1 Introduction↩︎

The Hecke triangle groups \(G_q\) are an infinite family of lattices in \(\mathop{\mathrm{SL}}(2,\mathbb{R})\) parametrized by an integer \(q \geq 3\). The group \(G_q\) is generated by the two matrices \[\label{eq:ST} S = \left(\begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix}\right), \quad T_q = \left(\begin{matrix} 1 & \lambda_q \\ 0 & 1 \end{matrix}\right) ,\tag{1}\] where \(\lambda_q = 2\cos(\pi / q)\). Each parabolic element of \(G_q\) fixes a unique cusp in \(\mathbb{P}^1(\mathbb{R}) = \mathbb{R}\cup \{\infty\}\), and each hyperbolic element fixes a pair of distinct points in \(\mathbb{P}^1(\mathbb{R})\). Basic longstanding open questions of interest about \(G_q\) include characterizing the set of cusps [1] and the set of hyperbolic fixed points as subsets of \(\mathbb{P}^1(\mathbb{R})\).

The difficulty of characterizing the cusps of \(G_q\) seems to depend mainly on the degree of the invariant trace field of \(G_q\), defined by \[K_q = \mathbb{Q}(\mathop{\mathrm{Tr}}(A^2) : A \in G_q) = \begin{cases} \mathbb{Q}(\lambda_q), \; & q \; \text{ odd,} \\ \mathbb{Q}(\lambda_q^2), \; & q \; \text{ even,} \end{cases}\] which is invariant under commensurability, see for instance [2]). The cusps of \(G_q\) form a single orbit \(G_q \cdot \infty\). By an inductive argument using the generators in (1 ), we have \[\label{eq:cusps} G_q \cdot \infty \subset \lambda_q \mathbb{Q}(\lambda_q^2) \cup \{\infty\} .\tag{2}\] When \(q\) is odd, \(\lambda_q \mathbb{Q}(\lambda_q^2) \cup \{\infty\}\) is equal to \(\mathbb{P}^1(K_q)\). If \(K_q = \mathbb{Q}\), that is, \(q \in \{3, 4, 6\}\), then \(G_q\) is commensurable with \(\mathop{\mathrm{SL}}(2,\mathbb{Z})\) and it is easy to show that the containment in (2 ) is an equality. When \(K_q\) is quadratic, equivalently \(q \in \{5, 8, 10, 12\}\), there are several proofs showing that equality holds in these cases as well, using descent arguments or a study of the SAF-invariant [3], [4], [5], [6].

When \(K_q\) has degree at least \(3\), the question of characterizing the cusps of \(G_q\) is wide open, and it is known that the cusps are strictly contained in \(\lambda_q \mathbb{Q}(\lambda_q^2) \cup \{\infty\}\), see [7] and references therein. An important new phenomenon in this case is the existence of special hyperbolic elements of \(G_q\), which have eigenvalues in \(K_q\). In contrast, a typical hyperbolic element of \(G_q\) has eigenvalues in a quadratic extension of \(K_q\). Fixed points of special hyperbolic elements of \(G_q\) lie in \(\lambda_q \mathbb{Q}(\lambda_q^2)\) and thus provide new examples of \(G_q\)-orbits contained in \(\lambda_q \mathbb{Q}(\lambda_q^2)\). Only a few examples of such orbits are known [7], [8], [9], [10], and in all known examples \(K_q\) has degree \(3\) or \(4\). One can also obtain lower bounds on the number of orbits of \(G_q\) in \(\lambda_q \mathbb{Q}(\lambda_q^2)\) with algebraic methods, by studying the reduction of \(G_q\) modulo \(2\) or the class number of \(K_q\) [7], [11], [10], [12].

However, in all cases where \(K_q\) has degree at least \(3\), it was previously unknown whether the number of orbits of \(G_q\) in \(\lambda_q \mathbb{Q}(\lambda_q^2)\) was finite. Surprisingly, for at least one value of \(q\), special hyperbolic elements are abundant enough to produce infinitely many orbits.

Hyperbolic elements in \(G_q\) with a common power have the same fixed points, and conjugating a hyperbolic element in \(G_q\) only moves its fixed points within their respective \(G_q\)-orbits. Thus, Theorem 1 tells us there are infinitely many non-conjugate maximal cyclic subgroups of special hyperbolic elements in \(G_q\) when \(q = 18\).

Let \(\mathcal{O}_{K_q}\) be the ring of integers in \(K_q\), and let \(\mathcal{O}_{K_q}^\ast\) be its unit group. We searched for new examples of special hyperbolic elements of \(G_q\) by computing the \(\lambda_q\)-continued fraction expansions of many elements of \(\lambda_q \mathcal{O}_{K_q}\) and \(\lambda_q \mathcal{O}_{K_q}^\ast\). The examples we found (excluding \(q = 18\)) are shown in Table 4 in terms of periodic \(\lambda_q\)-continued fractions, and Table 3 contains some of the examples we found for \(q = 18\). In [10], it was conjectured that there are exactly \(2\) orbits for \(G_7\) in \(\mathbb{P}^1(K_q)\), represented by \(\infty\) and \(\lambda_7^2 - 1\), and distinguished by residue classes modulo \(2\). A counterexample to this conjecture is provided in [9]. A similar conjecture for the unit group \(\mathcal{O}_{K_7}^\ast\) appears in [13]. We expect that the structure of the \(G_q\)-orbits in \(\lambda_q \mathbb{Q}(\lambda_q^2) \cup \{\infty\}\) is much more complicated, even for \(q = 7\). For example, all but \(27\) (rational) integers in the interval \([1,10^6]\) are cusps of \(G_7\). The first few exceptional integers are

and all \(27\) are special hyperbolic fixed points that do not lie in the orbit of \(\lambda_7^2 - 1\). Regarding \(\mathcal{O}_{K_7}^\ast\), letting \(\lambda_7^\prime= -\lambda_7^2 + 2\), the unit \(\lambda_7^7 (-\lambda_7^\prime)^{-23}\) is a special hyperbolic fixed point that does not lie in the orbit of \(\lambda_7^2 - 1\). Our counterexample of \(671\) is also referenced in the survey [2].

Part of our motivation for studying the Hecke triangle groups \(G_q\) comes from the dynamics of billiards in the regular \(q\)-gon and straight-line flows on the translation surfaces \((X_q,\omega_q)\) obtained by unfolding these tables. We refer to [7] for more details about the following discussion. The surface \((X_q,\omega_q)\) is obtained from one or two copies of a regular \(q\)-gon by gluing pairs of opposite parallel sides by translations. The derivatives of orientation-preserving affine automorphisms of \((X_q,\omega_q)\) form the Veech group \(V_q\), which in this case is a lattice in \(\mathop{\mathrm{SL}}(2,\mathbb{R})\) conjugate to \(G_q\) or an index \(2\) subgroup of \(G_q\). As a consequence, these surfaces satisfy the Veech dichotomy [14]: every straight-line flow is either periodic or uniquely ergodic. However, the Veech dichotomy does not tell us which directions are periodic. The set of periodic directions for \((X_q,\omega_q)\) is precisely the set of cusps of \(V_q\), which is in turn equal to the set of cusps of \(G_q\) up to acting by an element of \(\mathop{\mathrm{SL}}(2,\mathbb{R})\).

The group \(\mathop{\mathrm{GL}}^+(2,\mathbb{R})\) acts on the moduli space of all translation surfaces of a given genus. By applying an appropriate element \(M \in \mathop{\mathrm{GL}}^+(2,\mathbb{R})\) to \((X_q,\omega_q)\), one can arrange that the periodic directions for \(M(X_q,\omega_q)\) lie in \(\mathbb{P}^1(K_q)\) [7], [15]. With this normalization, the directions in \(\mathbb{P}^1(K_q)\) are precisely the directions for which the straight-line flow has vanishing SAF-invariant. Roughly speaking, the SAF-invariant measures the algebraic obstruction to being periodic. Periodic straight-line flows have zero SAF-invariant, but it is well-known that the converse is not true. A striking source of uniquely ergodic flows with zero SAF-invariant are flows invariant under a special affine pseudo-Anosov homeomorphism. The first known examples are the Arnoux-Yoccoz pseudo-Anosovs [16], and other families of examples are constructed in [17], [18]. These pseudo-Anosovs are affine automorphisms whose derivative is a special hyperbolic element of the associated Veech group.

In the language of translation surfaces, Theorem 1 tells us that on the unfolding of the regular \(18\)-gon, special affine pseudo-Anosovs are abundant. Moreover, this conclusion only depends on the commensurability class of the Veech group.

In particular, for a suitable \(M \in \mathop{\mathrm{GL}}^+(2,\mathbb{R})\), the Veech group of \(M(X_{18},\omega_{18})\) acts on \(\mathbb{P}^1(K_{18})\) and there are infinitely many distinct orbits in \(\mathbb{P}^1(K_{18})\) for this action. To the best of our knowledge, Theorem 2 provides the first known examples of lattice Veech groups that act on the projective line over their invariant trace field with infinitely many orbits. For a lattice Veech group, the invariant trace field is equal to its trace field, the field generated over \(\mathbb{Q}\) by the traces [19]. Note that by [4], for any lattice Veech group with a rational or quadratic trace field \(K\), after normalizing every element of \(\mathbb{P}^1(K)\) is a cusp and there are only finitely many orbits of cusps. We also remark that the special pseudo-Anosovs in Theorem 2 lie in infinitely many distinct conjugacy classes in the mapping class group \({\rm Mod}_g\), where \(g\) is the genus of \(X\).

Our proof of Theorem 1 is obtained via explicit constructions. For example, letting \(T = T_{18}\), we will see that \[\label{eq:G18infinite} \left(S T^4 S T^{-1} S T^{-4} S T \right)^k S T^2 S T^{-2} S T^{-2} \left(S T S T^4 S T^{-1} S T^{-4} \right)^k S T^2\tag{3}\] is a special hyperbolic element of \(G_{18}\) for all \(k \geq 0\). In the next section, we will provide several other infinite families of special hyperbolic elements. For any \(q\), the eigenvalues of a special hyperbolic element of \(G_q\) lie in the unit group \(\mathcal{O}_{K_q}^\ast\). When \(q = 18\), we have \(\mathcal{O}_{K_{18}}^\ast \cong \mathbb{Z}/2 \times \mathbb{Z}^2\). Despite this, all of the special hyperbolic elements we found in this case had eigenvalues lying in a single cyclic group up to sign. In particular, it turns out that the eigenvalues of the special hyperbolic elements in (3 ) are all powers of the unit \[\label{eq:K18unit} u_{18} = 2\lambda_{18}^4 - 4\lambda_{18}^2 + 1 .\tag{4}\] We observed similar coincidences for other values of \(q\), especially \(q = 7\). In all but one case, the eigenvalues of the special hyperbolic elements in \(G_7\) we found were powers of the unit \[\label{eq:K7unit} u_7 = \lambda_7^2 + \lambda_7\tag{5}\] up to sign, which leads us to suspect that similar constructions to (3 ) are possible for some other values of \(q\). A partial analogue of this phenomenon is known to arise in real quadratic fields via the classical continued fractions associated to \(G_3 = \mathop{\mathrm{SL}}(2,\mathbb{Z})\), see [20] and [21] for constructions. However, we emphasize that the associated hyperbolic elements of \(G_3\) are not special, indeed \(G_3\) does not contain any special hyperbolic elements.

More generally, Theorem 1, the occurrence of “rare” \(G_q\)-orbits as shown in Tables 1 and 2, and the behavior shown in Figure 1 (right), seem to suggest a positive answer to the following question.

Lastly, to emphasize how poorly understood the sets of cusps and hyperbolic fixed points of \(G_q\) still are, the following questions are open in all cases where \(K_q\) has degree at least \(3\).

1. Is there an element of \(\lambda_q \mathbb{Q}(\lambda_q^2)\) that is not fixed by any element of \(G_q\)?

2. Is there an (always terminating) algorithm that takes as input \(x \in \lambda_q \mathbb{Q}(\lambda_q^2)\) and outputs whether \(x\) is a cusp of \(G_q\)? A hyperbolic fixed point of \(G_q\)?

3. For \(q\) odd, does \(G_q\) have infinitely many integer cusps?

In [9] and [10], it is conjectured that question (1) has a negative answer for \(q = 7,9\). Based on our searches, we expect that question (1) also has a negative answer for \(q = 18\), and possibly for \(q = 14,30\) as well. Note that for each \(q\), a negative answer to (1) implies a positive answer to (2). For question (3), our searches suggest a positive answer for \(q = 7,9\).

1.0.0.1 Acknowledgements.

The author thanks Curt McMullen for many inspiring discussions on this topic. During the preparation of this paper, the author was supported by an NSF GRFP under grant DGE-1144152 and by an NSF MSPRF under grant DMS-2303185.

2 Special hyperbolic elements of \(G_{18}\)↩︎

We recall material about Hecke triangle groups, their action on \(\mathbb{P}^1(\mathbb{R})\) via Möbius transformations, and traces of matrix products in \(\mathop{\mathrm{SL}}(2,\mathbb{R})\). We then prove Theorem 1.

2.0.0.1 Stabilizers and conjugacy classes in \(G_q\).

A matrix \(A \in \mathop{\mathrm{SL}}(2,\mathbb{R})\) is hyperbolic if \(|\mathop{\mathrm{Tr}}(A)| > 2\), parabolic if \(|\mathop{\mathrm{Tr}}(A)| = 2\), and elliptic if \(|\mathop{\mathrm{Tr}}(A)| < 2\). Similarly for elements of \(\mathop{\mathrm{PSL}}(2,\mathbb{R})\). As a subgroup of \(\mathop{\mathrm{SL}}(2,\mathbb{R})\), the Hecke triangle group \(G_q\) acts on \(\mathbb{P}^1(\mathbb{R})\) by Möbius transformations with \(\pm I\) acting trivially. We denote by \(\overline{G}_q\) the image of \(G_q\) in \(\mathop{\mathrm{PSL}}(2,\mathbb{R})\).

The stabilizer in \(\overline{G}_q\) of a point in \(\mathbb{P}^1(\mathbb{R})\) is cyclic (Theorem 8.1.2 in [22]). The stabilizer of a hyperbolic fixed point is a maximal cyclic subgroup consisting of hyperbolic elements. Primitive hyperbolic elements are the generators of these stabilizers. By sending a hyperbolic element to its attracting fixed point, we obtain a bijection between primitive hyperbolic elements of \(\overline{G}_q\) and hyperbolic fixed points, and similarly a bijection between conjugacy classes of primitive hyperbolic elements in \(\overline{G}_q\) and orbits of hyperbolic fixed points. We record this discussion with the following lemma.

The group \(\overline{G}_q\) is isomorphic to a free product of two cyclic groups of orders \(2\) and \(q\), respectively. The images in \(\mathop{\mathrm{PSL}}(2,\mathbb{R})\) of the matrices \[S = \left(\begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix}\right), \quad U_q = S T_q = \left(\begin{matrix} 0 & -1 \\ 1 & \lambda_q \end{matrix}\right)\] realize the presentation \[\overline{G}_q = \langle S, U_q \mid S^2 = U_q^q = I \rangle .\] Denote \(T = T_q\) and \(U = U_q\). Each element in \(\overline{G}_q\) can thus be expressed uniquely as a reduced word in \(S\) and \(U\) (Theorem 4.1 in [23]). These reduced words are powers of \(S\) or \(U\) or have the form \[S^{\varepsilon_1} U^{a_1} S U^{a_2} \cdots S U^{a_{k-1}} S U^{a_k} S^{\varepsilon_k}\] for some \(k > 0\), \(1 \leq a_1,\dots,a_k \leq q - 1\), and \(\varepsilon_1,\varepsilon_k \in \{0,1\}\). A reduced word is cyclically reduced if it is a power of \(S\) or \(U\) or if \(\varepsilon_1 \neq \varepsilon_k\) above. Up to conjugation by \(S\), a cyclically reduced word is a power of \(S\) or \(U\) or has the form \[S U^{a_1} \cdots S U^{a_k}\] for some \(k > 0\) and \(1 \leq a_1,\dots,a_k \leq q - 1\). Two cyclically reduced words of this form are conjugate in \(\overline{G}_q\) if and only if they are cyclic permutations of each other (Theorem 4.2 in [23]). For convenience, we will use a weaker version of this fact for words in \(S,T\).

2.0.0.2 Trace relations.

Recall that for any matrix \(A \in \mathop{\mathrm{SL}}(2,\mathbb{R})\), \[A + A^{-1} = \mathop{\mathrm{Tr}}(A) I .\] It follows that for any \(A,B \in \mathop{\mathrm{SL}}(2,\mathbb{R})\), \[\label{eq:trab} \mathop{\mathrm{Tr}}(AB) + \mathop{\mathrm{Tr}}(AB^{-1}) = \mathop{\mathrm{Tr}}(A)\mathop{\mathrm{Tr}}(B) .\tag{9}\] We will be interested in the traces of certain matrix products of the form \[M_{k,\ell} = D^\ell C B^k A\] with \(A,B,C,D \in \mathop{\mathrm{SL}}(2,\mathbb{R})\) and \(k,\ell \in \mathbb{Z}\).

2.0.0.3 Families of special hyperbolic elements.

A hyperbolic element of \(G_q\) is special if its eigenvalues lie in \(K_q\). Denote \(\lambda = \lambda_q\). Let \(A = \left(\begin{matrix} a & b \\ c & d \end{matrix}\right) \in G_q\) be a special hyperbolic element, and let \(t,t^{-1} \in K_q\) be its eigenvalues. Note that \(c \neq 0\) since \(A\) does not fix the cusp \(\infty\). Since \(A\) is special, \(\mathop{\mathrm{Tr}}(A)^2 - 4 = (t - t^{-1})^2\) is a square in \(K_q\). Solving \(A \cdot x = (ax + b)/(cx + d) = x\), we see that the fixed points of \(A\) are given by \(x = ((a - d) \pm (t - t^{-1})) / 2c\). For \(q\) odd, since the matrix entries of \(G_q\) lie in \(\mathbb{Z}[\lambda]\), we have \(x \in \mathbb{Q}(\lambda) = \lambda \mathbb{Q}(\lambda^2)\). For \(q\) even, by an inductive argument using the generators \(S,T_q\), the matrix \(A\) has one of the forms \[\left(\begin{matrix} a & b_0\lambda \\ c_0\lambda & d \end{matrix}\right), \quad \left(\begin{matrix} a_0\lambda & b \\ c & d_0\lambda \end{matrix}\right),\] with \(a_0,b_0,c_0,d_0 \in \mathbb{Z}[\lambda^2]\) (see Corollary 1 in [1]). Since \(\mathop{\mathrm{Tr}}(A) \in K_q = \mathbb{Q}(\lambda^2)\) is nonzero, \(A\) is of the first form and \(x = ((a-d) \pm (t-t^{-1}))/2\lambda c_0 \in \lambda \mathbb{Q}(\lambda^2)\). Thus, fixed points of special hyperbolic elements of \(G_q\) lie in \(\lambda \mathbb{Q}(\lambda^2)\).

We now apply Lemma 7 to produce infinite families of orbits of special hyperbolic fixed points. Denote \(T = T_{18}\) and \(\lambda = \lambda_{18}\). For \(n_1,\dots,n_k\) a finite sequence of nonzero integers, define \[M(n_1,\dots,n_k) = S T^{n_k} \cdots S T^{n_1} \in G_{18} .\] Additionally, denote by \((n_1,\dots,n_k)^m\) the concatenation of \(m\) copies of \(n_1,\dots,n_k\). Lastly, recall the unit \(u_{18} = 2\lambda^4 - 4\lambda^2 + 1\) from (4 ) and denote \(u = u_{18}\). All of the computational checks in the proof of the following theorem were carried out with SageMath.

3 \(\lambda_q\)-continued fraction expansions↩︎

Many questions about the Hecke triangle groups \(G_q\) can be studied computationally using a family of continued fraction algorithms introduced by Rosen [1]. Throughout, we denote \(\lambda = \lambda_q\), \(K = K_q\), and \(T = T_q\). Rosen showed that any \(x \in \mathbb{R}\) can be expressed as a \(\lambda\)-continued fraction \[\label{eq:lambdaCF} x = [a_0,a_1,a_2,\dots] = a_0 \lambda - \frac{1}{a_1 \lambda - \frac{1}{a_2 \lambda - \cdots}}\tag{12}\] with \(a_0 \in \mathbb{Z}\), \(a_i \in \mathbb{Z}\setminus\{0\}\) for \(i \geq 1\), in a unique way by requiring \[\begin{align} x - a_0\lambda &\in (-\lambda/2, \lambda/2] \\ -1/(x - a_0\lambda) - a_1\lambda &\in (-\lambda/2, \lambda/2] \\ & \vdots \end{align}\] The cusps \(G_q \cdot \infty\) are precisely the elements of \(\mathbb{P}^1(\mathbb{R})\) that can be expressed as finite \(\lambda\)-continued fractions \([a_0,a_1,\dots,a_N]\). If there exists \(n \geq 1\) and \(i_0 \geq 0\) such that \(a_i = a_{i+n}\) for all \(i \geq i_0\), we say that the \(\lambda\)-continued fraction is preperiodic, and periodic if we can take \(i_0 = 0\). If \(x\) is a periodic \(\lambda\)-continued fraction, we denote this by \[x = \overline{[a_0,a_1,\dots,a_n]}\] where \(n+1\) is the minimal period, and \(x\) is fixed by \[M_x = ST^{-a_n} \cdots ST^{-a_0} \in G_q .\]

For all \(x \in \mathbb{P}^1(\mathbb{R})\), we have \[S \cdot (-x) = -(S \cdot x), \quad T \cdot (-x) = -(T^{-1} \cdot x) .\] It follows that the set of cusps and the set of hyperbolic fixed points are preserved under negation. Since \(\lambda \mathbb{Q}(\lambda^2)\) is invariant under negation, the set of special hyperbolic fixed points is also preserved under negation. Since \(\lambda/2\) is a cusp [1], for any periodic \(\lambda\)-continued fraction, negation simply negates its periodic part. Suppose \(x = \overline{[a_0,a_1,\dots,a_n]}\) is fixed by a hyperbolic element \(M_x = S T^{-a_n} \cdots S T^{-a_0}\), and let \(x^\prime= \overline{[a_n,a_{n-1},\dots,a_0]}\). Since \(M_x\) and \(M_x^{-1}\) have the same fixed points and \(S M_x^{-1} S^{-1} = S T^{a_0} \cdots S T^{a_n}\), we see that \(M_x\) also fixes \(S \cdot (-x^\prime) = 1/x^\prime\). The periodic \(\lambda\)-continued fractions obtained by negating and reversing the periodic part do not necessarily lie in the same \(G_q\)-orbit. In this way, we may obtain \(1\), \(2\), or \(4\) distinct \(G_q\)-orbits.

Below, we summarize the computer searches we carried out to find new \(G_q\)-orbits of special hyperbolic fixed points. All of our computations were done using SageMath, which supports arithmetic in number fields and arbitrary precision numerical computations.

3.0.0.1 \(G_7\)-orbits.

Let \(q = 7\). Then \(\lambda\) has one other Galois conjugate with absolute value greater than \(1\), given by \(\lambda^\prime= -\lambda^2 + 2\). The ring of integers of \(K_q\) is \(\mathbb{Z}[\lambda]\), and the positive unit group is generated by \(\lambda, -\lambda^\prime\). We counted the number of elements of each \(G_7\)-orbit in \(\mathbb{Z}\cap [1,10^6]\) by computing the associated \(\lambda\)-continued fraction expansions, and similarly for a large subset of the positive unit group and the ring of integers. The results are reported in Table 1. All of the \(\lambda\)-continued fraction expansions we computed were either finite or preperiodic.

We found several new \(G_7\)-orbits of special hyperbolic fixed points, and Table 4 lists the associated periodic \(\lambda\)-continued fractions, up to negation and reversal. We obtain a lower bound on the number \(N_7\) of \(G_7\)-orbits in \(\mathbb{P}^1(K_q)\) of \[N_7 \geq 1 + 1 + 4 + 4 + 4 + 4 + 2 + 4 = 24\] Note that only \(13\) of these \(24\) orbits appear in the Table 1. In particular, even up to negation and reversal of periodic parts, the orbit of \[\frac{531}{7} \lambda^2 + \frac{402}{7} \lambda - \frac{319}{7} = \overline{[169,2,1,-1,-1,-2]}\] does not appear. We found this orbit of special hyperbolic fixed points by checking all periodic \(\lambda\)-continued fractions of the form \(\overline{[a_1,a_2,\dots,a_6]}\) with \(\prod_{j=1}^6 a_i = \pm 2^2 \cdot 13^2\). Lastly, for all but the last row in the \(q = 7\) section of Table 4, the eigenvalues of the associated special hyperbolic elements are powers of the unit \[u_7 = \lambda_7^2 + \lambda_7 .\] We were unable to prove a version of Theorem 8 in the \(q = 7\) case though.

\(G_9\)-orbits.↩︎

Let \(q = 9\). Again, the ring of integers of \(K_q\) is \(\mathbb{Z}[\lambda]\), and the positive unit group is generated by \(\lambda\) and \(-\lambda^\prime= \lambda^2 - 2\). Table 2 counts the number of elements of each \(G_9\)-orbit in a large subset of the rational integers, the positive units, and the ring of integers, and Table 4 lists the associated periodic \(\lambda\)-continued fractions. We get a lower bound on the number \(N_9\) of \(G_9\)-orbits in \(\mathbb{P}^1(K_9)\) of \[N_9 \geq 1 + 4 + 4 + 4 = 13 .\] All of the \(\lambda\)-continued fraction expansions we computed were either finite or preperiodic.

3.0.0.2 \(G_{18}\)-orbits.

Let \(q = 18\). We carried out similar searches as in the \(G_7\) and \(G_9\) cases, for elements of the form \[\begin{align} \lambda n, & \quad n \in \mathbb{Z}\cap [1,10^6] \\ \lambda \lambda_1^a \lambda_2^b, & \quad a,b \in \mathbb{Z}\cap [-10^2,10^2] \\ \lambda^3 a + \lambda^5 b, & \quad a,b \in \mathbb{Z}\cap [-10^3,10^3] \end{align}\] where \(\lambda_1 = \lambda^2 - 1\) and \(\lambda_2 = \lambda^2 - 2\). In this case, we found a much larger number of orbits, so we only report the associated periodic \(\lambda\)-continued fractions up to negation and reversal in Table 3. All of the \(\lambda\)-continued fraction expansions we computed were either finite or preperiodic. Moreover, all of the periodic \(\lambda\)-continued fractions \(\overline{[a_1,a_2,\dots,a_n]}\) we found satisfied the following properties:

• \(n\) is divisible by \(4\)

• for \(1 \leq j \leq n\), there is \(k \geq 0\) such that \(a_j = \pm 2^k\)

• the number of positive and negative \(a_j\)’s is equal

• \(\prod_{j=1}^n a_j = 2^n\)

Note that only finitely many periodic parts of each length satisfy the above properties. We systematically checked all periodic parts of length \(4\), \(8\), and \(12\) satisfying the above properties, and listed the periodic parts up to negation and reversal that yield special hyperbolic fixed points in Table 3.

3.0.0.3 \(G_q\)-orbits for other \(q\).

For other values of \(q\), we did some ad hoc searches to find additional special hyperbolic fixed points, which are reported in Table 4 in terms of periodic \(\lambda\)-continued fractions. For space reasons, we sometimes use exponential notation \(n^k\) to denote \(k\) consecutive occurrences of \(n\). Many of the orbits in Table 4 have previously appeared in the literature [7], [9], [10], and two of the examples with \(q = 16,30\) were recently independently discovered in [8].

For \(q = 7,9,18\), our computations provide strong evidence that every element of \(\lambda\mathbb{Q}(\lambda^2)\) is fixed by a nontrivial element of \(G_q\). This was previously conjectured in the cases \(q = 7,9\) in [9] and [10]. For other values of \(q\), it seems more difficult to investigate this question empirically.

To illustrate some of this evidence, suppose for simplicity that \(K_q\) has class number \(1\). Then every element of \(\mathbb{P}^1(K_q)\) can be expressed as \([a : b]\) with \(a,b \in \mathcal{O}_{K_q}\) relatively prime. Define a height on \(\mathbb{P}^1(K_q)\) by \[h([a : b]) = \prod_{\sigma : K_q \hookrightarrow \mathbb{R}} \left(|\sigma(a)| + |\sigma(b)|\right)\] where the product is over the real embeddings of \(K_q\). Note that \(K_q\) is totally real. The function \(h\) is well-defined since any other expression for \([a : b] \in \mathbb{P}^1(K_q)\) differs by simultaneous multiplication of \(a,b\) by a unit \(u \in \mathcal{O}_{K_q}^\ast\), which satisfies \(\prod_{\sigma} |\sigma(u)| = 1\). For elements \(x \in \lambda\mathbb{Q}(\lambda^2)\), we will consider the height of \(x/\lambda\). We refer to [5] for more discussion on heights in the context of triangle groups and Veech groups.

Figure 1 illustrates the behavior of \(\log(h)\) of a “typical” element of \(\lambda \mathbb{Q}(\lambda^2)\) under the \(\lambda\)-continued fraction algorithm. The cases \(q = 7,9,18\) all behave as in the left image in Figure 1. The cases \(q = 14,30\) behave as in Figure 2. The remaining cases (where \(K_q\) has degree at least \(3\)) all seem to behave as in the right image in Figure 1.

Email: kgwinsor@gmail.com

Department of Mathematics, Stony Brook University, Stony Brook, New York, USA

References↩︎