we have \(\sum _{x \in G(p,\ell )} f_j(x)^2 = n\). Hence, we may bound the right-hand side further

$$\begin{aligned} \sup _{x \in G(p,\ell )} \sum _{j \ne 0} |\lambda _j(\ell ^t)|f_j(x)^2\le & {} \sum _{j \ne 0} |\lambda _j(\ell ^t)| \nonumber \\ {}{} & {} + \sup _{x,y \in G(p,\ell )} \sum _{j \ne 0} |\lambda _j(\ell ^t)|\left( f_j(x)^2-f_j(y)^2 \right) .\nonumber \\ \end{aligned}$$

By applying Cauchy–Schwarz and making use of (4.3) and (4.2), we conclude \(l^{t} \ll _{\epsilon } n^{2+\epsilon }\) or \(t \le (2+o(1))\log _{\ell }(n)\). In particular, the diameter is bounded by one plus the same quantity, the former getting absorbed into \(o(\log _{\ell }(n))\).