The integral homology of $ PSL_2$ of imaginary quadratic integers with nontrivial class group

INTEGERS WITH NON-TRIVIAL CLASS GROUP

ALEXANDERD.RAHMANDMATHIASFUCHS

Abstrat. WeshowthataellularomplexdenedbyFlögeallowstodeterminetheintegralhomol-

ogyoftheBianhigroups PSL

2 ( O −m )

^,where

O −m

^{istheringofintegersofanimaginaryquadrati}

numbereld

Q ˆ√

− m ˜

forasquare-freenaturalnumber

m

^.

Intheasesofnon-trivial lassgroup, wehandlethediultiesarisingfromtheuspsassoiatedto

thenon-trivialideallassesof

O −m

^{. Weusethis toomputeintheases}

m = 5, 6, 10, 13

^and

15

^the

integralhomologyofPSL

2 ( O ^−m )

^{,whihbeforewasknownonlyintheases}

m = 1, 2, 3, 7

^and

11

^with

triviallassgroup.

Contents

1. Introdution 1

2. Flöge'somplex,ontratibility anda spetralsequene 3

2.1. Theequivariantspetral sequeneingroup homology 5

2.2. Thehomology ofthe nite subgroups intheBianhi groups 8

2.3. Themassformula forthe equivariant Euler harateristi 10

3. Computations oftheintegralhomology of PSL

2 O _Q [ ^√ − m ]

11

3.1.

m = 13

¹²

3.2.

m = 5

¹⁸

3.3.

m = 10

²⁰

3.4.

m = 6

²⁴

3.5.

m = 15

²⁷

4. Appendix: Theequivariant retration 29

Referenes 31

1. Introdution

Theobjets of study ofthis paperarethe

PSL 2

^-groups

Γ

^{of theringof integers}

O − m := O Q[ √

− m ]

ofan imaginaryquadrati numbereld

Q[ √

− m ]

^,where

m

^isasquare-freepositiveinteger. Wehave

O − m = Z[ω]

^with

ω = √

− m

^for

m

^{ongruent to 1 or 2 modulo 4, and}

ω = − ¹ ₂ + ¹ ₂ √

− m

^for

m

ongruent to3 modulo4.

The arithmeti groups understudy have often been alled Bianhi groups, beause Luigi Bianhi

[6℄ omputed fundamental domains for them asearly as in1892. Theyat on

PSL ₂ (C)

^{'s symmetri}

Date:April9,2011.

2000 Mathematis Subjet Classiation. 11F75,Cohomologyofarithmeti groups. 22E40,Disretesubgroupsof

Liegroups. 57S30,Disontinuousgroupsoftransformations.

The rst named author is supported by a researh grant of the Ministère de l'Enseignement

Supérieur et de la Reherhe, and partially supported by DFH-UFA grant CT-26-07-I and by a

DAAD(GermanAademiExhangeServie)grant.

TheseondnamedauthorwassupportedbyaGKGruppenundGeometriepost-dotoralfellowshipoftheDFG.

Figure1: Resultsinthegrouphomologywithsimpleintegeroeients

H _q (

^PSL

₂ ( O ₋ 5 ); Z) ∼ =

 



 

Z ² ⊕ Z/3 ⊕ (Z/2) ² , q = 1, Z ⊕ Z/4 ⊕ Z/3 ⊕ Z/2, q = 2, Z /3 ⊕ ( Z /2) ^q , q > 3;

H _q (

^PSL

₂ ( O ₋ 10 ); Z ) ∼ =

 



 

Z ³ ⊕ (Z/2) ² , q = 1, Z ² ⊕ Z/4 ⊕ Z/3 ⊕ Z/2, q = 2, Z /3 ⊕ ( Z /2) ^q , q > 3;

H _q (

^PSL

₂ ( O ₋ 15 ); Z) ∼ =

 



 

Z ² ⊕ Z/3 ⊕ Z/2, q = 1, Z ⊕ Z/3 ⊕ Z/2, q = 2, Z/3 ⊕ Z/2, q > 3;

H _q (

^PSL

₂ ( O ₋ 13 ); Z) ∼ =

 

 

 

 

 

 

 

 

 



Z ³ ⊕ (Z/2) ² , q = 1, Z ² ⊕ Z/4 ⊕ (Z/3) ² ⊕ Z/2, q = 2,

(Z/2) ^q ⊕ (Z/3) ² , q = 4k + 3, k > 0, (Z/2) ^q , q = 4k + 4, k > 0, (Z/2) ^q , q = 4k + 1, k > 1, ( Z /2) ^q ⊕ ( Z /3) ² , q = 4k + 2, k > 1;

H _q (

^PSL

₂ ( O − 6 ); Z) ∼ =

 

 

 

 

 

 

 



 

 

 

 

 

 

 

Z ² ⊕ Z/3 ⊕ Z/2, q = 1, Z ⊕ Z/4 ⊕ Z/3 ⊕ (Z/2) ² , q = 2, Z/3 ⊕ (Z/2) ^2k+2 , q = 6k + 3, Z/3 ⊕ (Z/2) ^2k+1 , q = 6k + 4, Z/3 ⊕ (Z/2) ^2k+4 , q = 6k + 5, Z/3 ⊕ (Z/2) ^2k+3 , q = 6k + 6, Z/3 ⊕ (Z/2) ^2k+2 , q = 6k + 7, Z/3 ⊕ (Z/2) ^2k+5 , q = 6k + 8.

spae, the hyperboli three-spae

H

^{. Interest in this ation rst arose when Felix Klein and Henri}

PoinaréstudiedertaingroupsofMöbiustransformationswithomplexoeients[14,18℄,layingthe

groundwork for the study of Kleiniangroups. Thelatter arenowadays dened asdisrete subgroups

of

PSL ₂ (C)

^{. Eah non-oompat arithmeti Kleinian group is} ommensurable with some Bianhi group [15℄. Thus, the Bianhi groups play a key role in the study of arithmeti Kleinian groups. A

wealth of information ontheBianhigroups an be foundinthepertinent monographs[8,9,15℄.

Poinaré gaveanexpliitformulafortheirationon

H

^{. However,thevirtual}ohomologialdimension ofarithmetigroupswhih arelattiesin

SL ₂ (C)

^{istwo,soitisdesirabletorestritthisproperation}

on

H

^{to a ontratible ellular} two-dimensional spae. Moreover, this spae should be onite. In priniple, this has been ahieved by Mendoza [16℄ and also by Flöge[10℄, using redution theory of

Minkowski, Humbert, Harder and others. Their two approahes have inommon thatthey onsider

two-dimensional

Γ

-equivariant retrats whih are oompat and are endowed with a natural CW- struture suh thatthe ation of

Γ

^{isellular and thequotient isa niteCW-omplex.}

Using Mendoza'somplex, Shwermer and Vogtmann [20℄ alulated theintegral group homology in

theasesoftriviallassgroup

m = 1, 2, 3, 7, 11

^{,andVogtmann[24℄omputedtherationalhomologyas}

thehomologyofthequotient spaeinmanyasesofnon-triviallassgroup. Theintegralohomology

intheases

m = 2, 3, 5, 6, 7, 10, 11

^{hasbeen determinedbyBerkove [5℄,basedonFlöge's}presentation ofthegroupswithgenerators andrelations. Aompletely dierent methodto obtaingrouppresenta-

tions hasbeenhosenbyYasaki[26℄,whohasimplementedanalgorithm ofGunnells [12℄to ompute

the perfet forms modulothe ation of

GL ₂ ( O − m )

^{and obtain the faets of theVoronoï polyhedron}

arising froma onstrution ofAsh[3℄.

Itisthe purposeofthe presentpaperto showhowFlöge'somplex anbeusedto obtaintheintegral

homologyofBianhigroupsalsowhenthelassgroupisnon-trivial. Weobtaintheresultsdisplayedin

gure1. Thusfor

q > 2

^{,thetorsionin}

H _∗ (

^PSL

₂ ( O ₋ 5 ); Z )

^{isthesameasthatin}

H _∗ (

^PSL

₂ ( O ₋ 10 ); Z )

^,

analogous to the ohomology results of Berkove [5℄. The free part of these homology groups is in

aordane withthe rational homology resultsof Vogtmann[24℄.

In theasesof non-trivial ideal lassgroup, thereis a dierenebetween theapproahesof Mendoza

and Flöge. We usethe upper-half-spae model of

H

^{and identify its boundary with}

C ∪ ∞ ∼ = CP ¹ .

The elements of the lass group of the number eld are in bijetion with the

Γ

^{-orbits of the usps,}

where theuspsare

∞

^{andthe elementsof thenumbereld}

Q √

− d

,thought ofaselementsof the

anonialboundary

CP ¹

^{. Theuspswhihrepresenta}non-trivialelement ofthelassgroup areom- monlyalledsingular points. WhilstMendozaretratsawayfromallusps,Flögeretrats awayonly

from thenon-singular ones. Rather than the spae

H

^{itself,he onsiders the spae}

H b

^{obtained from}

H

^{by adjoining the}

Γ

^{-orbits of the singular points. We onsider an analogous} equivariant retration of

H ˆ

^{suh thatits retrat}

X

^{ontains the singular points. Nowit turns out that the quotient spae}

of

X

^by

Γ

^{is ompat, and}

X

^{is a suitable ontratible}

2

-dimensional

Γ

^{-omplex also inthe ase of}

non-trivial lassgroup.

Withan implementation inPari/GP [2℄,due to therst named author, of Swan's algorithm [23℄ we

obtain afundamentalpolyhedronfor

Γ

ⁱⁿ

H

^{. Inthe ases onsidered, Bianhihasalready omputed}

this polyhedron, sowe have aontrol of theorretnessoftheimplementation.

In the ases

m = 5, 6

^and

10

^{, Flöge has omputed the ell} stabilizers and ell identiations; and withour Pari/GP program,we redoFlöge'somputationsanddo thesameomputation intheases

m = 13

^and

15

^{. We use the} equivariant Euler harateristi to hek our omputations. Then we followthelinesofShwermerandVogtmann[20℄,enountering aspetralsequenewhihdegenerates

onthe

E ³

^{-page, inontrasttotheasesoftriviallassgroup whereitdoessoalreadyonthe}

E ²

^-page.

This is beause of the singular points in our ell omplex

X

^{, whih have innite} stabilizers. So we have some additional use of homologial algebra to obtain the homology of the Bianhi group. We

give the full details for our homology omputation inthe ase

m = 13

^{. We thengive slightly fewer}

details intheases

m = 5, 6, 10

^and

15

^.

Theauthors would like to thankPhilippe Elbaz-Vinent andBill Allombertfor manyhelpful disus-

sionsand hintson the tehniquesand thereferee for helpful omments.

Thisartile isdediated toFritzGrunewald(1949-2010).

2. Flöge's omplex, ontratibility and a spetral sequene

Denote thehyperboli three-spae by

H ∼ = C × R ₊ ^∗

^{. We will not useits smooth struture,only its}

struture asahomogeneous

SL ₂ (C)

^{-spae. The ationis given bytheformula}

a b

c d

· (z, r) :=

(d − cz)(az − b) − r ² ca

| cz − d | ² + r ² | c | ² , r

| cz − d | ² + r ² | c | ²

;

where

a b c d

∈ SL ₂ (C)

^{. As usual, we extend the ation of}

SL ₂ (C)

^{to the boundary}

CP ¹

^{whih we}

identifywith

{ r = 0 } ∪ ∞ ∼ = C ∪ ∞

^{. Theationpasses}ontinuouslytotheboundary,whereitredues

to the usualation by Möbiustransformations

a b c d · z = ₋ ^az _cz+d ^{− b} .

^As

− 1 ∈ SL ₂ (C)

^{ats trivially, the}

ationpassesto

PSL ₂ (C)

^{. Now, xa} square-free

m ∈ N

^,let

O − m

^{betheringofintegersin}

Q[ √

− m ]

^,

and dene

Γ = PSL ₂ ( O − m )

^{. When the lass number of}

Q[ √

− m ]

^{is one, then lassial redution}

theoryprovidesanaturalequivariantdeformationretratof

H

^{whihisaCW-omplex. Thisomplex}

isdened asfollows. One rstonsiders theunionof allhemispheres

S _µ,λ :=

( (z, r) :

z − λ

µ

2

+ r ² = 1

| µ | ² )

⊂ H ,

for anytwo

µ, λ

^with

µ O − m + λ O − m = O − m

^{. Then oneonsiders thespae above the}hemispheres

B :=

(z, r) : | cz − d | ² + r ² | c | ² > 1

^{for all}

c, d ∈ O − m , c 6 = 0

^{suh that}

c O − m + d O − m = O − m

and itsboundary

∂B

^inside

H

^{. For nontriviallassgroup, thefollowing denitionomes to work.}

Denition 1. A point

s ∈ CP ¹ − {∞}

^{is alled a singular point if for all}

c, d ∈ O − m

^,

c 6 = 0

^,

c O − m + d O − m = O − m

^{we have}

| cs − d | > 1

^.

Thesingular pointsmodulotheationof

Γ

^on

CP ¹

^{areinbijetionwiththenontrivialelementsof}

thelassgroup[22℄. In [10℄,Flögeextendsthe hyperboli spae

H

^{to alarger spae}

H b

^asfollows.

Denition 2. As a set,

H ⊂ b C × R ^>0

^{is the losure under the}

Γ

^{-ation of the union}

B b := B ∪ {

^{singular points}

}

^{. The topology isgenerated by thetopology of}

H

^{together withthefollowing}

neighborhoods of the translates

s

^{of singular points:}

U b _ǫ (s) := { s } ∪

s 0

− 1 s ^{− 1}

·

(z, r) ∈ H : r > ǫ ^{− 1} .

Remark 3. The matrix

s 0

− 1 s ⁻¹

maps the point at innity into

s

^{, thus giving the point}

s

^the

topology of

∞

^{. The} neighborhood

U b _ǫ (s)

^{is sometimes alled a horoball beause in the upper-half}

spae modelitis aEulidean ball,but withthe hyperboli metri ithasinniteradius.

Thespae

H b

^{isendowedwiththenatural}

Γ

^{-ation. NowtheessentialaspetofFlöge'sonstrution}

isthefollowing onsequene of Flöge'stheorem[11, 6.6℄,whihwe append astheorem28.

Corollary 4. There is a retration

ρ

^from

H b

^{onto the set}

X ⊂ H b

^{of all}

Γ

-translates of

∂ B b

^{, i. e.}

there is a ontinuous map

ρ : H → b X

^{suh that}

ρ(p) = p

^{for all}

p ∈ X

^{. The set}

X

^{admits a natural}

struture asa ellular omplex

X ^•

^{on whih}

Γ

^{ats ellularly.}

Remark 5. (1) Weshowwiththelemmabelowthat

ρ

^isahomotopyequivalene,withoutgiving a ontinuouspath of maps

H → b H b

^onneting

ρ

^{to the identity on}

H b

^.

(2) The map

ρ

^is

Γ

-equivariant beause its bers are geodesis. But we do not make useof this fat, aswe donot need to showthat the homotopy type of

Γ \ H b

^{is thesame asthat of}

Γ \ X

^.

This would be useful in the ase of trivial lass group, i. e. the ase of a proper ation, to

ompute the rational homology

H _∗ (Γ; Q) ∼ = H _∗ (Γ \H ; Q)

^.

(3) We willprovide

X ^•

^{witha ellular struture whihis ne enough to makethe ell} stabilizers xthe ells pointwise.

Lemma 6. Let

Y

^{be a CW-omplex whih admits an inlusion}

i

^{intoa ontratible topologial spae}

A

^{, suh that}

i

^{is a} homeomorphism between

Y

^{withits ellular topology and the image}

i(Y )

^withthe

subset topology of

A

^{. Let}

p : A → Y

^{be a ontinuous map with}

p ◦ i = id _Y

^{. Then}

p

^{is a homotopy}

equivalene.

Proof. For all

n ∈ N

^{, the indued maps on the homotopy groups}

(id _Y ) _∗ = (p ◦ i) _∗ : π _n (Y ) → π _n (Y )

fator through

π _n (A) = 0

^{, hene are the zero map; and}

π _n (Y ) = 0

^{. Denote by}

c

^{the onstant map}

from

A

^{to the one-point spae. Then}

c ◦ i

^{is a morphism of} CW-omplexes, and the zero maps it induesonthehomotopygroupsareisomorphisms. ThusbyWhitehead'sTheorem,

c ◦ i

^isahomotopy

equivalene. As

A

^is ontratible, theomposition

(c ◦ i) ◦ p = c

^{is a homotopy} equivalene, so the

same holdsalready for

p

^.

Taking

Y = X

^,

A = H b

^,

p = ρ

^{,and using lemma8,we obtain aruial fatfor our} omputations.

Corollary 7.

X ^•

^is ontratible.

Thefollowing isan observation onFlöge'sonstrution.

Lemma 8. The spae

H b

^{isontratible.}

Proof. Oneanidentifytheboundaryof

H ∼ = { (z, r) ∈ C × R | r > 0 }

^with

CP ¹ ∼ = C ∪∞ ∼ = { r = 0 }∪∞

^.

Byviewingthesingular pointsaspartofthe boundary,we arrive atan upperhalf-spae modelof

H b

^.

Nowonsider

H 1 := { (z, r) ∈ H b : r > 1 }

^{withthesubspae topologyof}

H b

^{. Theideaoftheproofisto}

onsider a vertial retration onto

H 1

^{, and to showby an expliit argument that preimages of open}

sets areopen. Flöge[11,Korollar5.8℄ suggestsusing themap

φ : H × b [0, 1] → H b , ((z, r), t) 7→

( (z, r)

^{for all}

t ∈ [0, 1],

^if

r > 1 (z, r + t(1 − r)),

^if

r < 1.

Letusnowhekthatthisisaontinuousfamilyofontinuousmaps. Considertheolletionofopen

ballswithrespettotheEulidean metrion

C × R ₊

^{assoon astheyareeitherontainedin}

C × R ^∗ ₊

^,

or touh the boundary

C × { 0 }

^{ina uspin}

H − H b

^{. Thisis a basisfor thetopology of}

H b

^{. Consider}

one suhopenball

B

^{,and itspreimageundersome}

φ _t

^,

t ∈ [0, 1)

^{. Thiseitherliesentirely in}

H

^{,and is}

open,or ithasboundarypoints. Inthelatterase, onsider theinverseof

φ _t

^on

H − H b 1

^,givenby

φ ⁻ _t ¹ = z, ₁ ^{r −} ₋ ^t _t

if this is in

H b

^{. Suppose there is a usp}

s

^with

s ∈ H − H b

^and

φ _t (s, 0) ∈ B

^{. As}

B

^{is open, we nd}

β > 0

^and

δ > 0

^suhthat

(s, t + β)

^and

(s + δ, t)

^arein

B

^{. Sine}

( φ _t s, ₁ ^β _{− t}

= (s, t + β) ∈ B φ _t (s + δ, 0) = (s + δ, t) ∈ B ,

we know that

(s, ₁ ^β _{− t} )

^and

(s + δ, 0)

^{are in the preimage of}

B

^under

φ _t

^{. We dedue that the whole}

horosphere of Eulidean diameter min

{ β, δ }

^{touhing at the usp}

s

^{is inluded in the preimage of}

B

^{. Thus eah point of the preimage has a} neighborhood entirely ontained in the preimage, whih therefore also isopen. Theontinuity at

t = 1

^{aswell asthe ontinuity inthevariable}

t

^followfrom

very similararguments. Thespae

H 1

^ishomeomorphi to

C × R ₊

^,thus ontratible.

2.1. The equivariant spetral sequene ingroup homology.

Corollary7givesus aontratibleomplex

X ^•

^onwhih

Γ

^{atsellularly. Asaonsequene,theinte-}

gral homology

H _∗ (Γ; Z)

^{an beomputed asthe} hyperhomology

H _∗ (Γ; C _• (X))

^of

Γ

^withoeients

in the ellular hain omplex

C _• (X)

^{assoiated to}

X

^{. This} hyperhomology is omputable beause thereisaspetralsequeneasin[7, VII℄whih isalso theoneusedin[20℄. Itisthespetralsequene

assoiated tothe double omplex

Θ ^Γ _• ⊗ ZΓ C _• (X)

^{omputing the}hyperhomology, where we denoteby

Θ ^Γ _•

^{thebar resolution ofthe group}

Γ

^{. Thisspetralsequenean berewritten (see [20,1.1℄) to yield}

E _p,q ¹ = M

σ ∈ Γ \ X ^p

H q (Γ σ ; Z ) = ⇒ H p+q (Γ; Z ),

where

Γ _σ

^{denotes the stabilizer of (the hosen} representative for) the

p

^-ell

σ

^{. We have stated the}

above

E ¹

^{-termwith trivial}

Z

^{-oeients in}

H _q (Γ _σ ; Z)

^{,beause we usea} fundamental domain whih is strit enough to give

X

^{a ell struture on whih}

Γ

^{ats without inversion of ells. We shall also}

makeextensive useofthe desription ofthe

d ¹

-dierential given in[20℄.

The tehnial dierene to the ases of trivial lass group, treated by[20℄, is that the stabilizers of

the singular points are free abelian groups of rank two. In partiular, the

Γ

^{-ation on our omplex}

X ^•

^{is not a proper ation inthe sense thatall} stabilizers are nite. Asa onsequene, the resulting spetralsequenedoesnot degenerateon the

E ²

^{-levelasitdoesinShwermerandVogtmann'sases.}

So we ompute a nontrivial dierential

d ²

^{, making some additional use of homologial algebra, in}

partiular thebelowlemma andits orollary.

Remark 9. Itwouldbepossibletoshiftthetehnial diultyawayfromhomologialalgebra,using

atopologialmodiationofouromplex. Inourasesoflassnumbertwo,thereisonesingularpoint

in the fundamental domain, representing the nontrivial element of the lass group. Its stabilizer is

freeabelianofranktwo,andontributesthehomologyofatorustothezerotholumnofthe

E ²

^-term

of our spetral sequene:

H ₁ (Z ² ; Z) ∼ = Z ²

^,

H ₂ (Z ² ; Z) ∼ = Z

^and

H _q (Z ² ; Z) = 0

^for

q > 2

^{. One ould}

modify our omplex inorder to make the

Γ

^{-ation on it proper, by replaing eah singular point by}

an

R ²

^{withtheformer stabilizer}

Z ²

^{now ating properly. Thenthe} nontriviality of our dierential is equivalent totheexisteneofanontrivialhomology relationinduedbyadjoining thetorus

R ² /Z ²

^to

thefundamental domain.

Thefollowing lemmawill beuseful foromputing our

d ²

-dierential inthesituationswhere yles for

Γ σ

^{are given in terms of thestandard resolution of}

Γ

^{instead of}

Γ σ

^{. In order to state it, let}

Γ σ

be anite subgroup of

Γ

^,let

M

^{be a}

ZΓ _σ

^{-module, and}

ℓ : Γ/Γ _σ → Γ

^a set-theoretial setion of the quotient map

π : Γ → Γ/Γ _σ

^{. Further, denote the standard bar resolution of a disrete group}

Γ

^by

Θ ^Γ _•

^{. It will be onvenient to view}

Θ _• ^Γ

^{as a omplex of}

ZΓ

^{-right modules resp.}

ZΓ _σ

^{-right modules.}

Thus,

Θ ^Γ _q

^{is dened as thefree}

Z

^{-modulegenerated bythe}

(q + 1)

^-tuples

(γ ₀ , . . . , γ _q )

^{of elements of}

Γ

^{withtheation given by}

(γ ₀ , . . . , γ _q ).γ = (γ ₀ γ, . . . , γ _q γ)

^{andthe same boundary operator asinthe}

left modulease, namely

∂ = P q

i=0 ( − 1) ⁱ d _i

^where

d _i (γ ₀ , . . . , γ _q ) = (γ ₀ , . . . , γ b _i , . . . , γ _q )

^.

Lemma 10. The setion

ℓ

^{denes a map of}

ZΓ _σ

^-omplexes

ˆ

ε _ℓ : Θ ^Γ _• −→ Θ ^Γ _• ^σ

of degree zero whih is a retration of the resolution

Θ ^Γ _•

^{of the group}

Γ

^{to the resolution}

Θ ^Γ _• ^σ

^of

Γ _σ

^.

For eah

γ ∈ Γ

^,

ℓ(π(γ))

^{isin the sameorbit of the}

Γ σ

-right-ation on

Γ

^as

γ

^{, so}

(ℓ(π(γ))) ^{− 1} γ ∈ Γ σ

^.

The map

ε ˆ _ℓ

^{is indued on}

Θ ^Γ ₀ = ZΓ

^by

Γ −−→ ^{ε ℓ} Γ _σ → ZΓ _σ , γ 7→ (ℓ(π(γ))) ^{− 1} γ

and isontinued as a tensorprodut

ε ˆ _ℓ = ε _ℓ ⊗ ... ⊗ ε _ℓ = ε ^⊗ _ℓ ⁽ⁿ⁺¹⁾

^on

Θ ^Γ _n

^.

Remark 11. (1) Sine thegroup

Γ _σ

^{atsfrom the right,the map}

ε _ℓ

^is

ZΓ _σ

^-linear.

(2) Notethattheresultingisomorphism inhomologyfrom

H _∗ (Θ ^Γ _• ⊗ ZΓ σ M)

^to

H _∗ (Θ ^Γ _• ^σ ⊗ ZΓ σ M)

^is

independent of the hoie of

ℓ

^{,and onsistent withthe anonial}isomorphisms of both sides with

H _∗ (Γ _σ ; M)

^.

(3) Note that in the above lemma, it is not neessary to require

ℓ(π(1)) = 1

^{. This would imply}

that

ε _ℓ

^{isthe identityon}

Θ _• ^{Γ σ}

^{. However,wewill hoose}

ℓ(π(1)) = 1

^{forsimpliity.}

(4) In expliitterms, themap

ε _ℓ

^{is desribed asfollows:}

ε _ℓ : ZΓ → ZΓ _σ ,

X

γ ∈ Γ

a _γ γ = X

γ σ ∈ Γ σ

X

ρ ∈ Γ/Γ σ

a _{γ σ ℓ(ρ)} γ _σ ℓ(ρ) 7→ X

γ σ ∈ Γ σ

X

ρ ∈ Γ/Γ σ

a _{γ σ ℓ(ρ)}

γ _σ ,

where the

a _γ

^{areoeients from}

Z

^{. The map}

ε _ℓ

^{restrits to the identity on}

ZΓ _σ

^{and gives}

an isomorphism of

Z

^{-modules from}

Z[ℓ(ρ)Γ _σ ]

^to

ZΓ _σ

^{for every}

Γ _σ

^-orbit

ℓ(ρ)Γ _σ

^.

Proof (ofthe lemma). Infat, thestatement holdsforanyhainmap

ε ˆ

^intheplaeof

ε ˆ _ℓ

^thatsatises

thefollowing onditions. Theyareeasily heked to holdfor themaps

ε ˆ _ℓ

^.

(1)

ε ˆ

^is

ZΓ _σ

^-linear.

(2) The augmentation

Θ ^Γ ₀ → Z

^{isthe omposition of}

ε ˆ

^{with the}augmentation

Θ ^Γ ₀ ^σ → Z

^.

Then the statement follows from the omparison theorem [25, 2.2.6℄ of fundamental homologial

algebra. In fat, the properties imply that

ε ˆ

^{is a hain map of} resolutions lifting the identity on

Z

^.

An inverse is given bythe anonialinlusion

Θ ^Γ _• ^σ → Θ ^Γ _•

^{,and sine theompositionis unique up to}

hainhomotopyequivalene,it must be homotopi to theidentity.

The group

Γ _σ

^{ats diagonally from the right on}

Θ ^Γ ₁ = ∼ ZΓ ⊗ Z ZΓ

^{, and trivially on}

Z

^{, so we an}

onsider

Θ ^Γ ₁ ⊗ ZΓ σ Z

^{. Denote theommutator quotient map}

Γ σ → (Γ σ ) ^ab ∼ = H 1 (Γ σ )

^by

a 7→ a

^.

Corollary 12. Consider a yle for

H ₁ (Γ _σ ; Z)

^{of the form}

P

i (a _i ⊗ Z b _i ) ⊗ ZΓ σ 1 ∈ Θ ^Γ ₁ ⊗ ZΓ σ Z

^where

a _i , b _i ∈ ZΓ

^{. Assumethat all}

a _i

^and

b _i

^{are elements of}

Γ

^{. The ensuing homology lass isthen given by}

X

i

ε _ℓ (b _i )ε _ℓ (a _i ) ^{− 1} ∈ (Γ _σ ) ^ab .

Bythelinearityofthedesribedmap,thisoversthegeneralaseastheylesoftheformappearing

intheorollarygeneratethesubmoduleofallyles. Notethattheyleonditionon

P

i

(a _i ⊗ ^Z b _i ) ⊗ ZΓ σ 1

says that

P

i

(b _i − a _i ) ⊗ ZΓ σ 1 = 0,

^{whih meansthat}

P

i a _i

^{isequivalentto}

P

i b _i

^modulo

ZΓ _σ

^.

Proof (ofthe orollary). Using lemma10,we apply themap

(ε _ℓ ⊗ Z ε _ℓ ) ⊗ ZΓ σ 1 : (ZΓ ⊗ Z ZΓ) ⊗ ZΓ σ Z −→ (ZΓ _σ ⊗ Z ZΓ _σ ) ⊗ ZΓ σ Z

to get

X (ε _ℓ ⊗ Z ε _ℓ ⊗ ZΓ σ 1)(a _i ⊗ Z b _i ⊗ ZΓ σ 1) = X

(ε _ℓ (a _i ) ⊗ Z ε _ℓ (b _i )) ⊗ ZΓ σ 1.

Denote theaugmentation

ZΓ _σ → Z

^by

ε

^{. As}

a _i ∈ Γ

^{, we have}

ε _ℓ (a _i ) ∈ Γ _σ

^{whihis invertible in}

ZΓ _σ

^,

and

ε(ε _ℓ (a _i )) = 1

^{. Sotheabove termequals}

X 1 ⊗ ^Z ε _ℓ (b _i )(ε _ℓ (a _i )) ^{− 1}

⊗ ZΓ σ ε(ε _ℓ (a _i )) = X

1 ⊗ ^Z ε _ℓ (b _i )(ε _ℓ (a _i )) ^{− 1}

⊗ ZΓ σ 1,

where we take into aount thatthe ation of

ZΓ _σ

^on

ZΓ _σ ⊗ ^Z ZΓ _σ

^{isthe diagonal right ation, and}

that of

ZΓ _σ

^on

Z

^{is thetrivial ation}

a · 1 = ε(a)

^for

a ∈ ZΓ _σ

^{. In bar notation, we thus obtain the}

yle

P ε _ℓ (b _i )(ε _ℓ (a _i )) ^{− 1}

⊗ ZΓ σ 1

^{,whih ismapped to}

X

i

ε _ℓ (b _i )ε _ℓ (a _i ) ^{− 1} ∈ (Γ _σ ) ^ab

by themapdesribed in[7, page 36℄;it iseasy to hekthatan isomorphism

H ₁ (Θ ^G _• ⊗ G Z) ∼ = G ^ab

^is

desribed by

(1 ⊗ g) ⊗ G 1 = [g] ⊗ G 1 7→ g

^{alsointhe ase where}

Θ ^G _•

^{is atedon by}

G

^{fromtheright.}

Moreover, this isomorphism isnatural withrespetto group inlusions.

Another property of the spetral sequene is that a part of it an be heked whenever the ge-

ometry of the fundamental domain and a presentation of

Γ

^{are known. AsFlöge shows, an inspe-}

tion of the omplex

X

^{and the assoiated stabilizer groups and} identiations yields, together with

[1, theorem4.5℄,apresentationof

Γ

^{bymeansofgeneratorsandrelations. Wewillusethe}presentation omputed byFlögefor

m = 5, 6, 10

^{and thatomputed bySwan [23℄ for}

m = 15

^.

Remark 13. Letus lookat the lowterm short exat sequene

0 ^// E _0,1 ^∞ // Γ

^ab

^{ρ //} E ^∞ _1,0 // 0

of the spetral sequene. We have

E ^∞ _1,0 = H ₁ (Γ \ X) = (π ₁ (Γ \ X))

^{ab, and the projetion}

ρ

^{is the}

abelianizationofthe map

Γ → π ₁ (Γ \ X)

^{givenasfollows. Chooseaxedbasepoint}

x ∈ X

^{. For every}

γ ∈ Γ

^{,hooseaontinuouspath in}

X

^from

x

^to

γx

^{. Thisgivesawell-dened loopin}

Γ \ X

^sine

X

^is

ontratible.

The abelianization of

Γ

^{an be} immediately dedued from its presentation. Thus, we an ompute thegroup

E _0,1 ^∞ = E _0,1 ³

^{asthe kernel of the projetion}

ρ

^{andhek this withtheresult obtained from}

detailed analysisofthe

d ²

-dierential.

2.2. The homologyof the nite subgroups in the Bianhi groups.

Inorder to ompute the

E ¹

^{-termof thespetralsequene introdued insetion2.1, we will need the}

isomorphism lasses ofthe homologygroups ofthe stabilizers.

Lemma14(Shwermer/Vogtmann[20℄). Theonlyisomorphismlassesofnitesubgroupsin

PSL ₂ ( O )

are the yli groupsof orders two and three, the trivialgroup, theKlein four-group

D 2 ∼ = Z/2 × Z/2

^,

the symmetri group

S 3

^andthe alternatinggroup

A 4

^.

The homology with trivial

Z

respetively

Z/n

^-oeients,

n = 2

^or

3

^{, of these groups is}

H q (Z/n; Z) ∼ = 8

> <

> :

Z, q = 0, Z/n, q odd, 0, q even, q > 0;

H q (Z/n; Z/n) ∼ = Z/n, q ∈ N ∪ {0};

H q (D 2 ; Z) ∼ = 8

> >

<

> >

:

Z, q = 0,

(Z/2) ^q+3 2 , q odd, (Z/2) ^{q 2} , q even, q > 0;

H q (D 2 ; Z/2) ∼ =(Z/2) ^q+1 H q (D 2 ; Z/3)=0, q > 1;

H q (S 3 ; Z) ∼ = 8

> >

> >

> <

> >

> >

> :

Z, q = 0, Z/2, q ≡ 1 mod 4, 0, q ≡ 2 mod 4, Z/6, q ≡ 3 mod 4, 0, q ≡ 0 mod 4, q > 0;

H q (S 3 ; Z/3) ∼ = 8

> >

> >

> <

> >

> >

> :

Z/3, q = 0, 0, q ≡ 1 mod 4, 0, q ≡ 2 mod 4, Z/3, q ≡ 3 mod 4, Z/3, q ≡ 0 mod 4, q > 0;

H q (S 3 ; Z/2) ∼ =Z/2, q ∈ N ∪ {0};

H q (A 4 ; Z) ∼ = 8

> >

> >

> >

> >

> <

> >

> >

> >

> >

> :

Z, q = 0,

(Z/2) ^k ⊕ Z/3, q = 6k + 1, (Z/2) ^k ⊕ Z/2, q = 6k + 2, (Z/2) ^k ⊕ Z/6, q = 6k + 3, (Z/2) ^k , q = 6k + 4, (Z/2) ^k ⊕ Z/2 ⊕ Z/6, q = 6k + 5, (Z/2) ^k+1 , q = 6k + 6.

H q (A 4 ; Z/2) ∼ = 8

> >

> >

> >

> >

> <

> >

> >

> >

> >

> :

Z/2, q = 0, (Z/2) ^2k , q = 6k + 1, (Z/2) ^2k+1 , q = 6k + 2, (Z/2) ^2k+2 , q = 6k + 3, (Z/2) ^2k+1 , q = 6k + 4, (Z/2) ^2k+2 , q = 6k + 5, (Z/2) ^2k+3 , q = 6k + 6.

H q (A 4 ; Z/3) ∼ =Z/3, q ∈ N ∪ {0}.

Using the UniversalCoeient Theorem, we seethat indegrees

q > 1

^{, the homology withtrivial}

Z/4

^{oeientsisisomorphitothehomologywithtrivial}

Z/2

^{oeientsforthegroupslistedabove.}

Thestabilizersofthe pointsinside

H

^{arenite andheneoftheabovelistedtypes. The} stabilizersof thesingular points areisomorphito

Z ²

^{,whihhas homology}

H _q (Z ² ; Z) ∼ =

8

> <

> :

0, q > 3, Z, q = 2, Z ² , q = 1.

The maps indued on homology by inlusions of thestabilizers determine the

d ¹

-dierentials of the spetralsequene fromsetion2.1.

Observation 15. Thethreeimagesin

H ₂ ( D 2 ; Z /2)

^ofthenon-trivialelement of

H ₂ ( Z /2; Z /2)

^under

themapsinduedbytheinlusionsofthethreeorder-2-subgroups of

D 2

^arelinearlyindependent,but thethree imagesof the non-trivial element of

H 2 ( Z /2; Z /4)

^{are linearlydependent in}

H 2 ( D 2 ; Z /4)

^.

Morepreisely,thereisaanonialbasisfor

H ₂ ( D 2 ; Z/2) ∼ = ( Z/2) ³

^{oming fromtheresolutionfor}

D 2

usedby[20℄,assoiated tothedeomposition

D 2 = ∼ Z/2 × Z/2

^{. Oneheksbydiretalulationthat}

inthis basis,theinlusions ofthe three order-2-subgroups in

D 2

^{induethe images}

8

<

: 0 ,

0

@ 1 0 0

1 A 9

=

;

,

8

<

: 0 ,

0

@ 1 1 1

1 A 9

=

;

,and

8

<

: 0 ,

0

@ 0 0 1

1 A 9

=

;

in

H ₂ ( D 2 ; Z/2)

^;

and inthebasisoming fromthe same resolution usedfor

Z/4

^{oeients these imagesare}

8

<

: 0 ,

0

@ 1 0 0

1 A 9

=

;

,

8

<

: 0 ,

0

@ 1 0 1

1 A 9

=

;

,and

8

<

: 0 ,

0

@ 0 0 1

1 A 9

=

;

in

H ₂ ( D 2 ; Z/4)

^.

The dierene between the ases

Z/2

^and

Z/4

^{omes from the behavior of the kernels of the}

dierential maps.

Lemma16(Shwermer/Vogtmann[20℄). Let

C ∈ { Z }∪{ Z /n : n = 2, 3, 4 }

^{. Considergrouphomology}

withtrivial

C

^{-oeients. Then the following hold.}

(1) Any inlusion

Z/2 → S 3

^{indues an injetion on homology.}

(2) An inlusion

Z/3 → S 3

^{indues an injetion on homology in degrees ongruent to}

3

^or

0 mod 4

^{, and iszero otherwise.}

(3) Any inlusion

Z/2 → D 2

^{indues an injetion on homology in alldegrees.}

(4) An inlusion

Z /3 → A 4

^{indues injetions on homology in all degrees.}

(5) In thease

C ∈ { Z }∪{ Z/n : n = 2, 3 }

^{, aninlusion}

Z/2 → A 4

^{induesinjetionsonhomology}

in degrees greater than

1

^{, and iszero on}

H 1

^.

In the ase

C = Z/4

^{, the same holdsin homology degrees}

q 6 = 2

^.

An inlusion

Z/2 → A 4

^{indues the zero map on}

H ₂ ( − ; Z/4)

^.

Sketh of proof. ShwermerandVogtmann provethisfor

C = Z

^{,andleave ittothereaderinthease}

C = Z/2

^{. Details for the latterase an be found in [19℄. In thefollowing, we are going to give the}

main arguments.

(1) This follows for alloeients fromthefatthat thegroup extension

1 → Z /3 → S 3 → Z /2 → 1

^{hasthe propertythatanyinlusion,}

Z /2 → S 3

^{,omposedwithits}

quotient mapisthe identityon

Z/2

^.

(2) The assertion is trivial for

C ∈ { Z /2, Z /4 }

^{beause then}

H q ( Z /3; C) = 0

^for

q > 1

^by

theUniversalCoeientTheorem. For

C = Z/3

^{,oneomputesthe}Lyndon/Hohshild/Serre spetralsequenewith

Z/3

^{-oeientsassoiatedtotheextension}

1 → Z/3 → S 3 → Z/2 → 1

^.

Its

E ²

^-term

E _p,q ² = H _p (Z/2; H _q (Z/3; Z/3))

^{is onentrated in theolumn}

p = 0

^{;speial are}

has to be taken with the ation of

Z/2

^on

H _q (Z/3; Z/3)

^{. So}

E _p,q ² ∼ = E _p,q ^∞

^{, and the assertion}

follows from a omputation of the map

H _q (Z/3; Z/3) → H ₀ (Z/2; H _q (Z/3; Z/3))

^{, i. e. the}

projetion onto theoinvariants.

(3) Similar to (1), this is an immediate onsequene of thefatthat

1 → Z/2 → D 2 → Z/2 → 1

splits.

(4) Similarto(1)and(3),thisfollowsforalloeientsfromthefatthatanyinlusion

Z/3 → A 4

omposed with the unique quotient map

A 4 → Z /3

^{is an} isomorphism, hene indues an isomorphism inhomology.

(5) Theassertionistrivialfor

C = Z/3

^beausethen

H _q (Z/2; C) = 0

^for

q > 1

^{. For}

C ∈ { Z/2, Z/4 }

^,

one onsiders the fatorization ofthe inlusion

Z/2 → A 4

^as

Z/2 → D 2 → A 4

^{where therst}

map isone out of threepossible inlusions

Z/2 → D 2

^{,denoted by}

α, β, γ

^{. By (3),}

α, β

^and

γ

indue injetions. Furthermore, one onsiders the spetralsequene with

C

^{-oeients asso-}

iated to the extension

1 → D 2 → A 4 → Z/3 → 1

^{. Similar to the aseonsidered in (2), the}

E ²

^-term

E ² _p,q = H p ( Z /3; H q ( D 2 ; C))

^{is onentrated inthe olumn}

p = 0

^{, thus}

E _p,q ² ∼ = E _p,q ^∞

^,

and themap

H _q ( Z /2; C) → H _q ( A 4 ; C)

^{iswritten astheomposition}

H _q (Z/2; C) → H _q ( D 2 ; C) → H ₀ (Z/3; H _q ( D 2 ; C)) ∼ = H q ( A 4 ; C)

where the rst map is

α _∗ , β _∗

^or

γ _∗

^{and the seond one is the projetion onto the}

Z/3

^-

oinvariants. From this, the statement an be diretly dedued for

q 6 = 2

^{. For the ase}

q = 2

^{, denote the generator of}

H ₂ (Z/2; C)

^by

x

^{. The ation of}

Z/3

^on

D 2

^{omes from}

onjugation within

A 4

^{and permutes the three} non-trivial elements. There is an automor- phism

φ

^of

D 2

^{given by the ation of a generator of}

Z/3

^{, suh that}

φ ◦ α = β

^{. Then}

φ _∗ (α _∗ (x)) = (φ ◦ α) _∗ (x) = β _∗ (x)

^and

φ _∗ (β _∗ (x)) = γ _∗ (x)

^{. For}

C = Z /4

^, observation 15 impliesthat

α _∗ (x) = γ _∗ (x) − β _∗ (x) = γ _∗ (x) − (φ ² ) _∗ (γ _∗ (x))

^,andthus

α _∗ (x)

^isinIm

(1 − φ _∗ ) =

Im

(1 − (φ ² ) _∗ )

^{,heneiszerointheoinvariants. Therefore,thesameholdsfor}

β _∗ (x)

^and

γ _∗ (x)

^,

and the assertion follows. For

C = Z/2

^{, one omputes with the help of} observation 15 that

α _∗ (x) 6∈

^Im

(1 − φ _∗ )

^{;thus, thesame holds for}

β _∗ (x)

^and

γ _∗ (x)

^{and theassertion follows.}

2.3. The mass formula for the equivariant Euler harateristi.

We will usetheEuler harateristi to hekthegeometry of thequotient

Γ \ X

^{. Reallthefollowing}

denitions and proposition, whih we inlude for thereader's onveniene.

Denition 17 (Euler harateristi). Suppose

Γ ^′

^{is a} torsion-free group. Then we dene its Euler harateristi as

χ(Γ ^′ ) = X

i

( − 1) ⁱ dim H _i (Γ ^′ ; Q).

Suppose further that

Γ ^′

^{is a} torsion-free subgroup of nite index in a group

Γ

^{. Then we dene the}

Euler harateristi of

Γ

^as

χ(Γ) = χ(Γ ^′ ) [Γ : Γ ^′ ] .

Thisis well-dened beause of [7,IX.6.3℄.

Denition 18 (Equivariant Euler harateristi). Suppose

X

^isa

Γ

^{-omplex suh that}

(1) every isotropy group

Γ _σ

^{is of nite homologial type;}

(2)

X

^{has onlynitely many ellsmod}

Γ

^.

Thenwe dene the

Γ

-equivariant Euler harateristi of

X

^as

χ _Γ (X) := X

σ

( − 1) ^dimσ χ(Γ _σ ),

where

σ

^{runs over the orbit r}epresentatives of ells of

X

^.

Proposition19([7,IX.7.3e'℄). Suppose

X

^isa

Γ

^{-omplexsuhthat}

χ _Γ (X)

^{isdened. If}

Γ

^isvirtually

torsion-free, then

Γ

^{is of nite homologial type and}

χ(Γ) = χ _Γ (X).

Let now

Γ

^{be PSL}

₂ O _Q [ ^√ − m ]

. Then the above proposition applies to

X

^{taken to be Flöge's}

(or still, Mendoza's)

Γ

-equivariant deformation retrat of

H

^{, beause}

Γ

^{is virtually} torsion-free by Selberg'slemma. Using

χ(Γ _σ ) = _card(Γ ¹

σ )

^for

Γ _σ

^{nite,thefatthatthesingularpointshavestabilizer}

Z ²

^{,and the}torsion-freeEuler harateristi

χ(Z ² ) = X

i

( − 1) ⁱ rank _Z (H _i Z ² ) = 1 − 2 + 1 = 0,

we gettheformula

χ(Γ) = X

σ

( − 1) ^dimσ 1 card(Γ _σ ) ,

where

σ

^{runs over theorbit} representativesof ellsof

X

^withnitestabilizers.

Proposition 20. The Euler harateristi

χ(Γ)

^vanishes.

Remark 21. Thus,the formula

0 = X

σ

( − 1) ^dimσ 1 card(Γ _σ ) ,

allows to hek the joint data of the geometry of the fundamental domain, ell stabilizers and ell

identiations.

Proof of proposition20. Denote by

ζ _K

^{the Dedekind}

ζ

^{-funtionassoiatedto thenumbereld}

K := Q √

− m

.Brown[7, below(IX.8.7)℄ dedues thefollowing from Harder'sresult [13,p. 453℄:

χ(SL n ( O K )) = Y n

j=2

ζ K (1 − j),

soespeiallywehave

χ(SL ₂ ( O K )) = ζ _K ( − 1).

^As

Γ

^{isaquotientof}

SL ₂ ( O K )

^{byagroup ofordertwo,}

itfollows [4℄that

χ(Γ) = 2 · χ(SL ₂ ( O K )) = 2 · ζ _K ( − 1).

Using the funtional equation of

ζ _K

^{[17℄ and the fat that}

K

^{has no real embeddings beause it is}

imaginaryquadrati, we get

ζ _K ( − 1) = 0.

Remark 22. One an prove the above proposition without using the Dedekind zetafuntion. This

alternative proof applies to any onite arithmetially dened subgroup

Γ

^of

PSL(2, C)

^{. Let}

Γ ^′

denote a torsion-free subgroup of

Γ

^{of nite index. It is the main theorem of Harder's artile on}

theGauss-Bonnet theorem[13℄ thatthe Euler harateristi of

Γ ^′

^{is itsovolume withrespetto the}

Euler-Poinaréform

µ

^on

H

^,i.e.

χ(Γ ^′ ) = R

Y dµ

^,where

Y

^isafundamentaldomainfortheationof

Γ ^′

on

H

^{. Thisextends the lassial}Gauss-Bonnet theorem fromthetheory ofthe Euler-Poinaré form, see [21, paragraph 3℄ (where the theorem is hidden as the existene assertion of the Euler-Poinaré

measure) to non-oompat but onite disrete subgroups. The measure

µ

^isa fundamental datum assoiatedto thesymmetrispae, withoutreferenetoanydisretegroup. In[21, paragraph3,2a℄ it

isshown that

µ = 0

^{on any}odd-dimensional spae. Sine dim

H = 3

^{,we have}

χ(Γ ^′ ) = χ(Γ) = 0

^.

3. Computations of the integral homology of PSL

2 O _Q [ ^√ − m ]

Throughoutthissetion, theationonthehomologyoeientsistrivialbeausethestabilizersx

theellspointwise. Wemean

Z

^{-oeientswhereverwedonotmentiontheoeients.} Throughout, we labelthesingular pointinthefundamentaldomain by

s

^{and usethenotation}

⊗ σ := ⊗ Z[Γ σ ] .

We write

D 2

^{for the Klein four group,}

S 3

^{for the}permutation group on threeobjets and

A 4

^{for the}

alternating groupon four objets.

We have

Γ =

^PSL

₂ ( O _Q [ ^√ − m ]) =

^PSL

² (Z[ω])

^with

ω := √

− m

^{in the ases}

m = 5, 6, 10, 13

^{. The}

oordinates in hyperboli spae of the verties of the fundamental domains have been omputed by

Bianhi [6℄. There, they are listed up to omplex onjugation for

m = 5, 6, 15

^{; and for}

m = 10, 13

^,

thereader hasto divideout thereetion alledriessioneimpropria byBianhi.

o ^′ j ^′

v ^′′

y ^′′

z ^′ a ^′′

b ^′′

x ^′ u ^′

a ^′ b ^′ c ^′′

v ^′ y ^′ c ^′

o e

f g h

j s t

u v

w x

y z

a b c

Figure 2: The

fundamentaldo-

mainfor

m = 13

3.1.

m = 13

^{. Wemake the following denitions.}

A := ±

„ 9 7ω ω − 10

«

, B := ±

„ − 2 − ω 2 − ω 4 2 + 1ω

«

, C := ±

„ − 1 − ω 8 − ω 3 1 + 2ω

« ,

D := ±

„ 5 2ω ω − 5

«

, E := ±

„ − ω 6

2 ω

«

, J := ±

„ 1

− 1

« ,

S := ±

„ − 1

1 1

«

, K := ±

„ 11 + 4ω − 17 + 7ω

− 8 + ω − 10 − 3ω

«

, M := ±

„ 4 − 2ω 12 + ω 4 + ω − 4 + 2ω

« ,

U := ±

„ 1 ω 1

«

, V := ±

„ − ω 6 − ω 2 2 + ω

«

, W := ±

„ 14 − ω 13 + 6ω 2ω − 12 + ω

« ,

P := V ⁻¹ D, T := P ⁻¹ S ² , R := T U ⁻¹ S ² U.

Verties with the same letter in the fundamental domain displayed in

gure 2 are identied by the ation of

Γ

^{, for instane,}

y

^{is identied}

with

y ^′

^and

y ^′′

^{and so on. This yields relations between the matri-}

es in the same way as shown by [10℄. Amongst these relations, we

will use

T = CKCA(CKC) ^{− 1}

^,

V ^{− 1} = CAC ^{− 1} M

^and

S ² = BS ^{− 1} BS

in our further alulations, in partiular in the omputation of the

d ²

^-

dierential. Note that the 2-ell

(y ^′′ , a ^′′ , u ^′ , x ^′ , b ^′′ , v ^′′ )

^{is identied with the 2-}

ell

(y, a, u, x, b, v)

^{, hene only one of them an be in the} fundamental do- main. The matrix

U

^{ats as a vertial} translation by

− ω

^{. F}urthermore, we will use the identiations

C · x ^′ = x

^,

U · j ^′ = j

^,

C · y ^′ = y

^and

K · z = z ^′

^.

Amongsttheedgeidentiations,wewilluse

CAC ^{− 1} · (c, z) = (c ^′ , z)

^,

V ^{− 1} · (s, c) = (s, c ^′ ), CAC ^{− 1} · (b, x) = (b ^′ , x)

^,

V ^{− 1} · (b, v) = (b ^′ , v ^′ )

^,

P · (y, v) = (y ^′ , v ^′ )

^,

S ² · (a, y) = (a ^′ , y ^′ )

^,and

B · (a, u) = (a ^′ , u)

^{. There areseventeenorbitsof verties,whih have thefollowing} stabilizers.

Γ o = h J | J ² = 1 i ∼ = Z/2,

Γ a = h S ⁻¹ BS | (S ⁻¹ BS) ² = 1 i ∼ = Z/2, Γ b = Γ c = h M | M ² = 1 i ∼ = Z/2,

Γ u = h B | B ² = 1 i ∼ = Z/2, Γ v = h D | D ² = 1 i ∼ = Z/2,

Γ f = h D, E | D ² = E ² = (DE) ² = 1 i ∼ = D 2 ,

Γ h = h E, AU ⁻¹ JU | E ² = (AU ⁻¹ JU) ² = (EAU ⁻¹ JU) ² = 1 i ∼ = D 2 , Γ e = h A, U ⁻¹ JU | A ³ = (U ⁻¹ JU) ² = (AU ⁻¹ JU) ² = 1 i ∼ = S 3 , Γ g = h J, T | J ² = T ³ = (JT ) ² = 1 i ∼ = S 3 ,

Γ t = h R, U ⁻¹ SU | R ² = (U ⁻¹ SU) ³ = (RU ⁻¹ SU) ² = 1 i ∼ = S 3 , Γ w = h B, S | B ² = S ³ = (BS) ² = 1 i ∼ = S ³ ,

Γ j = h S | S ³ = 1 i ∼ = Z/3,

Γ x = Γ z = h CAC ⁻¹ | (CAC ⁻¹ ) ³ = 1 i ∼ = Z/3, Γ y = h T | T ³ = 1 i ∼ = Z/3,

Γ s = h V, W | V W = W V i ∼ = Z ² .

There aretwenty-eight orbitsof edges.

Theedge stabilizersisomorphito

Z/3

^{aregiven on the hosen}representatives as

Γ _(e,x ′ ) = h A | A ³ = 1 i ∼ = Z/3,

Γ _(x,z) = h CAC ⁻¹ | (CAC ⁻¹ ) ³ = 1 i ∼ = Z/3, Γ (g,y) = h T | T ³ = 1 i ∼ = Z/3,

Γ _(j,w) = h S | S ³ = 1 i ∼ = Z/3,

Γ _(t,j ′ ) = h U ⁻¹ SU | (U ⁻¹ SU ) ³ = 1 i ∼ = Z/3,

Γ _(y ′ ,z ′ ) = h KCA(KC) ⁻¹ | (KCA(KC) ⁻¹ ) ³ = 1 i ∼ = Z/3,