p-adic Schwarzian triangle groups of Mumford type

Fumiharu Kato

We define a certain p-adic analogue of classical Schwarzian triangle groups related to Mumford’s uniformization of analytic curves and give a complete classification of it.

1. Introduction

1.1. Uniformization of orbifolds and triangle groups. The rich geometric structure of uniformized analytic varieties over non-archimedean fields has been studied by many authors, and already has a long history. Mumford [Mum72] showed that an analytic curve X defined over a non-archimedean field K with a split multiplicative analytic reduction can be uniformized as X ≅ Γ\(ℙ_{K}^{1,an} − L_{Γ}), where Γ is a finitely generated free discrete subgroup of PGL(2, K) and L_{Γ} is the set of limit points of Γ. An equally important example is the uniformization of a curve which is an étale covering of a Mumford curve, studied by van der Put [vdP83]. These are the most practical and reasonable analogues of uniformizations of complex analytic curves.

Historically, however, the theory of uniformization in complex analysis arose from interplay between geometric and function-theoretic viewpoints. This is apparent if one considers the orbifold uniformization of ℙ_{ℂ}^{1,an} with finitely many points marked by positive integers (= the ramification degrees). The link between geometry and function theory stems from the behavior of the multivalued function z, inverse to the uniformization map, which is written as a ratio z = u_{1}/u_{2} of two linearly independent solutions of a Fuchsian differential equation with rational exponents (cf. [Yos87]). For example, if the orbifold is ℙ_{ℂ}^{1,an} with precisely the three points 0, 1, ∞ marked by e_{0}, e_{1}, e_{∞} ∈ ℤ_{>0}, then the corresponding differential equation is the Gaussian hypergeometric equation

(1.1.1)  x(1 − x)\frac{d^{2}u}{dx^{2}} + \{c − (a + b + 1)x\}\frac{du}{dx} − abu = 0

(where a, b, c ∈ ℚ with |1 − c| = 1/e_{0}, |c − a − b| = 1/e_{1}, and |a − b| = 1/e_{∞}). Depending on whether 1/e_{0} + 1/e_{1} + 1/e_{∞} > 1, = 1 or < 1, the image of z = u_{1}/u_{2} is isomorphic to ℙ_{ℂ}^{1,an}, ℂ or ℍ, respectively, and z maps the upper-half plane onto the interior of a triangle region with the angles π/e_{0}, π/e_{1} and π/e_{∞}. The corresponding orbifold fundamental group has a representation into PGL(2, ℂ) with discrete image, the so-called Schwarzian triangle group Δ(e_{0}, e_{1}, e_{∞}), whose action on the universal covering space is visible in terms of complex reflections (cf. [Mag74, Chap. II]).

The problems which arises from carrying out such a program in the p-adic situation are mainly topological. It is perhaps appropriate here to remind the reader of the fact that in rigid analysis an étale covering map is not necessarily a topological covering map (= the locally topologically trivial map). Consequently, in contrast to complex analysis, we have many simply connected domains, even one dimensional; for instance, the complement of finitely many points in ℙ_{K}^{1,an} is simply connected. In particular, a reasonable definition of the orbifold fundamental groups is highly non-trivial.

In [And98], Y. André studied the rather special class of étale covering maps which are composites of topological coverings followed by finite étale (not necessarily topological) coverings. He observed that the covering maps of this kind give rise to a reasonable concept of orbifold fundamental groups (denoted by π_{1}^{orb} in [And98]) in the p-adic situation, and discussed the relation with differential equations; one of his result phrases it as follows [And98, §6]: Consider the orbifold X (cf. [And98, 5.1] for the precise definition) which is supported on ℙ_{ℂ_{p}}^{1,an} with n-points ζ_{i} marked by positive integers e_{i} (1 ≤ i ≤ n). Then:

There exists a canonical fully faithful functor of categories

(1.1.2) \{Continuous representations ρ of π_{1}^{orb}(X , \overline{x}) into GL(r, ℂ_{p}) with discrete coimage\} ⟶ \{Algebraic regular connections on ℙ_{ℂ_{p}}^{1} − \{ζ_{i} | 1 ≤ i ≤ n\} of rank r such that the local monodromy at each ζ_{i} is of fi- nite order dividing e_{i}\} .

Moreover, the essential image of this functor consists of the connections ∇ enjoying the following condition (Global Monodromy Condition):

(1.1.3) There exists a connected rigid analytic curve S and a finite morphism φ: S → X ramified above precisely the points ζ_{i} with ramification indices dividing e_{i}, such that the connection φ^{∗}∇ on S admits a full set of multivalued analytic solutions on S (and, moreover, on the Berkovich space associated to S).

In particular, if n = 3, r = 2 and the image Γ of ρ in PGL(2, ℂ_{p}) is discrete, then Γ can be regarded as a p-adic analogue of the Schwarzian triangle group Δ(e_{0}, e_{1}, e_{∞}). If so, Γ gives the “projective monodromy” for the connection ∇ defined by the functor (1.1.2), which is nothing but the one associated to the Gaussian hypergeometric equation (1.1.1). In [And98, §9], André discussed such groups, called p-adic triangle groups, and gave a complete list of the so-called arithmetic p-adic triangle groups, which are constructed through the Cherednik-Drinfeld theory of uniformization of Shimura curves, starting from Takeuchi’s list of arithmetic triangle groups. Notably, he deduced that there exists no arithmetic p-adic triangle groups for p > 5. The uniformizations of the orbifolds X corresponding to these groups are given by the Drinfeld upper-half plane or its étale coverings.

1.2. Results of this paper. In this paper, we will discuss (not necessarily arithmetic) p-adic triangle groups Γ as above, especially in the case that the corresponding uniformization is given by the space ℙ_{ℂ_{p}}^{1,an} − L_{Γ}. We call such a group Γ a p-adic triangle group of Mumford type; we can define it in simpler terms (without involving π_{1}^{orb}) as follows:

Definition. A finitely generated discrete subgroup Γ of PGL(2, ℂ_{p}) is said to be a p-adic (Schwarzian) triangle group of Mumford type if Γ\(ℙ_{ℂ_{p}}^{1,an} − L_{Γ}) ≅ ℙ_{ℂ_{p}}^{1,an} and the uniformization map

ϖ_{Γ}: (ℙ_{ℂ_{p}}^{1,an} − L_{Γ}) ⟶ ℙ_{ℂ_{p}}^{1,an}

is ramified above precisely three points.

If Γ is a finite subgroup, then L_{Γ} = ∅, and the map ϖ_{Γ} is the one which comes from the algebraic quotient map ℙ_{ℂ_{p}}^{1} → ℙ_{ℂ_{p}}^{1}/Γ. Hence the spherical (i.e., 1/e_{0} + 1/e_{1} + 1/e_{∞} > 1) p-adic triangle groups of Mumford type amount to the same as the classical ones. Our main theorem gives the complete classification of the other p-adic triangle groups of Mumford type:

Theorem. (1) A non-spherical p-adic triangle group Γ of Mumford type of index (e_{0}, e_{1}, e_{∞}) exists if and only if p and the unordered triple (e_{0}, e_{1}, e_{∞}) occur in the left two columns of Table 1, where, in the last row, the integers l, m, n obey the following condition;

(∗) lmn ≥ 4, and at least two of l, m, n are odd.

In particular, p-adic triangle groups of Mumford type do not exist if p > 5.

(2) In each of the cases in Table 1, the conjugacy class of Γ in PGL(2, ℂ_{p}) is uniquely determined. Moreover, Γ is isomorphic to the abstract group given in the right column of the table, where Z_{n} denotes the cyclic group of order n, D_{n} the dihedral group of degree n (≅ Z_{n} ⋊ Z_{2}), etc. In the case (3, 2l, 4m) (resp. (2l, 2m, 2n)), the subgroups D_{4} and D_{2} in S_{4} are chosen such that D_{2} ⊂ D_{4} (resp. both D_{2} are chosen to be equal to each other).

Table 1: List of p-adic triangle groups of Mumford type

p  Index  Abstract structure
p = 5  (2, 3, 5l)  (l ≥ 2)  A_{5} ∗_{D_{5}} D_{5l}
p = 3  (2, 5, 3l)  (l ≥ 2)  A_{5} ∗_{D_{3}} D_{3l}
(2, 4, 3l)  (l ≥ 2)  S_{4} ∗_{D_{3}} D_{3l}
(2, 3l, 3l)  (l ≥ 2)  A_{4} ∗_{Z_{3}} Z_{3l}
p = 2  (3, 5, 2l)  (l ≥ 3 and l: odd)  A_{5} ∗_{D_{2}} D_{2l}
(3, 2l, 4m)  (lm ≥ 2 and l: odd) D_{4m} ∗_{D_{4}} S_{4} ∗_{D_{2}} D_{2l}
(3, 3, 2l)  (l ≥ 3 and l: odd)  A_{4} ∗_{D_{2}} D_{2l}
(2m + 1, 2l, 2l)  (m ≥ 1, l ≥ 2)  D_{2m+1} ∗_{Z_{2}} Z_{2l}
(2l, 2m, 2n)  (l, m, n satisfy (∗)) D_{2l} ∗_{D_{2}} D_{2m} ∗_{D_{2}} D_{2n}

Remark. (1) In each of the cases in the table, at least one of the numbers e_{i} (i = 0, 1, ∞) is divisible by the residue characteristic p.

(2) The theorem shows that there are no Euclidean (i.e., 1/e_{0} +1/e_{1} +1/e_{∞} = 1) p-adic triangle groups of Mumford type. The reason for this is that the elliptic curve which covers an Euclidean orbifold always has a complex multiplcation and never be a Tate curve.

(3) The theorem and the well-known fact on automorphisms of Mumford curves [GvP80, VII.§1] imply that for p > 5 and for a Mumford curve X, if Aut(X)\X ≅ ℙ_{K}^{1}, then the quotient map X → ℙ_{K}^{1} ramifies above at least 4 points. Applying the classical Hurwitz formula, we therefore see that |Aut(X)| ≤ 12(g − 1) (where g is the genus of X), which partly recovers the Herrlich’s result [Her80b].

(4) Our list of p-adic triangle groups has a non-empty intersection with André’s list of p-adic arithmetic triangle groups, but does not include it, since André’s padic arithmetic triangle groups are not of Mumford type in general. In other words, André’s p-adic arithmetic triangle groups do not always come out with the uniformization by the spaces of form ℙ_{ℂ_{p}}^{1,an} − L_{Γ}, but by étale coverings of them.

1.3. Outline of the proof. Let us briefly view an outline of the proof.

(1.3.1) The proof basically relies on the following principle, which is well-known and can be proved by using standard arguments as in [GvP80] and [vdP97]: Let K be a finite extension of ℚ_{p}, and Γ a finitely generated discrete subgroup of PGL(2, K). Replacing K by a finite extension if necessary, we assume that the fixed points in ℙ_{K}^{1,an} of any element (≠ 1) in Γ are K-valued (cf. [GvP80, I.3.1 (1)]). Let L_{Γ} (resp. F_{Γ}) be the set of all limit points (resp. fixed points) in ℙ_{K}^{1,an} of Γ. If Γ is free, then we have L_{Γ} = \overline{F}_{Γ} (where \overline{·} stands for the topological closure), while in general we only have L_{Γ} ⊆ \overline{F}_{Γ}, and \overline{F}_{Γ} − L_{Γ} is a discrete set consisting of the fixed points of elliptic elements in Γ. Let 𝒯_{Γ} (resp. 𝒯_{Γ}^{∗}) be the subtree of the Bruhat-Tits tree generated by the vertices determined by three pairwise distinct points in L_{Γ} (resp. \overline{F}_{Γ}); cf. [Mum72, §1] and [GvP80, I.§2]. Clearly, 𝒯_{Γ} ⊆ 𝒯_{Γ}^{∗}. It is easy to see that Γ acts on 𝒯_{Γ} and 𝒯_{Γ}^{∗} without inversion. Set T_{Γ} = Γ\𝒯_{Γ} and T_{Γ}^{∗} = Γ\𝒯_{Γ}^{∗}. Let Ω_{Γ} = ℙ_{K}^{1,an} − L_{Γ}, and ϖ_{Γ}: Ω_{Γ} → Γ\Ω_{Γ} the quotient map. Then:

There exist canonical bijections, compatible with the quotient maps,

(1.3.2)

\left\{{Ramification points in \atop Ω_{Γ} of the map ϖ_{Γ}}\right\} ⟷ \{Ends of 𝒯_{Γ}^{∗} − 𝒯_{Γ} \}
↓  ↓
{Branch points in Γ\Ω_{Γ} } ⟷ {Ends of T_{Γ}^{∗} }.

Moreover, the decomposition group of a ramification point coincides with the stabilizer of the corresponding end.

For example, the case when Γ = G is a finite subgroup provides a nice prototype. As is well-known, the finite subgroup G is isomorphic to one of the following groups: cyclic group Z_{n} of order n > 1, dihedral group D_{n} ≅ Z_{n} ⋊ Z_{2} of degree n > 1, tetrahedral group A_{4}, octahedral group S_{4}, and icosahedral group A_{5} (cf. [Web99, Vol. II, §67]). Since the map ϖ_{G} is the one associated to the algebraic map ℙ_{K}^{1} → G\ℙ_{K}^{1} ≅ ℙ_{K}^{1}, we deduce that the quotient graph T_{G}^{∗} is a tree with two ends, if G is cyclic, or with three ends, otherwise.

(1.3.3) Now we assume for a general Γ as above that Γ\Ω_{Γ} ≅ ℙ_{K}^{1,an}. Then T_{Γ}^{∗} is a tree, and hence we have a lifting ι_{Γ}: T_{Γ}^{∗} ↪ 𝒯_{Γ}^{∗} by [Ser80, I.4.1, Prop. 17]. Let ε_{i} (i = 1, …, n) be the ends of ι_{Γ}(T_{Γ}^{∗}), and ζ_{i} the corresponding branch point in Γ\Ω_{Γ} ≅ ℙ_{K}^{1,an}. For each i the stabilizer Γ_{ε_{i}} of the end ε_{i} is a finite cyclic group; let γ_{i} be a generator of it, and e_{i} its order. Then ϖ_{Γ} gives the uniformization of the orbifold X = (ℙ_{K}^{1,an}, (ζ_{i}, e_{i}); i = 1, …, n). By [And98, 5.8] we have a continuous surjective homomorphism π_{1}^{orb}(X , \overline{x}) → Γ, from which it follows in particular that Γ is generated by γ_{i} (i = 1, …, n) (cf. [And98, 5.4]).

(1.3.4) In our situation, what to do is the classification of Γ such that T_{Γ}^{∗} is a tree with precisely three ends. The proof of the “only if” part of Theorem (1) goes like as follows: Suppose that T_{Γ}^{∗} is a tree with precisely three ends. The first main point of the proof is that we can find a finite subgroup G ⊆ Γ such that by the natural map

(1.3.5)  T_{G}^{∗} ⟶ T_{Γ}^{∗}

induced by the inclusion 𝒯_{G}^{∗} ↪ 𝒯_{Γ}^{∗}, the tree T_{G}^{∗} essentially determines the shape of T_{Γ}^{∗}; more precisely, the map (1.3.5) gives rise to a bijection between ends. We can actually do it by choosing G to be a subgroup whose abstract structure is “maximal” with respect to a certain order.

On the other hand, it is easily observed that the action of Γ on 𝒯_{Γ}^{∗} is also “visible” like in the classical situation. In fact, Γ is generated by three elliptic elements γ_{0}, γ_{1} and γ_{∞} of finite order (by (1.3.3)) of which the action on 𝒯_{Γ}^{∗} looks like a “reflection”. Here, for such an elliptic element γ the apartment connecting the two fixed points of γ plays the role of reflection mirrors (cf. Definition 2.3), and the fundamental domain T_{Γ}^{∗}, which is essentially identified with T_{G}^{∗}, gives the counterpart of the triangle regions.

Let θ_{0}, θ_{1} and θ_{∞} be the generators of G corresponding to the three ends of T_{G}^{∗} as in (1.3.3). Since there exist no two elements in Γ which have exactly one fixed point in common (because of the discreteness), each γ_{i} (i = 0, 1, ∞) has the same (up to conjugacy) fixed points as one of θ_{0}, θ_{1} and θ_{∞}; hence, changing indices of γ_{i}’s if necessary, we may assume γ_{i}^{l_{i}} = θ_{i} for some l_{i} ≥ 1. Now the second main point of the proof is that such a trick with l_{i} > 1 is only possible when the mirror of θ_{i} runs apart enough from any vertices in 𝒯_{G}^{∗} stablized by G; otherwise, in the tree 𝒯_{Γ}^{∗} such a vertex would be stabilized by either an infinite subgroup (which contradicts the discreteness) or by a finite subgroup which is “bigger” than G with respect to the order as above.

In order to know which mirror is separated enough from the vertex stablized by G, we need a detailed study about the trees 𝒯_{G}^{∗} and T_{G}^{∗} for the finite subgroup G. This will be done in the next section, where we will find that the picture of these trees differs depending on the residue characteristic p, especially when p is small (which effects the restraint p ≤ 5 in the theorem). Having such a knowledge of the mirror arrangement, we will be finally able to determine all the possible fundamental domains T_{Γ}^{∗} and the triple (e_{0}, e_{1}, e_{∞}) as desired. The results in Theorem (2) are by-products of the proof of this part.

(1.3.6) The “if” part of the proof will be done by constructing Γ explicitly. We will do it, roughly speaking, along the recipe which was already indicated in the “only if” part of the proof. Comparing our construction with the abstract construction of groups as in [Ser80, I.4.5], we will show that Γ is discrete, as well as that Γ has the desired abstract structure.

1.4. Notation and conventions. (1) Throughout this paper K denotes a finite extension of ℚ_{p}, O_{K} the integer ring, and π a prime element in O_{K}. We write [K: ℚ_{p}] = ef, where e is the ramification degree and q = p^{f} is the the number of elements in the residue field k = O_{K}/πO_{K}. We denote by ν: K^{×} → ℤ the normalized (i.e., ν(π) = 1) valuation.

(2) Let 𝒯_{K}^{BT} be the Bruhat-Tits tree attached to PGL(2, K). The geometric realization |𝒯_{K}^{BT}| is a metrized space with the metric denoted by d(·, ·). For two vertices Λ_{0} and Λ_{1}, [Λ_{0}, Λ_{1}] denotes the geodesic path connecting them. For u, v ∈ ℙ^{1}(K) (u ≠ v) and a vertex Λ ∈ 𝒯_{K}^{BT}, the apartment (resp. half-line) connecting u and v (resp. u and Λ) is denoted by ]u, v[ (resp. [Λ, u[). We will use the notation L_{Γ}, F_{Γ}, 𝒯_{Γ}, 𝒯_{Γ}^{∗}, T_{Γ}, T_{Γ}^{∗} and Ω_{Γ} as in (1.3.1) for any finitely generated discrete subgroup Γ ⊂ PGL(2, K).

2. The trees associated to finite groups

2.1. The aim of this section is to describe the trees 𝒯_{G}^{∗} and T_{G}^{∗} (= G\𝒯_{G}^{∗}) for G a finite subgroup in PGL(2, K), where K is taken so that any element (≠ 1) in G has at most K-valued fixed points in ℙ_{K}^{1}. The procedure of the description is very elementary and straightforward, but needs a lot of computation. Here we mainly present only the results, and exhibit necessary data in Appendix.

First we need to know the following fact: Let a = (a_{0}: a_{1}), b = (b_{0}: b_{1}), c = (c_{0}: c_{1}) and d = (d_{0}: d_{1}) be four pairwise distinct K-rational points of ℙ_{K}^{1}. We are interested in the arrangement of two apartments ]a, b[ and ]c, d[ in 𝒯_{K}^{BT}. Let us define the crossratio

(2.1.1)  R(a, b; c, d) = \frac{(a_{1}c_{0} − a_{0}c_{1})(b_{1}d_{0} − b_{0}d_{1})}{(a_{0}b_{1} − a_{1}b_{0})(c_{0}d_{1} − c_{1}d_{0})}.

2.2. Lemma. (1) If |ν(R(a, b; c, d))| = |ν(R(b, a; c, d))| = 0, then ]a, b[ and ]c, d[ intersects at exactly one vertex.

(2) If |ν(R(a, b; c, d))| = |ν(R(b, a; c, d))| ≠ 0, then ]a, b[ and ]c, d[ are disjoint with the distance |ν(R(a, b; c, d))|.

(3) If |ν(R(a, b; c, d))| ≠ |ν(R(b, a; c, d))|, then the intersection of ]a, b[ and ]c, d[ is the path [Λ(a, b, c), Λ(b, c, d)] of length max{|ν(R(a, b; c, d))|, |ν(R(b, a; c, d))|}, where Λ(z_{0}, z_{1}, z_{2}) for pairwise distinct three points z_{0}, z_{1}, z_{2} ∈ ℙ^{1}(K) is the unique vertex lying in the intersection of all ]z_{i}, z_{j}[ for i, j = 0, 1, 2, i ≠ j.

Proof. First we recall how to calculate Λ(z_{0}, z_{1}, z_{2}): Let Y_{i} (i = 0, 1, 2) be a homogeneous coordinate of z_{i}, and choose α_{i} ∈ K^{×} such that α_{0}Y_{0}+α_{1}Y_{1}+α_{2}Y_{2} = 0. Then Λ(z_{0}, z_{1}, z_{2}) is the similarity class of O_{K}α_{i}Y_{i} + O_{K}α_{j}Y_{j} for any i, j = 0, 1, 2, i ≠ j. By this, it is easily checked that d(Λ(a, b, c), Λ(b, c, d)) = |ν(R(a, b; c, d))|. Once it is checked, all the statements are clear, since the apartments ]a, b[ and ]c, d[ either do not intersect or intersect along a path (see Figure 1). □

[figure: d; c; b; a; Λ(b, c, d); Λ(a, b, c); c; d; a; b; Λ(a, b, c); Λ(b, c, d)]

Figure 1: Arrangement of the apartments ]a, b[ and ]c, d[

2.3. Definition. Let γ ∈ PGL(2, K) be an elliptic element (i.e., an element having two distinct eigenvalues with the same valuation) of finite order, and suppose that γ has two K-valued fixed points u, v ∈ ℙ^{1}(K). We set

M(γ) =]u, v[,

and call it the mirror of γ. It is clear that for elliptic elements γ and θ of finite order the equality M(γ) = M(θ) holds if and only if ⟨γ, θ⟩ is a cyclic group.

The element γ acts trivially on M(γ), i.e., M(γ) is contained in the fixed locus of γ (= the subtree in 𝒯_{K}^{BT} generated by vertices fixed by γ); but they do not coincide with each other in general:

2.4. Lemma. Let n be the order of γ, and set G = ⟨γ⟩.

(1) Let Λ_{0} ∈ M(γ). If (n, p) = 1, then G acts freely on the q −1 vertices adjacent to Λ_{0} not lying on M(γ).

(2) Suppose n = p^{r} for r ≥ 1, and set s = ν(ζ_{p^{r}} −1), where ζ_{p^{r}} is a primitive p^{r}-th root of unity. Then a vertex Λ ∈ 𝒯_{K}^{BT} is fixed by G if and only if 0 ≤ d(Λ, M(γ)) ≤ s.

Proof. We may assume that γ: z ↦ ζ_{n}z, where z is the inhomogeneous coordinate. (1) follows from the fact that the adjacent vertices are in canonical one-to-one correspondence with points in ℙ^{1}(k). (2) is due to an easy calculation collaborated with the straightforward lemma below. □

2.5. Lemma. Set Λ_{0} = [O_{K}X_{0} + O_{K}X_{1}] and Λ_{1} = [O_{K}(X_{0} + u_{0}X_{1}) + O_{K}πX_{1}] with u_{0} ≢ 0 mod π. Let Λ be a vertex such that the path [Λ_{0}, Λ] contains Λ_{1}. Then we have Λ = [O_{K}(X_{0} + (∑_{i=0}^{d−1} u_{i}π^{i})X_{1}) + O_{K}π^{d}X_{1}], where d = d(Λ, Λ_{0}). □

2.6. Definition. Let Γ be a finitely generated discrete subgroup of PGL(2, K). Let A be an apartment in 𝒯_{Γ}^{∗}, and Λ ∈ A a vertex. Then we say γ ∈ Γ folds A at Λ if γ · Λ = Λ and γ maps one of the edge in A emanating from Λ to the other such edge. The same terminology will be applied to a half-line ℓ and a vertex Λ ∈ ℓ which does not sit on the end point of ℓ.

Observe that, if A is mapped to a half-line (resp. a segment of finite length) by the quotient map 𝒯_{Γ}^{∗} → Γ\𝒯_{Γ}^{∗} , then A is folded at at least one (resp. two) vertices.

2.7. Terminology. In the sequel the expression order-n mirror means the apartment A which is the mirror of an element of order n, but not that of an element of order greater than n. Clearly the number of order-n mirrors in 𝒯_{G}^{∗} is equal to the number of maximal cyclic subgroups in G of order n.

2.8. Cyclic case: G ≅ Z_{n}. For any residue characteristic p the tree 𝒯_{G}^{∗} consists of only one apartment which is the mirror M(θ) of any element θ ≠ 1 in G. Since G acts on 𝒯_{G}^{∗} trivially, the quotient tree T_{G}^{∗} also consists of one straight-line whose ends corresponds to the two points above which ℙ_{K}^{1} → G\ℙ_{K}^{1} ramifies.

2.9. Dihedral case: G ≅ D_{m}. Let N = ⟨θ⟩ be the normal subgroup of G of index 2, and H_{0} = ⟨φ⟩ a subgroup of order 2. Then G = ⟨θ, φ | φθφ = θ^{−1}⟩; G has exactly m subgroups of order 2 given by H_{i} = ⟨θ^{i}φ⟩ for 0 ≤ i ≤ m − 1. The tree 𝒯_{G}^{∗} therefore contains one order-m mirror (corresponding to N) and m order-2 mirrors (corresponding to H_{i}’s). If m is odd, then all H_{i}’s are conjugated among others, while they fall into two conjugacy classes if m is even. The element φ of order 2 folds the apartment M(θ) to a half-line. If m is even, then θ^{m/2} folds each M(θ^{i}φ) to a half-line.

(2.9.1) p ≠ 2: All the mirrors M(θ) and M(θ^{i}φ) (0 ≤ i ≤ m − 1) intersect among others at precisely one vertex Λ^{∘} which is the unique vertex stabilized by G. We therefore have 𝒯_{G}^{∗} = M(θ) ⋃ ( ⋃_{i=0}^{m−1} M(θ^{i}φ)), and hence the quotient tree T_{G}^{∗} simply consists of three half-lines starting at one point.

(2.9.2) p = 2: All the mirrors M(θ) and M(θ^{i}φ) (0 ≤ i ≤ m − 1) are disjoint from among others. The tree 𝒯_{G}^{∗} is the union of all these mirrors and paths [Λ^{∘}, Λ_{i}] of length e connecting them, where Λ^{∘} ∈ M(θ) and Λ_{i} ∈ M(θ^{i}φ). Each path [Λ^{∘}, Λ_{i}] is pointwise fixed by the subgroup θ^{−i}Hθ^{i}. If m is even, it is further pointwise fixed by the subgroup ⟨θ^{m/2}, θ^{i}φ⟩ ≅ D_{2}.

The proofs of (2.9.1) and (2.9.2) are straighforward: We may assume that θ(z) = ζ_{m}z and φ(z) = z^{−1}, where z is an inhomogeneous coordinate of ℙ_{K}^{1}. Then apply Lemma 2.2 and Lemma 2.4. The quotient tree T_{G}^{∗} in p = 2 looks like as in Figure 2, where the arrow indicates the ends. The groups beside vertices (resp. edges, ends) indicate the stabilizers (⊆ G) of the them.

[figure: D_{m}; Z_{2}; Z_{2}; Z_{m}; Z_{2}; Z_{2}; p = 2, m: odd; Z_{2}; Z_{2}; Z_{m}; D_{2}; D_{2}; D_{2}; D_{2}; D_{2}; D_{2}; D_{m}; p = 2, m: even]
Figure 2: T_{G}^{∗} for G ≅ D_{m} in p = 2

The assertions in the following paragraphs can be proved in the same spirit: We can compute the arrangement of apartments by Lemma 2.2 using the combinatorial data presented in Appendix. The statements about fixed loci and fixers are straighforward by Lemma 2.4.

2.10. Tetrahedral case: G ≅ A_{4}. The subtree 𝒯_{G}^{∗} contains three order-2 mirrors and four order-3 mirrors. The order-2 mirrors are folded to half-lines, while the order-3 mirrors are not.

(2.10.1) p ≠ 2, 3: All the 3 + 4 mirrors intersect among others at precisely one vertex Λ^{∘}. Hence the tree 𝒯_{G}^{∗} is simply the union of all these mirrors.

(2.10.2) p = 3: The three order-2 mirrors intersect at one vertex Λ^{∘}, while the order-3 mirrors are disjoint among others and do not intersect the order-2 mirrors. Let Λ be the vertex nearest to Λ^{∘} in one of the order-3 mirrors. Then d(Λ^{∘}, Λ) = e/2, and [Λ^{∘}, Λ] is pointwise fixed by a subgroup isomorphic to Z_{3}. The tree 𝒯_{G}^{∗} is the union of all mirrors and the four paths (of the form [Λ^{∘}, Λ] as above) of length e/2.

(2.10.3) p = 2: The order-3 mirrors have a path σ_{0} of length e in common. Let Λ^{∘} be the middle point of σ_{0}. The order-2 mirrors are disjoint among others and do not intersect the order-3 mirrors. Let Λ be the vertex nearest to Λ^{∘} in one of the order-2 mirrors. Then d(Λ^{∘}, Λ) = e/2, and [Λ^{∘}, Λ] is pointwise fixed by a subgroup isomorphic to D_{2}. The path σ_{0} is fixed by the whole G. The tree 𝒯_{G}^{∗} is the union of all mirrors and the three paths (of the form [Λ^{∘}, Λ] as above) of length e/2.

[figure: A_{4}; Z_{3}; Z_{3}; Z_{2}; Z_{3}; Z_{3}; p = 3; Z_{3}; Z_{3}; Z_{2}; A_{4}; A_{4}; A_{4}; A_{4}; A_{4}; D_{2}; D_{2}; p = 2]
Figure 3: T_{G}^{∗} for G ≅ A_{4}

2.11. Octahedral case: G ≅ S_{4}. The tree 𝒯_{G}^{∗} contains six order-2 mirrors, four order-3 mirrors, and three order-4 mirrors. All these mirrors are folded to half-lines.

(2.11.1) p ≠ 2, 3: All the 6 + 4 + 3 mirrors intersect among others at precisely one vertex Λ^{∘}. Hence the tree 𝒯_{G}^{∗} is simply the union of all these mirrors.

(2.11.2) p = 3: The order-4 mirrors intersect among others at a unique vertex Λ^{∘}. The order-2 mirrors also pass through Λ^{∘}; there are four paths σ_{i} (i = 0, 1, 2, 3) of length e of form σ_{i} = [Λ^{∘}, Λ_{i}] such that σ_{i} ⋂ σ_{j} = Λ^{∘} for i ≠ j, and that each order-2 mirror contains precisely two of σ_{i}’s. The order-3 mirrors are disjoint among others, and each of them intersects one of σ_{i}’s at exactly the middle point. Hence the tree 𝒯_{G}^{∗} is the union of all these mirrors. Each σ_{i} is pointwise fixed by a subgroup isomorphic to D_{3}.

(2.11.3) p = 2: The order-3 mirrors intersect among others along a path σ_{0} of length e. Let Λ^{∘} be the middle point of σ_{0}. The order-4 mirrors are disjoint among others and do not intersect order-3 mirrors. Let Λ be the vertex nearest to Λ^{∘} in one of the order-4 mirrors. We have d(Λ^{∘}, Λ) = e/2. Let Λ_{1} be the middle point of [Λ^{∘}, Λ]. Then there are two order-2 mirrors to which the distance from Λ_{1} is 3e/4. The paths from Λ_{1} to these order-2 mirrors share a path [Λ_{1}, Λ_{2}] of length e/4. Moreover:

(a) The path σ_{0} is pointwise fixed by the normal subgroup ≅ A_{4}.

(b) The paths [Λ^{∘}, Λ] and [Λ_{1}, Λ_{2}] are pointwise fixed by a subgroup ≅ D_{4}.

(c) Each of the geodesic paths of length e/2 from Λ_{2} to those order-2 mirrors is pointwise fixed by a subgroup ≅ D_{2}.

[figure: S_{4}; D_{3}; D_{3}; D_{3}; D_{3}; Z_{4}; Z_{3}; Z_{2}; p = 3; S_{4}; D_{4}; D_{4}; D_{4}; D_{4}; D_{2}; D_{2}; D_{2}; D_{2}; D_{4}; D_{4}; A_{4}; A_{4}; A_{4}; Z_{3}; Z_{4}; Z_{2}; p = 2]
Figure 4: T_{G}^{∗} for G ≅ S_{4}

2.12. Icosahedral case: G ≅ A_{5}. In this case, the tree 𝒯_{G}^{∗} contains fifteen order-2 mirrors, ten order-3 mirrors, and six order-5 mirrors. All these mirrors are folded to half-lines.

(2.12.1) p ≠ 2, 3, 5: All the 15+10+6 mirrors intersect among others at precisely one vertex Λ^{∘}. Hence the tree 𝒯_{G}^{∗} is simply the union of all these mirrors.

(2.11.2) p = 5: The order-3 mirrors intersect among others at exactly one vertex Λ^{∘}. The order-2 mirrors also pass through Λ^{∘}; there are six paths σ_{i} (i = 0, 1, 2, 3, 4, 5) of length e/2 of form σ_{i} = [Λ^{∘}, Λ_{i}] such that σ_{i} ⋂ σ_{j} = Λ^{∘} for i ≠ j, and that each order-2 mirror contains two of σ_{i}’s. The order-5 mirrors are disjoint among others, and each of them intersects one of σ_{i}’s at exactly the middle point. Each σ_{i} is pointwise fixed by a subgroup isomorphic to D_{5}.

(2.11.3) p = 3: The order-5 mirrors intersect among others at exactly one vertex Λ^{∘}. The order-2 mirrors also pass through Λ^{∘}; there are ten paths σ_{i} = [Λ^{∘}, Λ_{i}] (i = 0, …, 9) of length e such that σ_{i} ⋂ σ_{j} = Λ^{∘} for i ≠ j, and that each order-2 mirror contains two of σ_{i}’s. The order-3 mirrors are disjoint among others, and each of them intersects one of σ_{i}’s at exactly the middle point. Each σ_{i} is pointwise fixed by a subgroup isomorphic to D_{3}.

(2.11.4) p = 2: The order-5 mirrors intersect among others at exactly one vertex Λ^{∘}. The order-3 mirrors also pass through Λ^{∘}; there are five paths σ_{i} = [Λ^{∘}, Λ_{i}] (i = 0, 1, 2, 3, 4) of length e such that σ_{i} ⋂ σ_{j} = Λ^{∘} for i ≠ j, and that each order-3 mirror contains two of σ_{i}’s. Let Λ′_{i} be the middle point of σ_{i}. Then there are precisely three order-2 mirrors to which the distance from Λ′_{i} is e/2. The geodesic paths from Λ′_{i} to these order-2 mirrors, which are pointwise fixed by a subgroup iromorphic to D_{2}, intersect among others at exactly Λ′_{i}. Moreover, each path σ_{i} is pointwise fixed by a subgroup isomorphic to A_{4}.

[figure: A_{5}; D_{5}; D_{5}; D_{5}; D_{5}; Z_{3}; Z_{5}; Z_{2}; p = 5; A_{5}; D_{3}; D_{3}; D_{3}; D_{3}; Z_{5}; Z_{3}; Z_{2}; p = 3; Z_{5}; Z_{3}; Z_{2}; A_{5}; A_{4}; A_{4}; A_{4}; A_{4}; D_{2}; D_{2}; p = 2]
Figure 5: T_{G}^{∗} for G ≅ A_{5}

3. Proof of the theorem

3.1. In the rest of this paper we devote ourselves to prove the main theorem. We first recall the following well-known facts: Let γ be an element in PGL(2, K). Then γ is said to be parabolic (resp. elliptic, resp. hyperbolic) if it has only one eigenvalue (resp. two distinct eigenvalues with equal valuation, resp. two distinct eigenvalues with different valuations). Let Γ be a discrete subgroup of PGL(2, K). Then:

(3.1.1) There exists no parabolic element other than 1.

(3.1.2) An element γ ∈ Γ is of finite order if and only if it is elliptic.

Suppose two elements θ and χ have exactly one common fixed point ∞ ∈ ℙ^{1}(K). We may assume θ = \binom{u 0}{0 u^{−1}} and χ = \binom{a b}{0 a^{−1}} with b ≠ 0. Then it is easy to see that θχθ^{−1}χ^{−1} is a parabolic element. Hence:

(3.1.3) No two elements in Γ have exactly one common fixed point in ℙ_{K}^{1}.

In particular, for elliptic elements γ, θ ∈ Γ, M(γ) ⋂ M(θ) contains a half-line if and only if ⟨γ, θ⟩ is a cyclic group (cf. Definition 2.3).

Let Γ ⊂ PGL(2, K) be a finitely generated discrete subgroup. Replacing K by a finite extension if necessary, we may assume that every element (≠ 1) in Γ has at most K-valued fixed points in ℙ_{K}^{1} (cf. [GvP80, I.3.1 (1)]). Then Γ acts on the Bruhat-Tits building 𝒯_{K}^{BT} without inversion. We assume that the quotient graph T^{∗} is a tree, which is certainly the case if Γ is a p-adic triangle group of Mumford type. Let ϱ_{Γ}: 𝒯_{Γ}^{∗} → T_{Γ}^{∗} be the quotient map of trees, and take a lifting ι_{Γ}: T_{Γ}^{∗} ↪ 𝒯_{Γ}^{∗} ([Ser80, I.4.1, Prop. 17]); i.e., a section of ϱ_{Γ}. We associate to any vertex Λ ∈ T_{Γ}^{∗} (resp. edge σ ⊆ T_{Γ}^{∗}) the stabilizer of ι_{Γ}(Λ) (resp. ι_{Γ}(σ)) in Γ, denoted by Γ_{Λ} (resp. Γ_{σ}), to make up the tree of groups (T_{Γ}^{∗}, Γ_{•}) ([Ser80, I.4.4, Def. 8]). Note that stabilizers of vertices and edges are finite subgroups since Γ is discrete. Since 𝒯_{Γ}^{∗} is a tree, we have an isomorphism

Γ ⟶^{∼} lim_{⟶}(T_{Γ}^{∗}, Γ_{•})

([Ser80, I.4.5, Thm. 10]), where the right-hand side is the associated amalgam product of the groups Γ_{•}.

3.2. Next let us recall the following facts: Let Γ be as above. Since T_{Γ}^{BT} = Γ\𝒯_{K}^{BT} is also a tree, we can take a lifting ι: T_{Γ}^{BT} → 𝒯_{K}^{BT} and do the same as above to get the associated tree of groups (T_{Γ}^{BT}, Γ_{•}). Let Λ be a vertex in T_{Γ}^{BT} with the stabilizer G = Γ_{Λ}. If Λ is the similarity class of a lattice M ⊂ K^{2}, then G acts on Proj Sym_{k}(M/πM) ≅ ℙ_{k}^{1}; i.e., we have the representation (not necessarily injective) G → PGL(2, k).

(3.2.1) We have a canonical bijection equivariant with the action of G,

{Edges σ in 𝒯_{K}^{BT} emanating from ι(Λ) } ⟷ ℙ^{1}(k).

(3.2.2) Consider the quotient map ℙ_{k}^{1} → G\ℙ_{k}^{1}. Let H ⊆ G be a subgroup. Then we have a canonical bijection,

{Edges σ in T_{Γ}^{BT} emanating from Λ with the stabilizer Γ_{σ} a conjugate of H in Γ} ⟷ {Points of G\ℙ^{1}(k) above which points in ℙ^{1}(k) are fixed by a conjugate of H} .

3.3. Notation. In the sequel the notation Λ ⊢ σ for Λ a vertex and σ an edge in a tree means that the edge σ emanates from Λ.

3.4. Definition. Let T ⊆ 𝒯_{Γ}^{∗} be a subtree, and Λ ∈ T a vertex. A half-line ℓ ⊂ 𝒯_{Γ}^{∗} starting at Λ is said to be perpendicular to T if for any edge σ ∈ T with Λ ⊢ σ and for the edge ς ∈ ℓ with Λ ⊢ ς we have ϱ_{Γ}(σ) ≠ ϱ_{Γ}(ς).

Since σ ⊣ Λ ⊢ ς, the condition ϱ_{Γ}(σ) ≠ ϱ_{Γ}(ς) is equivalent to: There exists no η ∈ Γ_{Λ} such that σ = η · ς.

Now we quote here a few necessary facts which are proved in [CKK99]:

3.5. Proposition ([CKK99], lemma 3.4). Let Λ ∈ 𝒯_{Γ}^{∗} be a vertex and σ ⊂ 𝒯_{Γ}^{∗} an edge emanating from Λ. Suppose there exists an element γ ∈ Γ_{Λ} such that |⟨γ⟩ · σ| > 1 (i.e. σ is not fixed by all powers of γ). Then there exists χ ∈ Γ_{Λ} (χ ≠ 1) such that any half-line ℓ starting at Λ converging to an end of M(χ) is perpendicular to ⟨γ⟩ · σ. □

3.6. Proposition ([CKK99], lemma 3.5). Let 𝒯 ⊆ 𝒯_{Γ}^{∗} be a subtree such that

(3.6.1)  {Ends of T} = {Ends of T_{Γ}^{∗}} ,

where T = ϱ_{Γ}(𝒯). Then every half-line in 𝒯_{Γ}^{∗} starting from a vertex of 𝒯 is not perpendicular to 𝒯. □

3.7. Definition. Let (T, G_{•}) be a tree of groups, and T′ ⊆ T a subtree. Then the induced tree of groups (T′, G_{•}) is said to be a contraction of (T, G_{•}) if (i) {Ends of T′} = {Ends of T} and (ii) for every vertex Λ in T − T′ the stabilizers of vertices on the path from Λ to Λ′ are ordered increasingly with respect to inclusion upon approaching T′, where Λ′ is the vertex in T′ nearest to Λ.

It is clear that if (T′, G_{•}) is a contraction of (T, G_{•}) then we have a canonical isomorphism

(3.7.1)  lim_{⟶}(T′, G_{•}) ⟶^{∼} lim_{⟶}(T, G_{•}).

3.8. Proposition ([CKK99], proposition 1). Let T ⊆ T_{Γ}^{∗} be a subtree enjoying the condition (3.6.1). Then (T, Γ_{•}) is a contraction of (T_{Γ}^{∗}, Γ_{•}). □

3.9. In what follows the subgroup Γ is assumed to be a non-spherical p-adic triangle group of Mumford type of index (e_{0}, e_{1}, e_{∞}). By (1.3.2) the quotient graph T_{Γ}^{∗} is a tree with three ends, corresponding to 0, 1 and ∞, and the stabilizer of the end corresponding to 0 (resp. 1, ∞) is a cyclic subgroup of order e_{0} (resp. e_{1}, e_{∞}).

3.10. Lemma. Under the hypothesis above there exists a finite non-cyclic subgroup G in Γ.

Proof. Otherwise, the group Γ_{Λ} is a finite cyclic group for any Λ ∈ T_{Γ}^{∗}. Then the mirror of any element (≠ 1) of any Γ_{Λ} is not folded, and is mapped to a straightline (isometric to ℝ). Hence, T_{Γ}^{∗} would have even number of ends. □

3.11. We choose a non-cyclic finite subgroup G ⊆ Γ which is maximal with respect to the following ordering: D_{2m+1} ≺ D_{2n} ≺ A_{4} ≺ S_{4} ≺ A_{5} for any positive integers m and n, and D_{2m+1} ≺ D_{2n+1} and D_{2m} ≺ D_{2n} for m ≤ n. We can actually find a maximal one since elements of finite order in Γ comprise a finite number of conjugacy classes ([GvP80, I.§3]). Let us denote by Φ: 𝒯_{G}^{∗} ↪ 𝒯_{Γ}^{∗} the natural inclusion of trees, and by ϕ: T_{G}^{∗} → T_{Γ}^{∗} the induced map. As we have seen in (2.9)∼(2.12) the tree T_{G}^{∗} is the union of three half-lines ℓ_{i} (i = 0, 1, ∞) starting at one vertex v such that ℓ_{i} ⋂ ℓ_{j} = {v} for i ≠ j. Take a section ι_{G}: T_{G}^{∗} ↪ 𝒯_{G}^{∗} of the quotient map ϱ_{G}: 𝒯_{G}^{∗} → T_{G}^{∗}. Set Λ = ι_{G}(v). Clearly we have G_{Λ} ⊆ Γ_{Φ(Λ)}.

3.12. Proposition. (1) The map ϕ induces an injective map,

(3.12.1)  {Edges σ ⊣ v in T_{G}^{∗} } ⟶ {Edges σ ⊣ ϕ(v) in T_{Γ}^{∗} } .

(2) The mirror of any elliptic element in Γ is mapped by ϱ_{Γ} to either a stright-line or a half-line.

(3) The map ϕ induces an isometry between T_{G}^{∗} and ϕ(T_{G}^{∗}). Moreover, the subtree ϕ(T_{G}^{∗}) in T_{Γ}^{∗} with the induced groups is a contraction of (T_{Γ}^{∗}, Γ_{•}).

Proof. The proof goes step by step.

Step 1. First we prove (1) except the case G ≅ A_{4} with p = 3. If G_{Λ} = G, then we have Γ_{Φ(Λ)} = G by our choice of G. In this case, the claim is clear by (3.2.2). Suppose G_{Λ} ≠ G; this occurs if G ≅ A_{5} (p = 2, 3, 5), G ≅ S_{4} (p = 2, 3), G ≅ A_{4} (p = 3), or G ≅ D_{l} (p = 2). If the map (3.12.1) is not injective, then there exists an element χ ∈ Γ_{Φ(Λ)} of order 2 or 3 which permutes the edges in Φ(ι_{G}(T_{G}^{∗})) emanating from Φ(Λ). We clearly have G_{Λ} ⋂⟨χ⟩ = {1} in Γ_{Φ(Λ)}. In case G ≅ D_{2m+1} and p = 2 (G_{Λ} ≅ Z_{2}), Γ_{Φ(Λ)} is dihedral of even degree or non-dihedral; in case G ≅ D_{2n} and p = 2 (G_{Λ} ≅ D_{2}), Γ_{Φ(Λ)} would be non-dihedral; both contradict our choice of G. For the other cases, we consider the local representation ρ: Γ_{Φ(Λ)} → PGL(2, k) around Φ(Λ) as in (3.2). As we see in Figure 3∼5, there exists a non-trivial subgroup H in G_{Λ} which fixes three points in ℙ_{k}^{1}, and hence acts on ℙ_{k}^{1} by the identity. Hence Γ_{Φ(Λ)} is a subgroup such that (a) G_{Λ} ⊂ Γ_{Φ(Λ)}, (b) Γ_{Φ(Λ)} ≺ G and (c) there exists a normal subgroup N ⊂ Γ_{Φ(Λ)} with H ⊆ N and N ⋂⟨χ⟩ = {1}. But, except G ≅ A_{4} with p = 3, we easily find that such a group does not exist, thereby a contradiction. If G ≅ A_{4} and p = 3, then Γ_{Φ(Λ)} ≅ D_{3} would occur; we postpone this case to Step 3 below.

Step 2. Next we prove (2). Let γ ∈ Γ be an elliptic element, and suppose on the contrary that M(γ) is mapped to a segment of finite length. Then it is folded at at least two vertices Λ and Λ′. We can take such vertices so that the path [Λ, Λ′] is mapped by ϱ_{Γ} bijectively onto a path. By Proposition 3.5 we have non-trivial χ ∈ Γ_{Λ} (resp. χ′ ∈ Γ_{Λ′}) such that each half-line starting at Λ (resp. Λ′) and converging to an end of M(χ) (resp. M(χ′)) is perpendicular to M(γ). We have therefore four half-lines; since T_{Γ}^{∗} has only three ends, at least one of them should be folded somewhere. Let [Λ′′, Λ′′′] be the minimal path which connects M(χ) and M(χ′). We define a subtree 𝒯_{0} of 𝒯_{Γ}^{∗} as follows:

— If these mirrors are not folded at Λ′′ or Λ′′′, define 𝒯_{0} to be the union of [Λ′′, Λ′′′], M(χ) and M(χ′).

— If, for instance, M(χ) is folded at Λ′′, then Γ_{Λ′′} is isomorphic either to D_{2m}, S_{4} or A_{5}, since it interchanges two pairs of ends. Then there exists one more mirror; we accordingly define 𝒯_{0} to be the union of [Λ′′, Λ′′′] and all such mirrors.

In any case the tree ϱ_{Γ}(𝒯_{0}) has at least one tip (= a vertex from which only one edge emanates); this is clear in the first case, and is easy in the second case by what we have proved in Step 1. The folding above the tip of ϱ_{Γ}(𝒯_{0}) yields another pair of half-lines perpendicular to 𝒯_{0}. Repeating this procedure, we can define inductively a sequences {𝒯_{n}}_{n=0}^{∞} of subtrees of 𝒯_{Γ}^{∗} with 𝒯_{i} ⊂ 𝒯_{i+1} such that

(a) the number of tips of ϱ_{Γ}(𝒯_{i+1}) is twice as large as that of ϱ_{Γ}(𝒯_{i}),

(b) any tips of ϱ_{Γ}(𝒯_{i+1}) is not in ϱ_{Γ}(𝒯_{i}). But this immediately yields a contradiction, since T = ⋃_{n=0}^{∞} ϱ_{Γ}(𝒯_{n}) is a subtree of T_{Γ}^{∗} with infinitely many ends. We therefore have proved (2).

Step 3. Suppose G ≅ A_{4} and p = 3 (G_{Λ} ≅ Z_{3}). If the map (3.12.1) is not injective, then Γ_{Φ(Λ)} would be D_{3}, as we already deduced in Step 1. Let χ be as in Step 1; it is easy to see that χ is of order 2. We have three mirrors (two order-2 and one order-3) which are distinct in 𝒯_{Γ}^{∗} up to the action of Γ. The order-2 mirror other than M(χ) is folded at the vertex fixed by G (see Figure 3). Since T_{Γ}^{∗} has only three ends, the mirror M(χ) also should be folded, but not at Φ(Λ) (because of (2.9)). Because the folding yields another half-line, we have at least four half-lines which is mapped by ϱ_{Γ} to pairwise distinct half-lines (by (2)), which is a contradiction. Hence the proof of (1) is finished.

Step 4. Finally we prove (3). It suffices by Proposition 3.8 to show that the map ϕ induces the bijection between ends of T_{G}^{∗} and of T_{Γ}^{∗}. If not, at least one half-line in Φ(ι_{G}(T_{G}^{∗})) is folded; by (1) these foldings do not occur at Φ(Λ). Then, by an argument similar to that in Step 2, we can find a sequences {𝒯_{n}}_{n=0}^{∞} of subtrees of 𝒯_{Γ}^{∗} satisfying (a) and (b), which is a contradiction. Now the proof of (3), and hence of the proposition is finished. □

3.13. By Proposition 3.12 we particularly deduce the following: For any vertex Λ ∈ 𝒯_{G}^{∗} we have G_{Λ} ⊂ Γ_{Φ(Λ)}, Γ_{Φ(Λ)} ≺ G, and that there exists no cyclic subgroup of Γ_{Φ(Λ)} of which every conjugate has only the trivial intersection with G_{Λ} (otherwise, 𝒯_{Γ}^{∗} has a half-line which is perpendicular to 𝒯_{G}^{∗}, and contradicts Proposition 3.6). We therefore find that

(a) Γ_{Φ(Λ)} = G_{Λ} if G_{Λ} = G,

(b) Γ_{Φ(Λ)} = G_{Λ} if G_{Λ} ≅ A_{4}, S_{4}, A_{5},

(c) Γ_{Φ(Λ)} ≅ D_{ml} for l ≥ 1 if G_{Λ} ≅ D_{m},

(d) Γ_{Φ(Λ)} ≅ Z_{ml} for l ≥ 1 if G_{Λ} ≅ Z_{m}.

In particular, for any θ ∈ G (θ ≠ 1) Φ(M(θ)) is the mirror of an element γ ∈ Γ such that γ^{l} = θ for some l ≥ 1 (by (3.1.3)). Let θ_{0}, θ_{1} and θ_{∞} be generators of the stabilizers of ends of 𝒯_{G}^{∗} which are mapped by ϱ_{G} to the ends different among others.

For each i = 0, 1, ∞ let γ_{i} be a generator of an end of ϕ(M(θ_{i})); by Proposition 3.12 (3) and (1.3.3) Γ is generated by γ_{0}, γ_{1} and γ_{∞}, and moreover, changing the indices if necessary, we may assume e_{i} = the order of γ_{i}. We also deduce that there exists positive integers l_{0}, l_{1} and l_{∞} such that γ_{i}^{l_{i}} = θ_{i}. Here l_{i} (for each i = 0, 1, ∞) is allowed to be > 1 only if the mirror M(θ_{i}) does not pass through a vertex Λ in 𝒯_{G}^{∗} with G_{Λ} = G, since Γ_{Φ(Λ)} = G_{Λ}. Hence:

3.14. Corollary. p ≤ 5.

Proof. As we saw in (2.9)∼(2.12) all the mirrors M(θ_{i}) pass through the unique vertex Λ^{∘} which is fixed by G if p > 5. □

3.15. Now the rest of the proof of “only if” part of Theorem (1) goes as follows: Suppose G ≅ A_{5}. Then we have three possibilities:

(1) If p = 5, then only the order-5 mirrors are separated from the vertex fixed by G (cf. (2.11.2)). By Lemma 2.4 we find that we can replace the index 5 by 5l (l ≥ 2) keeping the rules (a)∼(d). Hence we have (e_{0}, e_{1}, e_{∞}) = (2, 3, 5l) for l ≥ 2.

(2) If p = 3, then only the order-3 mirrors are separated from the vertex fixed by G (cf. (2.11.3)). By Lemma 2.4 we find that we can replace the index 3 by 3l (l ≥ 2) keeping the rules (a)∼(d). Hence we have (e_{0}, e_{1}, e_{∞}) = (2, 5, 3l) for l ≥ 2.

(3) If p = 2, then only the order-2 mirrors are separated from the vertex fixed by G (cf. (2.11.4)). We have (e_{0}, e_{1}, e_{∞}) = (3, 5, 2l) (l ≥ 2). But, if l is even, then the vertices Λ′_{i} (in the notation as in (2.11.4)) are fixed by the group generated by a subgroup H ≅ A_{4} and a subgroup Z ≅ Z_{4} with H ⋂ Z ≅ Z_{2} by Lemma 2.4. But this contradicts (b) above. Hence we have (e_{0}, e_{1}, e_{∞}) = (3, 5, 2l) for l ≥ 3 and l: odd.

All the rest cases are similar except the following two cases:

(3.15.1) Suppose G ≅ S_{4} and p = 2. Then by (2.11.3) order-4 mirrors and order-2 are separated from the vertex fixed by S_{4}. Hence we have (e_{0}, e_{1}, e_{∞}) = (3, 2l, 4m). But, if l is even, then the vertex Λ_{2} (in the notation as in (2.11.3)) is fixed by the group generated by a subgroup H ≅ D_{4} and a subgroup Z ≅ Z_{4} with H ⋂ Z ≅ Z_{2} due to Lemma 2.4. As we see immediately this group must be isomorphic to S_{4}. But this contradicts (c) above. Hence l must be odd.

(3.15.2) Suppose G ≅ D_{2n} and p = 2. Similarly as above, we get (e_{0}, e_{1}, e_{∞}) = (2l, 2m, 2n). But, if two of l, m, n are even, then by Lemma 2.4 the vertex Λ^{∘} (cf. (2.9.2)) is fixed by a group isomorphic to S_{4}, which is a contradiction. Hence we deduce at least two of l, m, n are odd.

Now the “only if” part of Theorem (1) is proved. The above argument also shows that by (3.7.1), Lemma 2.4 and Proposition 3.8 the subgroup Γ is (if it exists) isomorphic to the amalgam product given in Table 1. Moreover, since the conjugacy class of G in PGL(2, K) is unique (cf. [Web99, Vol. II, §67]), it follows that the conjugacy class of Γ is also unique.

Hence it remains only to show the “if” part of Theorem (1). This part involves a criterion of discreteness:

3.16. Lemma. Let Γ be a subgroup in PGL(2, K) which acts on a subtree 𝒯 of 𝒯_{K}. Suppose that the stabilizer of at least one vertex is finite. Then Γ is discrete. (Hence, in particular, the stabilizer of any vertex is finite.)

Proof. If {γ_{i}} ⊂ Γ is a sequence converging to 1, then, except finitely many γ_{i}’s, they are contained in the stabilizer of Λ, since the stabilizer of a vertex in PGL(2, K) is an open neighborhood of 1. □

3.17. The rest of the proof goes as follows: For each of the above cases as in Table 1 we choose three elements γ_{i} ∈ PGL(2, K) (i = 0, 1, 2) having K-valued fixed points such that at least two of M(γ_{i})’s are disjoint. Then we construct a tree 𝒯 as follows: Let T be the smallest connected subtree of 𝒯_{K} which contains all M(γ_{i}). Let Γ be the subgroup in PGL(2, K) generated by γ_{i}’s. Then 𝒯 is defined to be the union of all γ · T for γ ∈ Γ (it is easy to see that 𝒯 is connected (cf. [Ser80, I.4.5 Theorem 9])). Clearly Γ acts on Σ. Note that our construction of the tree 𝒯 is the same as that in [Ser80, I.4.5 Theorem 9]; the only difference is that the tree in [loc. cit.] are abstract one, while we constructed it inside 𝒯_{K}. If Γ and 𝒯 satisfies the hypothesis of Lemma 3.16, and if a fundamental domain (in T) has exactly three half-lines with groups of expected order, we have done.

The constructions of Γ with given (e_{0}, e_{1}, e_{∞}) of all cases are quite similar to among others; here we only indicate the construction in (2, 3, 5l)-case. This is the case in p = 5. Let G be a finite subgroup in PGL(2, K) isomorphic to A_{5}. It has three generators γ_{0}, γ_{1}, and θ_{∞} of order 3, 2, and 5, respectively. Let γ_{∞} be an element in PGL(2, K) of order 5l (l > 1) such that γ_{∞}^{l} = θ_{∞}. Then M(γ_{1}) and M(γ_{∞}) intersects at exactly one vertex Λ_{1}, and M(γ_{0}) is disjoint from M(γ_{∞}) (cf. (2.11.2)). Hence T as above is the union of these three mirrors. Moreover, we find by Lemma 2.4 that the subgroup Γ generated by γ_{0}, γ_{1} and γ_{∞} is isomorphic to A_{5} ∗_{D_{5}} D_{5l} as desired. Since the stabilizer of Λ_{0} is G (cf. [Ser80, I.4.5 Theorem 9]), Γ is discrete. Now by (2.11) those three mirrors are folded to three half-lines; the order of the groups attached to them are exactly (3, 2, 5l) as desired.

Now all the statements in the theorem have been proved.

A. Appendix: Combinatorial data

A.1. This appendix gives the tables of combinatorial data by which one can see how the mirrors of elements in finite subgroup in PGL(2, K) are arranged in 𝒯_{K}^{BT} so that one can draw the picture of 𝒯_{G}^{∗} (cf. §2). The basic principle is as follows: Suppose G is a finite subgroup in PGL(2, K) and γ, θ ∈ PGL(2, K) (γ ≠ 1, θ ≠ 1). We assume that fixed points a, b (resp. c, d) of γ (resp. θ) are in ℙ^{1}(K). Then the mirror of γ (resp. θ) is given by ]a, b[ (resp. ]c, d[). As we saw in Lemma 2.2 the correlation between M(g) and M(h) can be calculated by the cross-ratios R(a, b; c, d) and R(b, a; c, d). If we have the complete list of these values for every pair of generators of maximal cyclic subgroups in G, we therefore can perfectly describe 𝒯_{G}^{∗}.

A.2. Convention. For the subgroups G ≅ A_{4}, S_{4}, A_{5}, which are given in the standard forms as presented below, we will give a complete list of maximal cyclic subgroups by choosing generators. To two of these generators, say γ and θ, we associate an expression P (γ, θ) = P (θ, γ), which is either a number or an unordered pair of numbers. The meaning of P (γ, θ) is:

— If it is simply a number, then both |ν(R(a, b; c, d))| and |ν(R(b, a; c, d))| are equal to |ν(P (γ, θ))|.

— If it is a pair of numbers, say P (γ, θ) = {s, t}, then, as a set, {|ν(s)|, |ν(t)|} coincides with {|ν(R(a, b; c, d))|, |ν(R(b, a; c, d))|}.

A.3. Here, for the careful reader, we briefly mention how to calculate P (γ, θ): First, calculate the algebraic numbers R = R(a, b; c, d) and S = R(b, a; c, d) (faster by computer), which are sometimes fairly complicated. Then, look at R · S and R + S. If they are both p-adic integers, then so are R and S; if not, replace R and S by the inverse of them. Then, in our case, R · S and R + S will turn out to be p-adic integers. Now the two numbers R · S and R + S are always very simple, and can be easily valuated, from which the valuations of R and S can be easily calculated (actually, only the following two cases occur; either ν(R · S) = ν(R + S) or ν(R + S) = 0; see Figure 1 in §2).

Drawing 𝒯_{G}^{∗} in a picture is a tedious work; but as a hint for the interested reader we give it for the case G ≅ A_{5} and p = 2 in (A.6) below.

A.4. Case G ≅ A_{4} or S_{4}. Since A_{4} is a subgroup of S_{4}, the calculation can be mixed up. Let G be a subgroup isomorphic to S_{4}. By [Web99, §73] we may assume that G is generated by θ and χ with

θ = [ \begin{matrix}\sqrt{i} & 0 \\ 0 & \frac{1}{\sqrt{i}} \end{matrix} ] , χ = [ −\frac{1+i}{2} \frac{1+i}{2}^{\frac{1−i}{2} \frac{1−i}{2}} ] ,

where i denotes a primitive 4-th root of unity. We set ω = χθχθ^{2}. The group G has three cyclic groups of order 4 generated by each of θ, χ^{2}θ^{3}, χωθ^{2}, four cyclic groups of order 3 generated by each of χ, θχθ^{3}, θ^{2}χθ^{2}, θ^{3}χθ, and six cyclic groups of order 2, not coming from those of order 4, generated by each of ω, ωθ^{2}, ωχ, χ^{2}θ, ωχ^{2}, θχ^{2}. Table 2 presents the P (g, h) for any distinct two of these elements.

The group G has a subgroup isomorphic to A_{4} generated by θ^{2} and χ. It has four cyclic groups of order 3, ⟨χ⟩, ⟨θ^{2}χθ^{2}⟩, ⟨θχθ^{3}⟩, and ⟨θ^{3}χθ⟩, and three cyclic groups of order 2, ⟨θ^{2}⟩, ⟨χθ^{2}χ^{2}⟩, and ⟨χ^{2}θ^{2}χ⟩.

Table 2: P (g, h) for g and h generators of cyclic subgroups in G ≅ S_{4}. Here ♡ = {3, 6} and ♢ = {4, 12}.

χ^{2}θ^{3} χωθ^{2} χ θχθ^{3} θ^{2}χθ^{2} θ^{3}χθ ω ωθ^{2} ωχ χ^{2}θ ωχ^{2} θχ^{2}
θ  2  2  \sqrt{6} \sqrt{6}  \sqrt{6}  \sqrt{6}  2  2 2\sqrt{2} 2\sqrt{2} 2\sqrt{2} 2\sqrt{2}
χθχ^{2}  2  \sqrt{6} \sqrt{6}  \sqrt{6}  \sqrt{6} 2\sqrt{2} 2\sqrt{2} 2  2 2\sqrt{2} 2\sqrt{2}
χ^{2}θχ  \sqrt{6} \sqrt{6}  \sqrt{6}  \sqrt{6} 2\sqrt{2} 2\sqrt{2} 2\sqrt{2} 2\sqrt{2} 2  2
χ  ♡  ♡  ♡  2 2\sqrt{3} 2 2\sqrt{3} 2 2\sqrt{3}
θχθ^{3}  ♡  ♡ 2\sqrt{3} 2 2\sqrt{3} 2  2 2\sqrt{3}
θ^{2}χθ^{2}  ♡  2 2\sqrt{3} 2\sqrt{3} 2 2\sqrt{3} 2
θ^{3}χθ  2\sqrt{3} 2  2 2\sqrt{3} 2\sqrt{3} 2
ω  2  ♢  ♢  ♢  ♢
ωθ^{2}  ♢  ♢  ♢  ♢
ωχ  2  ♢  ♢
χ^{2}θ  ♢  ♢
ωχ^{2}  2

A.5. Case G ≅ A_{5}. By [loc. cit. §74] G is generated by θ and χ with

θ = [ \begin{matrix}ζ & 0 \\ 0 & 1 \end{matrix} ] , χ = [ \begin{matrix}ζ + ζ^{4} & 1 \\ 1 & −(ζ + ζ^{4}) \end{matrix} ] ,

where ζ is a primitive 5-th root of unity. We set φ = χθ^{−1}χθχθ^{−1}, which sends z ↦ −1/z. The group G has

— six cyclic subgroups of order 5

⟨θ⟩, ⟨θχ⟩, ⟨θχθ^{3}⟩, ⟨θ^{2}χθ^{2}⟩, ⟨θ^{3}χθ⟩, ⟨θ^{4}χ⟩,

— ten cyclic subgroups of order 3

⟨θχθ⟩, ⟨θ^{2}χ⟩, ⟨θ^{3}χ⟩, ⟨θ^{2}χθ⟩, ⟨θ^{3}χθ^{4}⟩,
⟨θχφ⟩, ⟨θ^{4}χφ⟩, ⟨θ^{2}χφθ^{4}⟩, ⟨θ^{2}χφθ^{2}⟩, ⟨θ^{3}χφθ⟩,

— and fifteen cyclic subgroups of order 2

⟨φθ^{i}⟩, ⟨θ^{−i}χθ^{i}⟩, ⟨θ^{−i}χφθ^{i}⟩ (i = 0, 1, 2, 3, 4).

Under these notation we have:

— P (g, h) = \sqrt{5} if g and h are generators of cyclic groups of order 5 (g ≠ h).

— P (g, h) = \sqrt{3} · 5^{1/4} if g (resp. h) is one of the generators of cyclic groups of order 5 (resp. 3).

All the rest are shown in the tables below:

Table 3: P (g, h) for g and h generators of cyclic subgroups in G ≅ A_{5} of order 3. Here ♡ = {3, 6}.

θ^{2}χ θ^{3}χ θ^{2}χθ θ^{3}χθ^{4} θχφ θ^{4}χφ θ^{2}χφθ^{4} θ^{2}χφθ^{2} θ^{3}χφθ
θχθ  ♡ ♡  3  3  ♡  ♡  ♡  3  ♡
θ^{2}χ  3  3  ♡  ♡  ♡  ♡  ♡  3
θ^{3}χ  ♡  3  ♡  ♡  3  ♡  ♡
θ^{2}χθ  ♡  3  ♡  ♡  ♡  ♡
θ^{3}χθ^{4}  ♡  3  ♡  ♡  ♡
θχφ  3  3  ♡  ♡
θ^{4}χφ  ♡  ♡  3
θ^{2}χφθ^{4}  3  ♡
θ^{2}χφθ^{2}  3

Table 4: P (g, h) for g (resp. h) a generator of cyclic group subgroups in G ≅ A_{5} of order 5 (resp. 2). Here ∗= 2 · 5^{1/4}.

φ φθ φθ^{2} φθ^{3} φθ^{4} χ θχθ^{4} θ^{2}χθ^{3} θ^{3}χθ^{2}
θ  2 2  2  2  2 ∗  ∗  ∗  ∗
θχ ∗ ∗  ∗  ∗  2 ∗  ∗  2  ∗
θχθ^{3} ∗ ∗  2  ∗  ∗ 2  ∗  ∗  2
θ^{2}χθ^{2} 2 ∗  ∗  ∗  ∗ ∗  2  ∗  ∗
θ^{3}χθ ∗ ∗  ∗  2  ∗ 2  ∗  2  ∗
θ^{4}χ ∗ 2  ∗  ∗  ∗ ∗  2  ∗  2
θ^{4}χθ χφ θχφθ^{4} θ^{2}χφθ^{3} θ^{3}χφθ^{2} θ^{4}χφθ
∗  ∗  ∗  ∗  ∗  ∗  θ
2  2  2  ∗  ∗  ∗  θχ
∗  ∗  2  2  ∗  ∗  θχθ^{3}
2  ∗  ∗  2  2  ∗  θ^{2}χθ^{2}
∗  ∗  ∗  ∗  2  2  θ^{3}χθ
∗  2  ∗  ∗  ∗  2  θ^{4}χ

Table 5: P (g, h) for g (resp. h) a generator of cyclic group subgroups in G ≅ A_{5} of order 3 (resp. 2).

φ  φθ φθ^{2} φθ^{3} φθ^{4} χ θχθ^{4} θ^{2}χθ^{3} θ^{3}χθ^{2}
θχθ  2  \sqrt{6} 2\sqrt{3} 2\sqrt{3} \sqrt{6} 2\sqrt{3} 2\sqrt{3} \sqrt{6}  \sqrt{6}
θ^{2}χ  2\sqrt{3} 2\sqrt{3} \sqrt{6}  2  \sqrt{6} 2\sqrt{3} \sqrt{6}  \sqrt{6}
θ^{3}χ  2\sqrt{3} \sqrt{6}  2  \sqrt{6} 2\sqrt{3} 2\sqrt{3} \sqrt{6}  \sqrt{6}  2\sqrt{3}
θ^{2}χθ  \sqrt{6} 2\sqrt{3} 2\sqrt{3} \sqrt{6}  2  \sqrt{6} \sqrt{6} 2\sqrt{3} 2\sqrt{3}
θ^{3}χθ^{4}  \sqrt{6}  2  \sqrt{6} 2\sqrt{3} 2\sqrt{3} \sqrt{6} 2\sqrt{3} 2\sqrt{3} 2\sqrt{3}
θχφ 2\sqrt{3} \sqrt{6} \sqrt{6} 2\sqrt{3} 2  2  2  \sqrt{6}  2\sqrt{3}
θ^{4}χφ 2\sqrt{3} 2 2\sqrt{3} \sqrt{6} \sqrt{6}  2  \sqrt{6} 2\sqrt{3}  \sqrt{6}
θ^{2}χφθ^{4}  \sqrt{6} 2\sqrt{3} 2 2\sqrt{3} \sqrt{6} \sqrt{6}  2  2  \sqrt{6}
θ^{2}χφθ^{2} 2 2\sqrt{3} \sqrt{6} \sqrt{6} 2\sqrt{3} 2\sqrt{3} \sqrt{6}  2  2
θ^{3}χφθ \sqrt{6} \sqrt{6} 2\sqrt{3} 2 2\sqrt{3} \sqrt{6} 2\sqrt{3} \sqrt{6}  2

θ^{4}χθ χφ θχφθ^{4} θ^{2}χφθ^{3} θ^{3}χφθ^{2} θ^{4}χφθ
2\sqrt{3} 2\sqrt{3}  2  \sqrt{6}  \sqrt{6}  2  θχθ
\sqrt{6}  2  \sqrt{6}  2  \sqrt{6}  \sqrt{6}  θ^{2}χ
2\sqrt{3} 2  \sqrt{6}  \sqrt{6}  2  2\sqrt{3}  θ^{3}χ
2\sqrt{3} \sqrt{6}  \sqrt{6}  2  2\sqrt{3}  2  θ^{2}χθ
\sqrt{6} \sqrt{6}  2  2\sqrt{3}  2  \sqrt{6}  θ^{3}χθ^{4}
\sqrt{6} 2\sqrt{3} 2\sqrt{3}  \sqrt{6}  2\sqrt{3}  \sqrt{6}  θχφ
2 2\sqrt{3} \sqrt{6}  2\sqrt{3}  \sqrt{6}  2\sqrt{3}  θ^{4}χφ
2\sqrt{3} \sqrt{6} 2\sqrt{3}  2\sqrt{3}  \sqrt{6}  2\sqrt{3} θ^{2}χφθ^{4}
\sqrt{6} 2\sqrt{3} \sqrt{6}  2\sqrt{3}  2\sqrt{3}  \sqrt{6} θ^{2}χφθ^{2}
2  \sqrt{6} 2\sqrt{3}  \sqrt{6}  2\sqrt{3}  2\sqrt{3} θ^{3}χφθ

Table 6: P (g, h) for g and h generators of cyclic group subgroups in G ≅ A_{5} of order 2. Here ⋆ = \{4, 4\sqrt{5}\} and ♢ = \{4, 12\}.

φθ φθ^{2} φθ^{3} φθ^{4} χ θχθ^{4} θ^{2}χθ^{3} θ^{3}χθ^{2}
φ  ⋆ ⋆ ⋆ ⋆ 2 ⋆  ♢  ♢
φθ  ⋆ ⋆ ⋆ ♢ ⋆  2  ⋆
φθ^{2}  ⋆ ⋆ ⋆ ♢  ♢  ⋆
φθ^{3}  ⋆ ⋆ 2  ⋆  ♢
φθ^{4}  ♢ ♢  ⋆  2
χ  ♢  ⋆  ⋆
θχθ^{4}  ♢  ⋆
θ^{2}χθ^{3}  ♢

θ^{3}χθ^{2} θ^{4}χθ χφ θχφθ^{4} θ^{2}χφθ^{3} θ^{3}χφθ^{2}

θ^{4}χθ χφ θχφθ^{4} θ^{2}χφθ^{3} θ^{3}χφθ^{2} θ^{4}χφθ
⋆  2  ♢  ⋆  ⋆  ♢  φ
♢  ⋆  ♢  2  ♢  ⋆  φθ
2  ♢  ⋆  ⋆  ♢  2  φθ^{2}
♢  ♢  2  ♢  ⋆  ⋆  φθ^{3}
⋆ ⋆  ⋆  ♢  2  ♢  φθ^{4}
♢  2  ⋆  ⋆  ⋆  ⋆  χ
⋆ ⋆  2  ⋆  ⋆  ⋆  θχθ^{4}
⋆ ⋆  ⋆  2  ⋆  ⋆  θ^{2}χθ^{3}
♢  ⋆  ⋆  ⋆  2  ⋆  θ^{3}χθ^{2}
⋆  ⋆  ⋆  ⋆  2  θ^{4}χθ
⋆  ♢  ♢  ⋆  χφ
⋆  ♢  ♢  θχφθ^{4}
⋆  ♢  θ^{2}χφθ^{3}
⋆  θ^{3}χφθ^{2}

A.6. Picture of 𝒯_{G}^{∗} with G ≅ A_{5} and p = 2. Mirrors are denoted by piecewise linear lines; both ends of them are indicated by arrows. Each generator of cyclic subgroups of order 3 is denoted twice at both ends of its mirror, while that of cyclic subgroups of order 2 is written once beside its mirror. The mirrors of generators of cyclic subgroups of order 5 are not drawn; they simply pass through the center of the picture. Every unit-segment is of length e/2.

[figure: θ^{3}χθ^{4}; θ^{2}χ; θχφ; θ^{2}χφθ^{2}; θ^{3}χθ^{4}; θ^{3}χφθ; θ^{2}χφθ^{4}; θ^{2}χθ; θχθ; θ^{2}χ; θ^{4}χφ; θ^{2}χφθ^{4}; θχθ; θχφ; θ^{3}χφθ; θ^{3}χ; θ^{2}χθ; θ^{4}χφ; θ^{3}χ; θ^{2}χφθ^{2}; φθ^{2}; θ^{4}χθ; θ^{4}χφθ; φ; χ; χφ; φθ^{4}; θ^{3}χθ^{2}; θ^{3}χφθ^{2}; φθ^{3}; θχθ^{4}; θχφθ^{4}; φθ; θ^{2}χθ^{3}; θ^{2}χφθ^{3}]
Figure 6: 𝒯_{G}^{∗} for G ≅ A_{5} and p = 2

Acknowledgments. The present work owes much to stimulating conversations with Professor Yves André. The author expresses gratitude to him. The author thanks Gunther Cornelissen and Aristeides Kontogeorgis for valuable discussions. Thanks are also due to Professor Jaap Top who pointed out a mistake in the first version of the main theorem. The author thanks Max-Planck-Institut für Mathematik Bonn for the nice hospitality.

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Graduate School of Mathematics, Kyushu University, Hakozaki Higashiku, Fukuoka 812-8581, Japan.

Current Address: Max-Planck-Institut für Mathematik, P.O.Box: 7280, D-53072, Bonn, Germany.