identity map on the fundamental group ofT2(this is essentially shown in [FM11,
Proof of Proposition 2.7, pp. 59–60]).
In our situation, such a lift to T2 of the homeomorphismhcan easily be ob-
tained. Namely, it follows from (3.29) that there exists a homeomorphismB:T2→
T2withπ
◦B=B◦π. Then by definitionBis a lift ofBbyπ. Now the translations u7→u+γ, γ ∈Γ, form the group of deck transformations of π which represents the fundamental group ofT2. Then it follows from (3.29) that the induced map of
B on the fundamental group is the identity (see the remarks after Lemma A.25 for a related discussion). Finally,B is a lift ofhby Θ, because we have
Θ◦B◦π= Θ◦π◦B= Θ◦B=h◦Θ =h◦Θ◦π, and so
Θ◦B=h◦Θ.
This shows that his indeed isotopic to idbCrel.P = post(f).
3.5. Covers of parabolic orbifolds
In this section we provide the proofs of Proposition 3.9 and Theorem 3.10. The main point is to prove existence of the map Θ in these statements. We will do this in an explicit geometric way that is also useful for visualizing examples of Latt`es maps (see Section 3.6).
We start with a given crystallographic groupGnot isomorphic toZ2. We first
consider the types (244), (333), and (236). The type (2222) is different and will be treated later. So let G be of type (244). Our goal is to find a holomorphic map Θ : C→Cb induced by G. We explain the construction in detail only in this case. For the types (333) and (236) the considerations are completely analogous and we will skip the details.
The group G has an invariant tiling made out of isometric copies of a right- angled Euclidean triangle with anglesπ/2,π/4, π/4 as shown in Figure 3.1. The triangles are colored black or white. The union of a white and a black triangle with a common edge forms a fundamental domain for the action of GonC. Let T be one of the white triangles in this tiling. We glue an isometric copyTwofT, colored
white, with another isometric copy Tb of T, colored black, along their boundaries
to form a pillow ∆ (see Section A.10). Then ∆ is a topological 2-sphere and can be identified with the quotient space C/G. The identificationT ∼=Tw induces an
orientation on ∆ (see Section A.10). We equip ∆ with the unique path metric that restricts to the Euclidean metric on the two copies ofT (see Figures 3.6, 3.7, and 3.8 for an illustration of ∆ for the different types ofGconsidered).
One can now define a map Θ∆: C→ ∆ as follows. The map Θ∆ sends each
white triangle T ⊂ C from the tiling as represented in Figure 3.1 to (the white triangle)Tw⊂∆ and each black triangleT ⊂∆ to (the black triangle)Tb⊂∆ by
an orientation-preserving isometry. Then Θ∆:C→∆ is a well-defined continuous
map.
Note that ifGis of type (333), then a similar construction does not lead to a well-defined map due to a rotational ambiguity which is not present for the types (244) and (236). In this case, one has to single out one of the common verticesvof Twand Tband impose the additional requirement that the piecewise isometry Θ∆
It is clear from the definition of Θ∆ that Θ∆= Θ∆◦g for eachg∈G. On the
other hand, if Θ∆(z) = Θ∆(w) forz, w∈C, then we may pick two trianglesT and
T′ of the same color in the tiling as represented by Figure 3.1 such thatz∈T and
w∈T′. There is a unique element g∈Gwithg(T) =T′. Theng(z), w∈T′ and
Θ∆(g(z)) = Θ∆(z) = Θ∆(w).
Since Θ∆is an isometry onT′ and hence injective onT′, it follows thatw=g(z).
This shows that Θ∆(w) = Θ∆(z) forz, w∈Cif and only if there existsg∈Gsuch
thatw=g(z). Hence Θ∆ is induced byG.
The 2-sphere ∆ is a polyhedral surface equipped with a locally Euclidean metric with three conical singularities (as shown in Figure 3.6). In particular, ∆ carries a natural conformal structure, with respect to which it is conformally equivalent to Cb. Moreover, Θ∆: C → ∆ is a continuous map that is a local isometry near
each point inCthat is not a preimage of one of the three cone points of ∆. Hence Θ∆:C → ∆ is a holomorphic map (all this is explained in greater generality in
Section A.10).
It follows from the uniformization theorem that we can find a conformal map ϕ: ∆→Cb that sends the vertices of the pillow to the points 0, 1,∞inCb. In fact, one can constructϕ quite explicitly by first mappingTw conformally to the upper
half-plane, andTbto the lower half-plane such that the vertices of the triangles are
sent to 0, 1,∞. If we define Θ = ϕ◦Θ∆:C→Cb, then Θ is a holomorphic map
induced byG.
We will verify the other properties of Θ as specified in Proposition 3.9 later in this section by a general argument. It applies to all types ofG and does not use our specific construction. It is still illuminating to see how these properties can be extracted from the tiling in Figure 3.1, at least on an intuitive level. First, it is clear that Θ∆, and hence also Θ, is a branched covering map: ifq∈∆ is arbitrary,
then an open ball V ⊂ ∆ centered at q of small enough radius ǫ > 0 is evenly covered by Θ∆ (see Definition A.7). Each componentU of Θ−∆1(V) is a Euclidean
disk of radiusǫ >0 centered at a pointz∈Θ−∆1(q). Each pointq′
∈V withq′
6
=q has precisely ddistinct preimages in U, where d= #Gz. So Θ∆ isd-to-1 nearz.
Hence deg(Θ∆, z) = deg(Θ, z) = #Gz forz ∈C. Since #Gz is the same for each
point z in a givenG-orbit and Θ∆ is induced by G, we can define a ramification
function α∆: ∆→Nby settingα∆(p) = #Gz forp∈∆, where we pick any point
z ∈ Θ−∆1(p) = Gz. Then α∆(p) = 2 if pis the corner of ∆ corresponding to the
common vertex ofTwandTbwith angleπ/2,α∆(p) = 4 for the other two cornersp
of ∆, andα∆(p) = 1 for all other pointsp∈∆. So if we setα:=α∆◦ϕ−1, thenαis
a finite ramification function onCb satisfying (3.7). The orbifold (Cb, α) is parabolic and has conical singularities at 0, 1,∞. Its signature is (2,4,4) corresponding to the type ofG.
By (3.7) the holomorphic branched covering map Θ :C →Cb is the universal orbifold covering map of (Cb, α). As was briefly discussed in Section 2.5, we can push forward the Euclidean metricd0 by Θ and obtain the canonical orbifold metric ω
on Cb (see Section A.10 for more details). If we equip Cb with this metric, then ϕ is in fact an isometry. So (Cb, ω) and the pillow ∆ are isometric, and one can view ∆ as a geometric realization of the orbifold (C, α). In particular, (Cb, ω) is locally isometric to C, except at the conical singularities of the orbifold (Cb, α) (i.e., the
3.5. COVERS OF PARABOLIC ORBIFOLDS 79
pointsp∈Cb withα(p)≥2), where (Cb, ω) is locally isometric to a Euclidean cone of angle 2π/α(p).
If (Cb,α) is an arbitrary orbifold with signature (2,e 4,4), then we can find a M¨obius transformationψ:Cb →Cb that matches up the three cone points of (Cb,α)e
and (Cb, α) such thatαe◦ψ=α. Thenψ◦Θ : C→Cb is the (holomorphic) universal orbifold covering map of (Cb,α). The geometric pictures remains the same: if wee
equip (Cb,α) with its (possibly rescaled) universal orbifold metrice eω, then (Cb,ω) ise
isometric to the pillow ∆.
We now turn to crystallographic groupsGof type (2222). In order to construct a holomorphic map Θ :C→Cb induced byG, we may assume thatGconsists of all isometries on C of the formz 7→ ±z+γ, where γ∈ Γ. Here Γ is the underlying (rank-2) lattice Γ ⊂Cof G. For a general group Gof type (2222) one considers Θ◦hwith suitable h∈Aut(C) to obtain a map induced byG (this reduction is based on Theorem 3.7 and related to the remarks at the beginning of Section 3.3). Holomorphic maps Θ :C →Cb induced by a groupG of this special form can be obtained from the Weierstraß ℘-function (see [Ah79, Section 7.3] for general background). Recall that the Weierstraß ℘-function for a given lattice Γ ⊂ C is defined as (3.31) ℘(z; Γ) = 1 z2 + X γ∈Γ\{0} 1 (z−γ)2− 1 γ2 forz∈C.
Then℘=℘(·; Γ) is an even meromorphic function onCwith the period lattice Γ. The function℘satisfies the differential equation
(3.32) (℘′)2= 4(℘
−e1)(℘−e2)(℘−e3).
The numberse1, e3, e3∈Chere are three distinct values (depending on Γ) with
(3.33) e1+e2+e3= 0.
The critical values of ℘: C → Cb are e1, e2, e3, ∞. Actually, ℘: C → Cb is a
holomorphic map satisfying
(3.34) deg(℘, z) =
2 if℘(z)∈ {e1, e2, e3,∞},
1 otherwise.
IfGis the crystallographic group corresponding to Γ as above, then℘is induced byG. Indeed,℘(w) =℘(z) if and only ifw=±z+γfor someγ∈Γ (the “only if” implication can be derived from the familiar fact that℘descends to a holomorphic map℘: T→Cb on the torusT=C/Γ with deg(℘) = 2).
As is well known and classical, one can reverse this procedure and start with the differential equation (3.32): if three distinct valuese1,e2,e3∈Cwith (3.33) are
given, then there exists a (unique) lattice Γ such that the corresponding function ℘=℘(·; Γ) satisfies (3.32). See [Ah79, Sections 7.3.3 and 7.3.4].
The general argument in the proof of Proposition 3.9 will show that℘:C→Cb
is a branched covering map, because ℘ is induced by G. It then follows from (3.34) that ℘ is the universal orbifold covering map of the orbifold (Cb, α), where α(p) = 2 for p ∈ {e1, e2, e3,∞} and α(p) = 1 for p ∈ Cb \ {e1, e2, e3,∞}. This
implies that the universal orbifold covering map Θ of any orbifold (Cb, α) with signature (2,2,2,2) can always be obtained from a Weierstraß℘-function followed by a suitable M¨obius transformation. Indeed, ifp1, . . . , p4∈Cb are the four distinct
points with α(pk) = 2 fork= 1, . . . ,4, then there exists a M¨obius transformation
ψ on Cb such that ψ({p1, . . . , p4}) = {e1, e2, e3,∞}, where e1, e2, e3 ∈C are three
distinct points satisfying (3.33) (first map p4 to ∞ by a M¨obius transformation
and then apply a suitable translation). Then we can find a lattice Γ such that the corresponding function ℘= ℘(·; Γ) satisfies (3.32) with these values e1, e2, e3. If
we define Θ =ψ−1
◦℘(·; Γ), then Θ is the universal orbifold covering map of the given orbifold (Cb, α).
Actually, one can make one more reduction here. Namely, in the last statement we may assume that the lattice has the form Γ =Z⊕Zτwithτ ∈Cand Im(τ)>0. This follows from the homogeneity property
℘(λz;λΓ) = 1
λ2℘(z; Γ), z∈C, λ∈C\ {0},
of the℘-function.
We will now describe a more geometric construction for the maps Θ and their associated orbifolds that arise here similar to the one given for the crystallographic groups of type (244). For this we fix τ ∈ C with Im(τ) > 0, and consider the crystallographic groupGof type (2222) given by all isometriesz7→ ±z+γ, where γ∈Γ :=Z⊕Zτ. Not all crystallographic groups of type (2222) have this form, but we restrict ourselves to the special case for simplicity. One can easily adjust the ensuing discussion to the general case by essentially precomposing all relevant maps on C with a suitable element h∈ Aut(C). Note that by the reduction discussed above we still get the same class of orbifolds from this restricted class of groups.
A fundamental domain for G is the Euclidean triangle T ⊂ C with vertices 0,1, τ. To obtain the quotient space C/G from T, we divide T into four similar triangles by connecting the midpoints of each side and fold T along the edges of these smaller triangles. In this way, we can build a tetrahedron ∆ in Euclidean 3-space as indicated in Figure 3.4 (the two halves of each side are identified). Here we allow the degenerate case that ∆ is a pillow. This happens precisely when T has a right angle.
We equip ∆ with the path metric induced by the Euclidean metric onT, and the orientation induced by the orientation onT ⊂C. Then ∆ is homeomorphic to a 2-sphere. It is a polyhedral surface with four conical singularities and carries a natural conformal structure.
The parallelogramP ⊂ C spanned by 1 and τ is a fundamental domain for the group of translationsGtr⊂Ggiven by all maps of the formz7→z+γ, where
γ∈Γ =Z⊕Zτ. We can divideP into two triangles that are isometric to T, and into 8 smaller triangles similar toT by the scaling factor 2 (see Figure 3.5).
As indicated in Figure 3.5, there is a map Θ∆:P → ∆ such that each small
triangle inP is mapped isometrically to a face of ∆. This map extends naturally to a continuous map Θ∆:C→∆ that respects the lattice translations in the sense
that Θ∆◦g= Θ∆for allg∈Gtr. Note that then Θ∆(z) = Θ∆(−z) forz∈C, and
it easily follows that Θ∆ is induced byG. In particular, we can identifyC/Gwith
the tetrahedron ∆ (this follows from Corollary A.23 (ii)).
By the uniformization theorem we can find a conformal mapϕ: ∆→Cb. Ifϕis such a map, then Θ =ϕ◦Θ∆:C→Cb is a holomorphic map induced byG. This (or
a direct geometric argument) implies that Θ and Θ∆ are branched covering maps.
3.5. COVERS OF PARABOLIC ORBIFOLDS 81
0 1
τ
T
∆
Figure 3.4. Folding a tetrahedron from a triangle.
P τ 1 0 ∆ Figure 3.5. Construction of Θ =℘.
If we assign to each of the four conical singularities of ∆ (where the cone angle isπ) the value 2 for a ramification function α∆ on ∆ and the value 1 to all other
points, then (∆, α∆) is an orbifold with signature (2,2,2,2) and we obtain the
relation (3.7) (for α∆ and Θ∆). So if we define α=α∆◦ϕ−1, then (Cb, α) is an
orbifold with the same signature as (∆, α∆) and (3.7) holds. Then Θ is the universal
orbifold covering map of (Cb, α). If we equip Cb with the corresponding universal orbifold metricω, then (Cb, ω) is isometric to ∆ (possibly up to scaling).
It follows from our earlier analytic discussion that if we set ϕ=ψ◦ϕ0 for a
suitable M¨obius transformationψand chooseτ ∈Cwith Im(τ)>0 appropriately, then the universal orbifold covering map Θ of any orbifold (Cb, α) with signature (2,2,2,2) can be written in the form Θ =ϕ◦Θ∆.
We can now give the proofs of Proposition 3.9 and Theorem 3.10.
Proof of Proposition 3.9. LetG be a crystallographic group not isomor- phic toZ2. ThenGis of type (244), (333), (236), or (2222). We have seen earlier in
of these types. For logical clarity we will not use any other facts from our previous discussion.
Since G acts cocompactly on C and Θ is induced by G, the image Θ(C) is compact. This implies that Θ is surjective, because Θ(C) is also open inCb, and so Θ(C) =Cb.
In order to see the second identity in (3.7), letz0 ∈ Cbe arbitrary and d:=
#Gz0. We have to show that Θ is locally d-to-1 near z0. Since G acts properly
discontinuously onC, the orbitGz0 is a discrete set inC. This implies that each
isometryg ∈G\Gz0 moves points near z0 a definite distance away fromz0. So if
we choose the radiusǫ >0 of the Euclidean disk B:=BC(z0, ǫ) sufficiently small,
thenB∩g(B)6=∅for some g∈Gif and only if g∈Gz0. It follows that
(3.35) g(B)∩h(B)6=∅ forg, h∈Gif and only ifh−1
◦g∈Gz0.
Now if u, v ∈ B and Θ(u) = Θ(v), then there exists g ∈ G with v = g(u), because Θ is induced byG. In particular,v∈B∩g(B), and sog∈Gz0 by (3.35).
We conclude that two points in B have the same image under Θ if and only if they belong to the sameGz0-orbit. NowGz0 is a cyclic group of orderdconsisting
of rotations around z0. Hence the orbit of each point z ∈ B \ {z0} consists of
preciselydpoints in B. This shows that Θ is indeed locallyd-to-1 near z0, and so
deg(Θ, z0) = #Gz0 as desired (for a similar argument in greater generality see the
proof of Proposition A.31 (ii)).
To see that Θ :C → Cb is a branched covering map, let z0 ∈ C be arbitrary.
We chooseB =BC(z0, ǫ) as before so that (3.35) holds. Since Θ is surjective, it is
enough to show thatw0:= Θ(z0) has a neighborhood that is evenly covered by Θ
as in Definition A.7.
The invariance property of Θ with respect to the cyclic rotation groupGz0 on
B implies that
Θ(z) =f((z−z0)d/ǫd) forz∈B,
wheref:D→Cb is an injective holomorphic map withf(0) =w0.
In particular, V := Θ(B) = f(D) is a topological disk containing w0. If we
defineϕ: B→Dbyϕ(z) = (z−z0)/ǫforz∈B and setψ:=f−1:V →D, thenϕ
andψare orientation-preserving homeomorphisms withϕ(z0) = 0,ψ(w0) = 0, and
(3.36) (ψ◦Θ◦ϕ−1)(u) =ud
for allu∈D.
Since Θ is induced byG, we have by (3.35) that Θ−1(V) = [
g∈G
g(B) = [
g∈I
g(B),
whereI⊂Gis a set that contains precisely one element from each left coset ofGz0
inG. Note that by (3.35) this means that the sets in the latter union are pairwise disjoint. For each setg(B) withg∈I, we have a relation as in (3.36) if we replace ϕ: B→Dwithϕ◦g−1:g(B)→D. This shows thatV is evenly covered by Θ. It
follows that Θ is indeed a branched covering map.
Since Θ is surjective and induced byG, there is a bijection between the set of G-orbits (i.e., C/G) and points inCb given byGz7→Θ(z). As we know, for points z ∈C in a givenG-orbit, the cardinality #Gz is independent of z. By using the