The Farrell--Tate and Bredon homology for PSL_4(Z) via cell subdivisions

VIA CELL SUBDIVISIONS

ANHTUANBUI,ALEXANDERD.RAHMANDMATTHIASWENDT

Abstra t. Weprovidesomenew omputationsofFarrellTateandBredon( o)homologyfor

arith-meti groups. For al ulationsofFarrellTateorBredonhomology,oneneeds ell omplexeswhere

ellstabilizersxtheir ells pointwise. Weprovidetwoalgorithms omputingane ient

subdi-visionofa omplextoa hievethisrigidityproperty. Applyingthesealgorithmsto available ell

omplexesfor

PSL

4

(Z)

provides omputationsofFarrellTate ohomologyforsmallprimesaswell astheBredonhomologyforthe lassifyingspa esofpropera tionswith oe ientsinthe omplex

representationring.

1. Introdu tion

Understandingthestru tureofthe ohomologyofarithmeti groupsisaveryimportantproblem

with relationsto number theory and various K-theoreti areas. Expli it ohomology omputations

usuallypro eed viathestudyof thea tionsofthearithmeti groupsontheirasso iatedsymmetri

spa es, and re ent years haveseen several advan es in algorithmi omputation of equivariant ell

stru tures for these a tions. To approa h omputations of FarrellTate and Bredon ( o)homology

of arithmeti groups, oneneeds ell omplexes having a rigidity property: Cell stabilizers must x

their ells pointwise. The known algorithms (using Voronoi de ompositions and su h te hniques,

f. e.g. [7,9℄) donot provide omplexes with this rigidity property, and both for the omputation

of FarrellTate ohomology (resp. the torsion at small prime numbers in group ohomology) of

arithmeti groupsaswellasforthe omputationofBredonhomology,thisla kofrigid ell omplexes

onstitutesasigni antbottlene k.

In theory, it is always possible to obtain this rigidity property via the bary entri subdivision.

However,the bary entri subdivision ofan

n

-dimensional ell omplex anmultiplythe numberof ellsby

(n + 1)!

andthus easily letthe memorysta koverow. We provide twoalgorithms, alled RigidFa etsSubdivision, f. Se tion2,andVirtuallySimpli ialSubdivision, f. Se tion3,aswellasa

ombinationofthem(HybridSubdivision, f. Se tion4)whi hsubdivide ell omplexesforarithmeti

groupssu hthat stabilizersxtheir ellspointwise,but onlyleadtoa ontrolledin rease(interms

of sizes of stabilizer groups) in thenumber of ells,avoiding anexplosion of thedata volume. An

implementationofthealgorithms, f.[5℄,showsthat aseslike

PSL

4

(Z)

anee tivelybetreatedwith it,using ommonlyavailablema hine resour es. Forthesakeof omparison, bary entri subdivison

appliedtothe ell omplexfor

PSL

4

(Z)

from[6℄wouldprodu e3540timesasmanytop-dimensional ellsasRigidFa etsSubdivisiondoesseeTable1.

1.1. Computations of FarrellTate ohomology. FarrellTate ohomology isa modi ationof

ohomologyofarithmeti groupswhi hisparti ularlysuitabletoinvestigatetorsionrelatedtonite

subgroups (in parti ular, the torsion in ohomologi al degrees abovethe virtual ohomologi al

di-mension). Whiletheknown ell omplexesforarithmeti groups andealverywellwiththerational

ohomologyandtorsionatprimeswhi hdonotdivideordersofnitesubgroups, omputationswith

these omplexes run into serious trouble for small prime numbers be ause the dierentials in the

relevantspe tralsequen earetoo ompli atedto evaluate. Thereisasuitablenewte hnique alled

torsionsub omplexredu tion, f.[14℄,whi hprodu essigni antlysmaller ell omplexesand

there-foresimpliestheequivariantspe tralsequen e al ulations. Toapply this simpli ation, however,

oneneeds ell omplexes withthe abovementionedrigidityproperty. ApplyingRigid Fa ets

Subdi-vision to a ell omplexfor

PSL

4

(Z)

, we have omputed the FarrellTate ohomology of

PSL

4

(Z)

, at the primes

3

and

5

. These results an be found in Theorem 10 and Proposition 14. Sin e the

Date:3rdO tober2018.

2010Mathemati sSubje tClassi ation. 11F75.

Table 1. Numbers of ells in the individual dimensions of the studied non-rigid

fundamental domain

X

for PSL

4

(Z)

from [6℄, its rigid fa ets subdivsion RFS

(X)

, itsvirtually simpli ial subdivision VSS

(X)

, its hybrid subdivision HyS

(X)

and its bary entri subdivisionBCS

(X)

. Dimension 0 1 2 3 4 5 6 #Orbitsin

X

2 2 2 4 4 3 1 #Cellsin

X

304 1416 2040 1224 332 36 1 #OrbitsinRFS

(X)

17 213 1234 3025 3103 1117 1 #CellsinRFS

(X)

4153 62592 268440 472272 370272 108096 96 #OrbitsinVSS

(X)

17 219 1508 5082 8456 6720 2040 #CellsinVSS

(X)

4153 71952 390936 974304 1238688 783360 195840 #OrbitsinHyS

(X)

17 213 1245 3095 3214 1180 12 #CellsinHyS

(X)

4153 62592 271416 483504 383520 114144 1152 #CellsinBCS

(X)

5353 110352 644136 1658304 2138688 1359360 339840

omputation pro eeds through a omplete des ription of the redu ed torsion sub omplex, we an

omputethetorsionabovethevirtual ohomologi aldimensioninalldegrees.

In the ases whi h are ee tively of rank one (

5

-torsion in

PSL

4

(Z)

), we an he k the results ofthe ohomology omputationusing torsionsub omplexredu tionby omparingto a omputation

usingBrown'sformula.

1.2. Computations of Bredon homology. For any group

G

, Baum and Connes introdu ed a mapfrom theequivariant

K

-homologyof

G

tothe

K

-theoryoftheredu ed

C

∗

-algebraof

G

, alled the assembly map. For many lasses of groups, it has been proven that the assembly map is an

isomorphism; and the BaumConnes onje ture laims that it is an isomorphism for all nitely

presented groups

G

( ounter-examples have been found only for stronger versions of the Baum Connes onje ture). The assembly map is known to be inje tive for arithmeti groups. For an

overviewonthe onje ture,seethemonograph[13℄.

Thegeometri -topologi alside of Baum and Connes'assembly map, namely the equivariant

K

-homology, anbedetermined usinganAtiyahHirzebru h spe tral sequen ewith

E

2

-page givenby theBredonhomology

H

Fin

n

(

E

G; R

C

)

of the lassifyingspa eE

G

for propera tionswith oe ients in the omplexrepresentationring

R

C

and withrespe t to thesystem

Fin

of nite subgroupsof

G

. ThisBredonhomology anbe omputedexpli itly,asdes ribedbySan hez-Gar ia[18,19℄.

WhileforCoxetergroupswithasmallsystemofgenerators[19℄andarithmeti groupsofrank

2

[15℄, general formulae for the equivariant

K

-homology have been established, the only known higher-rank ase to date is the example

SL

3

(Z)

in [18℄. Although there are by now onsiderably more arithmeti groupsforwhi h ell omplexeshavebeenworkedout[6,7,9℄,nofurther omputationsof

Bredonhomology

H

Fin

n

(

E

G; R

C

)

havebeendonesin e2008be ausetherelevant ell omplexes fail tohavetherigiditypropertyrequiredforSan hez-Gar ia'smethod. Wedis ussanexpli itexample,

f. Se tion 10, demonstrating that therigiditypropertyis essentialfor the omputation of Bredon

homologyand annotbe ir umventedbyadierentmethod.

Applyingrigidfa etssubdivision tothe ell omplexforE

PSL

4

(Z)

from[6℄,weobtain

H

Fin

n

(

E

PSL

4

(Z); R

C

) ∼

=































0,

n > 4,

Z,

n = 3,

0,

n = 2,

Z

4

_,

_{n = 1,}

Z

25

_{⊕ Z/2, n = 0.}

Thisis onsistentwiththerationalhomologyofB

PSL

4

(Z)

asinferredfrom theresultsof [6℄. As further onsisten y he ks, the authors havepaid attention that the homology of the ell omplex

remains un hangedunder our implementation of rigid fa ets subdivision, and that theequivariant

Organization of the paper. InSe tion 2,weprovidetheRigidFa etsSubdivisionalgorithm. In

Se tion 3, we provide the Virtually Simpli ial Subdivision algorithm. In Se tion 8, we apply the

RigidFa etsSubdivisionalgorithm toa

PSL

4

(Z)

, and omparetheresultwitha omputationusing Brown's onjuga y lasses ell omplex. FinallyinSe tion10,weprovidea ounterexampleinorderto

ontradi tthepossibilityto omputetheBredonhomologyfromanarbitrarynon-rigid ell omplex.

A knowledgements. WearegratefulforsupportbyGaborWiese'sFondsNationaldelaRe her he

Luxembourg grants (INTER/DFG/FNR/12/10/COMFGREP and AMFOR), whi h did fa ilitate

meetings forthis proje tviavisitsto Université duLuxembourg by therst author(thri e for one

month)andthethirdauthor. Therstauthorwasfunded byVietnamNationalUniversity-HoChi

MinhCity(VNU-HCM)undergrantnumberC2018-18-02.WewouldliketothankGrahamEllisfor

havingsupportedthedevelopmentoftherstimplementationofouralgorithmstheTorsion

Sub- omplexesSubpa kage forhis Homologi al AlgebraProgramming (HAP)pa kagein GAP.

Further thanksgoto SebastianS hönnenbe kforhavingprovided us ell omplexesfor ongruen e

subgroupsin

SL

3

(Z)

, used forben hmarkingpurposesin this paper,and espe iallyto Mathieu Du-tourSikiri¢forhavingprovidedthe ell omplexesfor

SL

3

(Z)

,Sp

4

(Z)

and

PSL

4

(Z)

inHAP. Thisarti leisdedi atedtothememoryofAledEllis.

2. The rigid fa etssubdivision algorithm

Inthis se tion, we dis uss therigid fa ets subdivision algorithm whi h rigidies equivariant ell

omplexes. The oreofthemethod is Algorithm2,whi h isexpe tedto run inreasonabletimefor

input omingfrom ell omplexes for arithmeti groups. The keyfa t whi h guarantees that rigid

fa etssubdivisionworks,isLemma7below.

Algorithm1Subdividetogetstabilizerswhi hxtheir ellspointwise

Input: An

n

-dimensional

Γ

-equivariantCW- omplex

X

withnite ellstabilizersandametri as inRemark2.

Output: Arigidi ationof

X

(thatis,anequivalent

Γ

- ell omplexonwhi hea hstabilizerxes its ellpointwise).

for

m

runningfrom

1

to

n

do

for

σ

runningthroughlifts of

m

- ellsin

Γ

\X

(m)

do

if

σ

isnotrigidthen

UseAlgorithm2or3tosubdivide

σ

intoapartition

P

,whi hisaunionofrigid

m

- ells, disjointuptoboundaries,withafundamentaldomain

F

forthe

Γ

σ

-a tionon

P

. Runthroughallthe

(m + 1)

- ells;iftheirboundaries ontain

σ

,

thenrepla e

σ

byitspartition

P

. Repla ethe ell

σ

by

F

in

Γ

\X

(m)

.

endif

endfor

endfor

Denition1. Followingthe notationin[4℄,weuse theterm

Γ

-equivariantCW- omplex,orsimply

Γ

- ell omplex,to meanaCW- omplex

X

onwhi hadis retegroup

Γ

a ts ellularly,i.e.,in su h a waythatthea tionindu esapermutationofthe ellsof

X

. Wesaythe ell omplexisrigid ifea h elementinthestabilizerofany ellxesthe ellpointwise.

Remark 2. The algorithm produ ing the subdivision of the

Γ

-equivariant CW- omplex

X

only modies ombinatorialdata,basedonthebary entri subdivisonofindividual ells. Werequire

X

to omewithageometri realization,equippedwithametri su hthatea hofthe ellsof

X

is onvex, therestri tionofthemetri toea h ellisCAT(0)andthe ellstabilizersa tbyCAT(0)-isometries.

Wearenotrequiring thatthe metri isCAT(0)onthe wholeCW- omplex. Note howeverthatthe

Denition 3. A rigidi ation

X

ˆ

of a

Γ

- ell omplex

X

isarigid

Γ

- ell omplex

X

ˆ

withthe same underlyingtopologi alspa eas

X

. Themappassingthroughtheunderlyingtopologi alspa eisthen a

Γ

-equivarianthomeomorphism

X → X

ˆ

of

Γ

-spa es. Notethat a

Γ

-equivarianthomeomorphismof

Γ

-spa esdoesnotneedtopreserveexisting ellstru ture,so

X

ˆ

isallowedtohavemore ellsthan

X

.

Theoutershelloftherigidfa etssubdivisionisAlgorithm1,whi hsubdivides(wheneverne essary)

representativesof ellorbitsusingAlgorithm2,respe tivelyAlgorithm3.

Proposition 4. Let

Γ

beadis retegroup,andlet

X

bea

Γ

-equivariantCW- omplex havingnitely many

Γ

-orbitsandnite ell stabilizers. Assumefurthermorethat

X

isequippedwith ametri asin Remark2 . Then Algorithm1ndsarigidi ation of

X

(with respe ttothe

Γ

-a tion). Itterminates innite time.

Proof. Thekeystepof thealgorithm is provedbyLemma 7 below; the restis aroutineindu tion.

Lemma7is thepointwhere the onvexityand isometryrequirementsare needed. Bytheniteness

assumptionsfororbitsand ellstabilizers,theloopsarealldeterministi overniteindexsets. Ea h

operationinsidethemtakesnitetime, f. Corollary8,when e the laim.



Observation 5. Theouter shell Algorithm 1 an be used with any subdivision algorithm for the

ells. Inparti ular,repla ing theuseofAlgorithm2bythebary entri subdivision inAlgorithm1,

the laims of Proposition 4 still hold. However, as mentioned in the introdu tion, the reason for

developingAlgorithm2istoredu e theblow-upin thenumberof ells,so astomakethealgorithm

pra ti allyappli ableto ell omplexesforhigher-rankarithmeti groups.

Wenowdis ussthea tualsubdivisionto rigidify ells,Algorithm2.

Algorithm2RigidFa etsSubdivision

Input: An

m

- ell

σ

with stabilizer group

Γ

σ

, with rigid fa es, equipped with a metri as in Remark2.

Output: A

Γ

σ

-equivariantsetofrigid

m

- ells,disjointuptoboundaries, onstitutingapartition

P

of

σ

, togetherwith a ontra tiblefundamental domain

F

(a single

m

- ell) forthea tionof

Γ

σ

on

P

.

•

Sortthe

(m − 1)

-fa esof

σ

intoorbits

{{g

jt

σ

j

}

t

}

j

underthea tionof

Γ

σ

,where

j

isindexing theorbitsand

t

isindexingthe ellsinside ea horbit.

•

Let

T

bethelist ontainingtheelement

g

11

σ

1

. while

#T < #{

orbitsof

(m − 1)

-fa esof

σ}

do

Chooseone ell

τ ∈ {g

jt

σ

j

}

in thenextorbit

{g

jt

σ

j

}

notyet representedin

T

, if theunionof

τ

withthe ellsin

T

hasvanishingnaiveEuler hara teristi then

if

{τ } ∪ T

is ontra tible(we anuseEllis'method [10℄to he kthis)then Constru ttheboundary

S

of

{τ } ∪ T

.

if

S

hasthenaiveEuler hara teristi ofan

(m − 2)

-spherethen if

#T < #{

orbitsof

(m − 1)

-fa esof

σ}

-1then

Addthe hosen ell

τ

to

T

. else

if

S

is simply onne tedthenAddthe hosen ell

τ

to

T

. endif endif endif endif endif endwhile

•

UseAlgorithm3to onstru tthe

m

- ell

F := |

S

τ ∈T

onvexenvelope(

τ

,bary enter(

σ

))|.

•

Let

Γ

pw

σ

bethesubgroupof

Γ

σ

whi hxesthe ell

σ

pointwise.

•

Then

P :=

S

16t6|Γ

σ

/Γ

pw

σ

|

g

1t

F

isthedesiredpartition of

σ

.

Figure 1. From left to right: 1) A full square with natural ell stru ture and

a tion by its stabilizer mirroring it onto itself. 2) Bary entri subdivision of the

square. 3) Rigid fa et subdivision of the square with respe t to the mirroring

a -tion. 4)VirtuallySimpli ialSubdivision ofthesquarewithrespe ttothemirroring

a tion;note that the fundamental domain

F

anbe hosenarbitrarilyby sele ting one ellamongea hofthepairs

{a

1

, a

2

}

,

{b

1

, b

2

}

and

{c

1

, c

2

}

.

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

c

2

c

1

b

2

a

2

b

1

a

1

Remark6. Essentially,Algorithm2produ esa onvexunionof ellsofthebary entri subdivision

whi h isafundamental domainfor the

Γ

σ

-a tion. Theslight ompli ationsarisefrom thefa t that wedonota tuallywantto omputethefullbary entri subdivision,togain omputationalfeasibility.

Lemma7(RigidFa etsLemma). Let

σ

bea ell (with stabilizer

Γ

σ

), whosefa es are allrigidand whi h is equippedwith a metri asin Remark 2. Let

Γ

pw

σ

bethe subgroupof

Γ

σ

whi h xes the ell

σ

pointwise. Then there is a fundamental domain

F

for the a tion of

Γ

σ

/Γ

pw

σ

on

σ

su h that

σ

is tessellatedby

|Γ

σ

/Γ

pw

σ

|

opiesof

F

.

Proof. Firstwehaveto he kthatthestatementiswelldenedinthesensethat

Γ

σ

/Γ

pw

σ

isagroup. Thisisthe asebe auseforall

g

in

Γ

σ

,forall

γ

in

Γ

pw

σ

,forall

x

in

σ

wehave

(g

−1

_{γg)x = g}

−1

_{(γ(gx)) =}

g

−1

_{(gx) = x.}

Therefore,asthekernelofthea tionof

Γ

σ

on

σ

,

Γ

pw

σ

is anormalsubgroup;andthere isashortexa tsequen eofgroups,

1 → Γ

pw

σ

→ Γ

σ

→ Γ

σ

/Γ

pw

σ

→ 1,

whi hmakesourstatementwelldened.

Supposethat

α

isoneofthefa etsof

σ

. Wearegoingtoprovethatthesizeoftheorbitof

α

,under thea tionof

Γ

σ

onthesetof fa etsof

σ

, is

|Γ

σ

/Γ

pw

σ

|

. Let

Γ

α

bethestabilizerof

α

. We laimthat

Γ

α

∩ Γ

σ

= Γ

pw

σ

. Thea tion of

Γ

σ

onthe ompa t set

closure(σ)

isbyhomeomorphisms;therefore, any element of

Γ

σ

xing

σ

pointwisealsoxes theboundary

∂σ

pointwise. Hen e

Γ

α

∩ Γ

σ

⊃ Γ

pw

σ

. Ontheotherhand,let

g ∈ Γ

α

∩ Γ

σ

. Thenbyassumptionontherigidityofthefa ets,

g

xesthe ell

α

pointwise. Sin ethe ell

σ

is onvexandthegroupa tsbyCAT(0)-isometries,thebary enterof

σ

preservesitsdistan estotheboundary

∂σ

underthea tionof

Γ

σ

on

σ

,andhen eremainsxed. As afurther onsequen eoftheCAT(0)isometry,thexedpointsetof

g

extends,bypreservationofthe distan es,fromthe onvexenvelopeof

α

andthebary enterof

σ

to thewhole ell

σ

. Hen e,

g

isan elementof

Γ

pw

σ

. Thus,we an on ludethat

Γ

α

∩ Γ

σ

= Γ

pw

σ

. When e,thesizeoftheorbitof

α

under thea tionof

Γ

σ

is

|Γ

σ

/Γ

pw

σ

|

. Furthermore,from

Γ

α

∩ Γ

σ

= Γ

pw

σ

,weseethat

Γ

σ

/Γ

pw

σ

a tsfreelyonthesetoffa etsof

σ

. So,we antakeonearbitraryrepresentative

α

k

forea horbitof fa ets, to uniteto afundamental domain for

Γ

σ

/Γ

pw

σ

on theset offa etsof

σ

. Takingthe onvexenvelope

e

k

of

α

k

and thebary enterof

σ

, weget afundamental domain

F :=

S

k

e

k

for

Γ

σ

/Γ

pw

σ

on

σ

. Bytheaboveorbit size al ulation,it

yieldsthedesiredtessellation.



Corollary 8. Algorithm 2 terminates after nitely many steps and produ es a rigid subdivision of

the ell

σ

.

Proof. Theexisten eofthefundamentaldomain

F

forthea tion

σ

isguaranteedbyLemma7. The ontra tibilityof thefundamental domain

T

andthesimply onne tednessofitsboundary

S

ensure thatmergingtheunionof ells

S

τ ∈T

onvexenvelope(

τ

,bary enter(

σ

))

intoone ell

F

anberealizedwiththeboundary onstru tion

∂F =

S

τ ∈T

τ ∪

S

s∈S

e(s),

where

e(s)

Algorithm3Constru tingtheunionof the onvexenvelopes

Input: Alist

T

of

j

- ells, onne tedbyadja en y,

andan

m

- ell

σ

su hthatall

τ ∈ T

arefa esof

σ

.

Output: A

(j + 1)

- ell

F

whi h hasthesameunderlying topologi alspa eas

S

τ ∈T

onvexenvelope(

τ

,bary enter(

σ

)).

Re ordthebary enterof

σ

asanewvertex,withstabilizer

Γ

σ

. Enumeratetheniteset

S := {ρ is (j − 1)

- ell

| ∃! τ ∈ T : ρ ∈ ∂τ }

.

Then

S

ontainsall the

(j − 1)

-fa es

ρ

of all

τ ∈ T

su h that

ρ

isnot a ommon fa e of any two

j

- ells

τ

1

, τ

2

∈ T

.

•

Forea h

s ∈ S

,takethe onvexenvelope

e(s)

of

s

andthebary enterof

σ

.

•

Re ord

e(s)

asanoriented

j

- ell,withboundary

∂e(s) = {s} ∪

[

ε∈∂s

{

onvexenvelope

(ε, barycenter(σ))}.

Forthestabilizers,were ord

Γ

e(s)

= Γ

s

∩ Γ

σ

and

Γ

onvexenvelope

(ε,barycenter(σ))

= Γ

ε

∩ Γ

σ

.

The boundary of the

(j + 1)

- ell

F

onsists of all the

j

-fa es

τ ∈ T

and

e(s)

(for all

s ∈ S

). Here,wehavetotake areofwhi hofthenewly onstru ted ells

e(s)

areonthesame

Γ

σ

-orbit. In ordertode idethis,we makeuse oftheir ommon vertex,the bary enter of

σ

:

•

Identify theorbits

e(s

1

)

and

e(s

2

)

ifandonlyif

∃ γ ∈ Γ

σ

: γs

1

= s

2

.

•

Attributearbitraryorientationsto

Γ

σ

-representativesof thenew ells

e(s)

, andspreadthem ontheir

Γ

σ

-orbitusingtheaboveidenti ations.

•

Returnthe

(j + 1)

- ell

F

withboundary

∂F =

[

τ ∈T

τ ∪

[

s∈S

e(s)

andstabilizer

Γ

σ

∩

T

τ ∈T

Γ

τ

,wheretheorbitsaresubje ttotheabovespe iedidenti ations.

3. VirtuallySimpli ialSubdivision

As a ompromise between Rigid Fa ets Subdivision (RFS) and Bary entri Subdivision (BCS),

we anmakeasimpli ationofRFS,whi hweshall allVirtuallySimpli ialSubdivision(VSS)and

detailinAlgorithm3below. Fora ell omplex

X

,VSS

(X)

isarenementofRFS

(X)

,andBCS

(X)

isafurther renementofVSS

(X)

,asillustratedinFigure 1.

Algorithm4VirtuallySimpli ialSubdivision

Input: An

m

- ell

σ

with stabilizer group

Γ

σ

, with rigid fa es, equipped with a metri as in Remark2.

Output: A

Γ

σ

-equivariantset ofrigid

m

- ells,disjointuptoboundaries, onsituting apartition

P

of

σ

, togetherwithafundamentaldomain

F

(aset of

m

- ells)forthea tionof

Γ

σ

on

P

.

•

Sortthe

(m − 1)

-fa esof

σ

intoorbits

{{g

jt

σ

j

}

t

}

j

underthea tionof

Γ

σ

,where

j

isindexing theorbitsand

t

isindexingthe ellsinside ea horbit.

•

UseAlgorithm3to onstru tthe

m

- ells

F

j

:=

onvexenvelope(

g

j1

σ

j

,bary enter(

σ

)),where

j

runs throughallorbits.

•

Let

Γ

pw

σ

bethesubgroupof

Γ

σ

whi hxesthe ell

σ

pointwise.

•

Then

P :=

S

j

S

16t6|Γ

σ

/Γ

pw

σ

|

g

jt

F

j

isthedesiredpartitionof

σ

.

•

Return

P

andtheset

F := {F

j

}

j

.

Corollary 9. Algorithm 3 terminates after nitely many steps and produ es a rigid subdivision of

the ell

σ

.

Proof. Theexisten eofthefundamental domain

F

forthea tion

σ

isguaranteedbyLemma 7. We donotneedanytopologi alpropertiesfor

F

,be auseits ellsalreadyhavethestru ture onstru ted

Table 2. Timespentonasinglepro essorforsubdividingavailable ell omplexes.

Arithmeti group

SL

3

(Z)

Γ

1

in

SL

3

(Z)

Γ

2

in

SL

3

(Z)

Sp

4

(Z)

PSL

4

(Z)

RigidFa etsSubdivision 8s 17s 26s 24s 69minutes

VirtuallySimpli ialSubdivision 9s 22s 33s 43s 111minutes

HybridSubdivision 9s 20s 31s 33s 33minutes

Bary entri Subdivision 13s 56s 104s 296s ?

4. Hybrid Subdivision

Thebulkofthepro essingtimeofRigid Fa etsSubdivision (RFS) andVirtuallySimpli ial

Sub-division (VSS) is being spent on the top-dimensional ells. We ansavea onsiderable amount of

pro essing timeas ompared to RFS ifwe donotmerge thetop-dimensional ells. And we ando

that by applying RFS until we are in (top minus 1) dimensions, and then a hievethe subdivision

using VSSonthe top-dimensional ells. Wewill allthis pro ess HybridSubdivision (HyS). It is

fasterthan bothRFS andVSSon large ell omplexes, and itprodu es onsiderablyless ellsthan

VSS.WenotehoweverthattheVSSpro essonthetop-dimensional ellsaddsafewlow-dimensional

ellswhen onne tingto thebary enter,so forinstan e inTable5, thelow-dimensionalnumbersof

ellsareslightlyhigherforHyS thanforRFS,ratherthanbeingequal.

5. Comparison of the various subdivision algorithms

As illustratedin Figure 1, all of the ells onstru tedby Rigid Fa etsSubdivision are also

on-stru tedbybary entri subdivision;andusually,RigidFa etsSubdivisionis oarserthanbary entri

subdivision. However,theworst- ase omplexityofRigidFa etsSubdivision isnotbetterthanthat

of bary entri subdivision: If

X

is a single

n

-simplex with the natural permutation a tion of the symmetri group

Σ

n+1

a ting on theverti es, then any rigidi ation will need to produ e at least

(n + 1)! = #Σ

n+1

top-dimensional ellsfor

X

. However,the pointisthat the average ell omplex forinteresting arithmeti groupshas many ofits ellsalreadyalmost rigid and, onlyveryfew with

maximallypossiblestabilizer. Therefore,the omplexityforthe asesofinterestissigni antlybetter

thanthat ofthe bary entri subdivision,aseviden ed byTable1. As wesee from Figure1, Rigid

Fa etsSubdivision isaminimalrigidi ation onlyforthetop-dimensional ells;inlowerdimensions

it reatessomesuperuous ellsatthe bary enters offa ets, and ells onne tingthosebary enters

withthe onstru tedfundamental domainfortheboundaryof thesubdivided ell.

InTable1,notethatwedonothavethenumbersoforbitsforthebary entri subdivision,be ause

thelatterwasnot onstru ted;onlythenumbersof ellshavebeen al ulated.Insu ha onstru tion,

theboundariesofthe

4

- ellswould spana

1658304 × 2138688

-matrix,and eventhoughthatmatrix is quite sparse, the authors havenot tried to store that amount of information on a ma hine. In

retrospe tivehowever, onsideringthattheamountof ellsthathavebeen onstru tedwithVirtually

Simpli ial Subdivision omes lose to the amount of ells expe ted for bary entri subdivision, it

turnsout thatthis issuewouldnotpreventapplyingbary entri subdivision(whi h inthe endwas

notdone,be auseTable2wasinterpretedbytheauthorsasanexponentialgrowthoftheruntime).

In ontrast,forSL

3

(Z)

,thebary entri subdivisionisalmostrea hed(seeTable3): Indimensions

0

and

1

,thesame ellstru tureisobtainedwithbothsubdivisionmethods, theonlydieren ebeing that thefour tetrahedrawhi hrepresentthe top-dimensional ells,aremergedinto onepolyhedron

when passing to the rigid fa ets subdivision, along three triangles of trivialstabilizers. Note also

that Soulé didin [22℄ arry outthe bary entri subdivision byhand. This means forthe

2

-torsion sub omplexofSL

3

(Z)

,whi hwewilldis ussinSe tion6,thatwe anextra titequivalentlybyhand from[22℄,orbyma hineafterrigidfa etssubdivision.

Table 3. Numbers of ells in the individual dimensions, of the studied non-rigid

fundamentaldomain

X

forSL

3

(Z)

, itsrigidfa etssubdivsionRFS

(X)

,itsvirtually simpli ial subdivision VSS

(X)

, its hybrid subdivision HyS

(X)

and its bary entri subdivision BCS

(X)

(thelatterthree oin ideforSL

3

(Z)

).

Dimension 0 1 2 3

# Orbitsin

X

1 1 2 1

# Cellsin

X

16 24 10 1

# OrbitsinRFS

(X)

5 11 8 1

# CellsinRFS

(X)

51 194 168 24

# OrbitsinVSS

(X)

,HyS

(X)

andBCS

(X)

5 11 11 4 # CellsinVSS

(X)

,HyS

(X)

andBCS

(X)

51 194 240 96

6. Example: FarrellTate ohomologyof

SL

3

(Z)

Fromthenon-rigid ell omplexdes ribingthea tionofSL

3

(Z)

onitssymmetri spa e,provided byMathieu Dutour Sikiri¢ in [8℄, weobtain, after applyingRigid Fa etsSubdivision, thefollowing

2

-torsionsub omplex,ina ordan ewithSoulé'ssubdivision [22℄.

stab(M) ∼

= S

4

stab(Q) ∼

= D

6

stab(O) ∼

= S

4

stab(N) ∼

= D

4

stab(P) ∼

= S

4

N’

M’

D

2

D

3

D

3

D

2

Z/2

Z/2

Z/2

D

4

Z/2

D

4

D

2

Z/2

Here, thethree edges

N M

,

N M

′

and

N

′

_M

′

have to be identied asindi ated by the arrows. All

of the seven triangles belong with their interior to the

2

-torsion sub omplex, ea h with stabilizer

Z/2

,ex ept fortheonewhi h ismarkedto havestabilizer

D

2

. Usingtorsion sub omplexredu tion ( f.[14℄),weredu e thissub omplexto

b

S

4

O

b

D

2

D

6

Q

Z/2

S

4

M

b

D

4

S

4

P

b

D

4

D

4

N

′

b

andthento

S

4

b

Z/2

S

b

4

D

4

S

b

4

whi histhegeometri realizationofSoulé'sdiagramof ellstabilizers. Thisyieldsthemod

2

Farrell ohomologyasspe iedin [22℄.

Table 4. Numbersof ellsin the

2

-torsion sub omplex forPSL

4

(Z)

before redu -tion,obtained after rigid fa ets subdivsion of the studied ell omplex,sorted into

isomorphismtypes of their stabilizers. Here,

G

1

and

G

2

are non-trivialgroup ex-tensions

1 → A

4

× A

4

→ G

1

→ Z/2Z → 1

and

1 → G

0

→ G

2

→ Z/2Z → 1

for

1 → (Z/2Z)

4

_{→ G}

0

→ Z/3Z → 1

Stabilizertype

A

4

G

1

(Z/2Z)

2

_S

4

Z/2Z D

8

(Z/2Z)

3

G

2

Z/4Z

Verti es

2

1

5

4

1

2

1

1

0

Edges

2

0

24

2

101

5

1

0

5

2- ells

0

0

27

0

326

0

0

0

4

3- ells

0

0

8

0

340

0

0

0

0

4- ells

0

0

0

0

116

0

0

0

0

Table5. Numbersof ellsintheindividualdimensions,ofthenon-rigidfundamental

domain

X

forSp

4

(Z)

des ribedin[12℄andimplementedbyDutorSikiri¢[8℄,itsrigid fa ets subdivsion RFS

(X)

, its virtually simpli ial subdivision VSS

(X)

, its hybrid subdivision HyS

(X)

anditsbary entri subdivisionBCS

(X)

.

Dimension 0 1 2 3 4

In

X

: # Orbits|# Cells 2|76 2|216 3|180 3|40 1|1 InRFS

(X)

: # Orbits|# Cells 8|185 31|800 54|1048 30|448 1|16 InVSS

(X)

: # Orbits|# Cells 8|185 39|1152 106|2536 122|2352 49|784 InHyS

(X)

: # Orbits|# Cells 8|185 35|864 72|1336 53|816 10|160 InBCS

(X)

: # Orbits|# Cells 11|513 90|3488 295|7904 368|7392 154|2464

presentanewimplementationof Torsion Sub omplexRedu tion,whi hwill over omethis problem

usingDis reteMorseTheory.

7. Example: FarrellTate ohomologyof

Sp

4

(Z)

at odd primes

Weshortlydis usstheresultsofapplyingrigidfa etssubdivisionandtorsionsub omplexredu tion

(for odd primes) to the non-rigid ell omplex des ribing the a tion of

Sp

4

(Z)

on the asso iated symmetri spa e

Sp

4

(R)/ U(2)

,des ribedin[12℄andimplementedbyMathieuDutourSikiri¢in [8℄. The

3

-torsionsub omplexfor

Sp

4

(Z)

isasinglevertexwhosestabilizergroupisthegroup

[72, 30]

in the GAP SmallGroup library. This group is of the form

C

3

×((C

6

× C

2

) ⋊ C

2

)

and its mod 3 ohomologyisisomorphi tothemod3 ohomologyof

C

3

× S

3

. Inparti ular,wegetisomorphisms

H

i

(Sp

4

(Z), F

3

) ∼

= H

i

(C

3

× S

3

, F

3

),

i > vcd(Sp

4

(Z)).

TheHilbertPoin aréseriesforthemod3 ohomologyof

C

3

× S

3

isgivenby

HP

C

3

× S

3

(T ; 3) =

(1 + T )(1 + T

3

₎

(1 − T

2

_{)(1 − T}

4

₎

.

In[3, (6.7)℄, theHilbertPoin aré-seriesfor

Sp

4

(Z)

hasbeen omputed. In thenotation oflo . it., thedieren eto theHilbertPoin aré-seriesfor

C

3

× S

3

aboveisgivenby

P (Γ

6

0

) − P (B

′

⋊ Z/2) =

1 + T

3

_{+ T}

4

_{+ T}

5

1 − T

4

−

1 + T + T

3

_{+ T}

4

1 − T

4

=

T

5

_{− T}

1 − T

4

= −T.

Inparti ular,theHilbertPoin aréseriesfor

C

3

× S

3

and

Sp

4

(Z)

agreeabovethevirtual ohomologi al dimensionof

Sp

4

(Z)

and thus the omputationviarigid fa ets subdivisionand torsion sub omplex redu tionagreeswiththe omputationofBrownsteinandLeein[3℄.

The

5

-torsionsub omplexfor

Sp

Table5providesanoverviewofthenumbersof ellsresp. orbitsinthe omplexes subdividedby

rigidfa etssubdivisionandbary entri subdivision,respe tively.

8. Example: FarrellTate ohomologyof

PSL

4

(Z)

at the prime 3

Applyingtherigidfa etssubdivisionalgorithm tothe

PSL

4

(Z)

-equivariant ell omplexfrom[6℄, extra tingthe

3

-torsionsub omplex,andredu ingit usingthemethods of[14℄,wegetthefollowing graph of groups

T

, de orated with the groupsstabilizing the ells that are the pre-images of the proje tiontothequotientspa e.

b

b

S

3

S

3

S

3

× S

3

S

3

× S

3

E

9

⋊ C

2

C

3

b

_S

3

b

E

9

⋊ C

2

Here,

E

9

⋊ C

2

denotesthe semi-dire tprodu t ofthe elementary abeliangroupof order

9

with

C

2

, where

C

2

a tsbyinversion( orrespondingto thegroup[18,4℄inGAP's SmallGroupslibrary). The ma hine omputationprovidedthefollowingsystemofmorphisms amongtheabove ellstabilizers.

The

E

9

⋊ C

2

edge stabilizer admits an isomorphism of groups

φ

(not the identity, though) to the

E

9

⋊ C

2

vertex stabilizer and an in lusion into the

S

3

× S

3

vertex stabilizer. Of the two

S

3

edge stabilizers, one has maps

diag(1, 1)

and

diag(1, 0)

to the two

S

3

× S

3

vertex stabilizers, and the otheronehasmaps

diag(1, −1)

and

diag(−1, −1)

tothetwo

S

3

× S

3

vertexstabilizers. The

C

3

edge stabilizeradmits anin lusionintothe

E

9

⋊ C

2

vertexstabilizer, andanin lusion intothe

S

3

vertex stabilizer.

Bythe properties of torsion sub omplex redu tion, the

PSL

4

(Z)

-equivariant ohomology of the

3

-torsion sub omplex is isomorphi to the

PSL

4

(Z)

-equivariant ohomology of the abovegraph of groups

T

. Similarly, the FarrellTate ohomologyof

PSL

4

(Z)

at the prime

3

is isomorphi to the FarrellTate ohomology of the abovegraph of groups. In the following, we evaluate the isotropy

spe tralsequen e

E

₁

p,q

=

M

σ∈T

p

H

q

(Stab(σ); F

3

) ⇒ b

H

p+q

(PSL

4

(Z); F

3

)

onvergingtoFarrellTate ohomology. Asweonly onsideragraph,thespe tralsequen eis

on en-tratedin thetwo olumns

p = 0, 1

. Thedierential

d

1

isindu edfrom thein lusions ofsubgroups, up to the sign oming from the hoi e of orientation of the graph. Sin e the spe tral sequen e is

only on entratedin the rsttwo olumns, wewillhave

E

2

= E

∞

. Sin eweare interestedin eld oe ients,therearenoextensionproblems tosolveat the

E

∞

-page.

Therelevant ohomologygroupsofthenitegroupsare:

• H

•

(C

3

; F

3

) ∼

= F

3

[x](a)

with

deg a = 1

and

deg x = 2

.

• H

•

(S

3

; F

3

) ∼

= F

3

[y](b)

with

deg b = 3

and

deg y = 4

.

•

BytheKünnethformula,

H

•

_(S

3

× S

3

; F

3

) ∼

= H

•

(S

3

; F

3

)

⊗2

.

•

By the Ho hs hildSerre spe tral sequen e,

H

•

_(E

9

⋊ C

2

; F

3

) ∼

= F

3

[x

1

, x

2

](a

1

, a

2

)

C

2

where

deg x

i

= 2

,

deg a

i

= 1

and

C

2

a tsbymultipli ationwith

−1

onallthegenerators.

Todes ribethe

d

1

-dierential,itisenoughtonotethattherestri tionmapasso iatedtothein lusion

C

3

֒→ S

3

isthein lusionof

C

2

-invariants.

Now,fortheevaluationofthespe tralsequen e,werstdealwiththeedgesatta hedtotheloop.

(1) Therestri tionmap

φ ⊕ (Res

S

3

C

3

◦ pr

∗

2

) : H

•

(E

9

⋊ C

2

; F

3

) → H

•

(E

9

⋊ C

2

; F

3

) ⊕ H

•

(C

3

; F

3

)

isinje tive, and the okernelis isomorphi to

H

•

_(C

3

; F

3

)

. Here

φ

denotesthe isomorphism

E

9

⋊ C

2

∼

= E

9

⋊ C

2

appearingasstabilizerin lusion intheredu edtorsionsub omplex. (2) Thein lusion ofthedihedralvertexgroupinto the y li edgegroupisaninje tion

Res

S

3

C

3

: H

•

_(S

given by the in lusion of the invariant elements for the

C

2

-a tion by

−1

. Therefore, the okernelis on entratedin degrees

1, 2 mod 4

(ex eptforthedegree

0

).

Therefore,we anredu ethe

E

1

-page ofthespe tralsequen e asfollows: from (1), wend that we anremovethetwosummandsfor

E

9

⋊ C

2

fromthe olumns

p = 0

and

p = 1

,respe tively;butin turn, wehaveto repla etherestri tionmapfor

E

9

⋊ C

2

֒→ S

3

× S

3

byamapfrom the ohomology of

S

3

× S

3

to the okernel of the restri tion map in (2). However, sin e the latter is on entrated in degrees 1and 2, the indu edrestri tion map is trivial in the generating degrees

3

and

4

, hen e it is trivial. Therefore, the

E

1

-page de omposes: one ontribution omes from the okernelof the restri tionmap

H

•

_(S

3

; F

3

) ֒→ H

•

(C

3

; F

3

)

,theother ontribution omesfromtheloop onne tingthe two opiesof

S

3

× S

3

viathe

S

3

-edges.

The ohomologyoftheloopis omputedasfollows:

(3) Therestri tionmaps

diag(1, 1), diag(1, 0), diag(1, −1)

and

diag(−1, −1) :

F

3

[x

4

, y

4

](a

3

, b

3

) ∼

= H

•

(S

3

× S

3

; F

3

) → H

•

(S

3

; F

3

) ∼

= F

3

[z

4

](c

3

)

aresurje tive. Onthekernelof

diag(1, 1)

,therestri tionof

diag(1, −1)

isstillsurje tive. On thekernelof

diag(−1, −1)

,therestri tionof

diag(1, 0)

isstillsurje tive.

Therefore, the ohomology of oneedge of the loopis killed already by therestri tion map from

any oneof thevertex groups. Number the

S

3

× S

3

-verti esby

1

and

2

, and the

S

3

-edges by

a

and

b

. Therestri tionfrom thevertexgroup

1

to theedge

a

issurje tive. Removingthis partfromthe spe tral sequen e,the restri tionfrom thevertexgroup

2

to theedge

a

is trivial,but westill have the restri tionto theedge

b

. This kills theedge ohomology

b

, showingthat the dierential

d

1

is surje tiveinthelooppartofthe

E

1

-page. Thekernelofthedierential onsiststhenexa tlyoftwo opiesofthekernelofarestri tionmap

Res

S

3

× S

3

S

3

.

Theorem10. TheFarrellTate ohomologyofthegroup

PSL

4

(Z)

(with oe ientsin

F

3

)indegrees

>

2

isgiven asfollows:

b

H

•

(PSL

4

(Z); F

3

) ∼

=



ker Res

S

3

× S

3

S

3



⊕2

⊕ coker H

•−1

(S

3

; F

3

) → H

•−1

(C

3

; F

3

)

.

This, inparti ularalso omputes the

3

-torsiongroup ohomology of

PSL

4

(Z)

abovethe virtual oho-mologi al dimension.

The okernelof the

d

1

-dierential in degree

0

and hen ethe rst ohomology

H

b

1

(PSL

4

(Z); F

3

)

is of

F

3

-rank

1

, omingfromthe loopof the

3

-torsion graph.

Remark 11. The kernel omes fromthe

p = 0

olumnof the

E

2

= E

∞

-page. The okernel omes fromthe

p = 1

olumnand onsequentlyhasashift. The upprodu tonthekernelistheoneindu ed from ohomologyof

S

3

× S

3

,and up-produ twith lassesinthe okernelisalways

0

.

InspiredbyGrunewald,we onsidertheHilbertPoin aréseriesoftheFarrellTate ohomologyof

PSL

4

(Z)

with oe ientsin

F

ℓ

:

HP

PSL

4

(Z)

(T ; ℓ) :=

∞

X

q=1

dim b

H

q

(PSL

4

(Z); F

ℓ

) · T

q

.

Corollary12. TheHilbertPoin aréseriesofthe

3

-torsionFarrellTate ohomologyof

PSL

4

(Z)

(for degrees

>

1

)isthen

HP

PSL

4

(Z)

(T ; 3) = T +

2(T

3

_{+ T}

4

_{+ T}

6

_{+ T}

7

₎

(1 − T

4

₎

2

+

T

2

_{+ T}

3

1 − T

4

.

Remark13. Notethat theabove al ulationdes ribestheFarrellTate ohomologyinall degrees,

notjustsomesmallones. Essentially,the omputer al ulationprodu estheredu edtorsion

sub om-plex (whi h en odesthe ohomologyfor all degrees). The spe tralsequen e is evaluated using the

up-produ tstru ture. Notethatthenitenessresultsforgrouphomologyimplythatthe up-produ t

stru tureforbothgroupandFarrellTate ohomologyisnitelygenerated.Usingsuitable

ommuta-tivealgebrapa kages,su h omputationsoftheringstru ture(andthereforeadditive omputations

forall ohomologi aldegrees) ouldprobablyalsobeautomated.

Wemakethefollowing onsiderationonthe ompatibilityofourresultforFarrellTate ohomology

withtheresultofDutourEllisS hürmann[6℄forgrouphomologyinlowdegrees. Theisomorphism

H

q

(PSL

4

(Z); Z) ∼

=



























0,

q = 1,

(Z/2)

3

,

q = 2,

Z

⊕ (Z/4)

2

⊕ (Z/3)

2

⊕ Z/5,

q = 3,

(Z/2)

4

⊕ Z/5,

q = 4,

(Z/2)

13

,

q = 5,



dim b

H

q

(PSL

4

(Z); F

3

) =



























1,

q = 1,

1,

q = 2,

3,

q = 3,

2,

q = 4,

0,

q = 5.

For this to be onsistent, the FarrellTate ohomology groups in degrees

1

and

2

need to vanish in group homology; so, these should beannihilated by dierentials from the orbitspa e. We have

eviden efor thisin degree

1

, sin etheloopin thegraphbe omes ontra tiblein the orbitspa eof thefull lo ally symmetri spa e. Indegree

3

, oneofthe summandsin

H

3

(PSL

4

(Z); Z)

isrationally non-trivialand must ome from the orbitspa e. This means that onlythe submodule

(Z/3)

2

an

omefromFarrellTate ohomology,andthethirddimensionthatweobserveindegree

3

FarrellTate ohomologymustbelongtothedegree

2

stabilizer ohomology lassthatisannihilatedbytheabove mentioneddierentialsfromtheorbitspa e.

FromTheorem10,wededu ethatthedegree

2

FarrellTate lass anonly omefrom

coker H

•−1

(S

3

; F

3

) → H

•−1

(C

3

; F

3

)

.

Then, this lass and its grouphomology ounterpartsit at position

p = 1, q = 1

in the respe tive equivariantspe tralsequen e,andhen etheannihilatingdierential,emanatingfromtheorbitspa e

homologymodule

Z ⊂ H

3

(PSL

4

(Z); Z)

sittingatposition

p = 3, q = 0

,mustbeofse onddegree. Indegrees

4

and

5

, thedimensionsalready agree viatheUniversal Coe ient Theorem, sohere weinferthatthesubmodule

(Z/3)

2

in degree

3

shoulda tually omefromFarrellTate ohomology, soitshouldbestabilizer ohomologythatisnothitbyhigherdegreedierentials.

9. Homologi al5-torsion in

PSL

4

(Z)

Weapplied therigidfa ets subdivision algorithm tothe

PSL

4

(Z)

-equivariant ell omplexof[6℄, extra tedthe

5

-torsionsub omplex,andredu editusingthemethodsof[14℄to thefollowinggraph

T

:

b

D

5

D

5

The

d

1

-dierential of the equivariant spe tral sequen e on

T

is zero, be ause the isomorphismsatedgeendandedgeorigin an elea hother. Thenthe

E

1

= E

∞

pageis on entrated in the olumns

p = 0

and

1

, withdimensionsover

F

5

being

1

in rows

q

ongruentto

3

or

4

mod

4

, andzerootherwise. Thisyields

Proposition14. WeobserveonFarrell ohomology:

dim

F

5

H

b

p+q

(PSL

4

(Z); F

5

) =



















1,

p + q ≡ 1 mod 4,

0,

p + q ≡ 2 mod 4,

1,

p + q ≡ 3 mod 4,

2,

p + q ≡ 4 mod 4.

We he kthisresultwitha omputationof

H

b

•

(PSL

4

(Z); F

5

)

usingBrown's omplex[2,last hap-ters℄. In this ase, it is standard that the set

{1, ζ

5

, ζ

3

2

, ζ

5

3

}

is an integral basis of

O

Q(ζ

5

)

and in parti ular

Z[ζ

5

] = O

Q(ζ

5

)

isaDedekindring.

We anthereforeuseReiner'sresult[17℄todetermine onjuga y lassesof

C

5

-subgroupsin

GL

4

(Z)

. Sin eboth

Z

and

Z[ζ

5

]

havetrivial lassgroup,thereisonlyoneisomorphism lassof

Z[C

5

]

-module with nontrivial a tion and

Z

-rank

4

. Hen e, there is a unique onjuga y lass of y li order

5

subgroupin

GL

4

(Z)

. Sin ethe enterof

GL

4

(Z)

isoforder

2

,thesameistruefor

PGL

4

(Z)

.

Nowthereisane essarymodi ationtodealwiththe ase

SL

4

(Z)

,alongthelinesofthedis ussion in [16℄. While onjuga y lasses of

C

5

-subgroups in

GL

4

(Z)

orrespond to isomorphism lasses of

Z[C

5

]

-modules,the onjuga y lassesof

C

5

-subgroupsin

SL

4

(Z)

orrespondtosu hmodulesequipped withanadditionalorientation,i.e.,a hoi eofisomorphism

det M ∼

=

V

4

Z

M ∼

= Z

. The onjuga y lass in

GL

4

(Z)

lifts to

SL

4

(Z)

, and the orresponding module has two dierent hoi es of orientation. TheGaloisgroup

Gal(Q(ζ

5

)/Q) ∼

= Z/4Z

a ts onthesetof orientedmodules. Thea tionex hanges the orientations. Therefore, there is one onjuga y lass of

C

5

-subgroup in

SL

4

(Z)

stabilized by

Z/2Z ֒→ Gal(Q(ζ

5

)/Q)

.

The entralizerofthis

C

5

-subgroupisthegroupofnorm-1unitsof

Z[ζ

5

]

,whi hbyDiri hlet'sunit theoremisisomorphi to

ker Z[ζ

5

]

×

→ Z

×

∼

= Z/10Z × Z.

Asin [16, Se tion 5℄,the normalizeris anextensionof the entralizerby ana tionof thestabilizer

of the orresponding oriented module in the Galoisgroup. We notedabove that the Galois group

Z/4Z

ex hanges the two orientations of the trivial module, hen e the stabilizer is the subgroup

Z/10Z

is by multipli ation with

−1

be ausethea tion of theGalois groupisvia theidenti ation

Z/4Z ∼

= Z/5Z

×

. Thea tionof

Z/2Z

on

Z

istrivial: thefull Galoisgroupa tson

Z

viaasurje tive homomorphism

Z/4Z → Z

×

_∼

_{= Z/2Z}

. Thestabilizerof theorientedmodule inthe Galoisgrouplies

inthekerneloftheabovea tion,as laimed. Therefore,thenormalizerisinfa toftheform

D

10

×Z

. Applyingthe formulas from [16, Se tion 3℄, theFarrellTate ohomology of thenormalizer is of

theform

F

5

[a

±2

2

](b

3

1

)

⊕2

⊕ F

5

[a

±2

2

](b

3

1

)

⊕2

−1

wherethelowersubs ript

−1

indi atesadegreeshiftby

−1

. TheHilbertPoin aréseriesforthepositivedegreesis

T

3

+2T

4

+T

5

1−T

4

=

T

3

(1+T )

2

1−T

4

.

The omputationsin [6℄show thatthe

5

-torsionin integralhomologyof

PSL

4

(Z)

of dimension1 indegrees

0, 3 mod 4

andtrivialotherwise. Bytheuniversal oe ienttheorem,thisagreeswiththe above omputation.

10. Ne essityof Rigidity forBredon homology

Fromanon-rigid ell omplex,i.e.,a ell omplexwhere ellstabilizers donotne essarilyxthe

orresponding ellpointwise,one an ompute lassi algrouphomologyviatheequivariantspe tral

sequen e with oe ients in theorientation module. Su h an orientation module, where elements

of the stabilizer groupa t by multipli ation with

1

or

−1

, depending on whether theypreserve or reverse theorientationof the ell, annot exist for Bredon homology. We make this pre ise in the

followingstatement:

Proposition 15. There is no module-wise variation of the Bredon module with oe ients in the

omplex representation ring and withrespe t tothe systemof nite subgroups su h that Bredon

ho-mology anbe omputedfrom anon-rigid ell omplex.

Proof. We provide a ounterexample in order to ontradi t the possibility to ompute the Bredon

homologyfrom anarbitrarynon-rigid ell omplex.

Considerthe lassi almodular group

PSL

2

(Z)

. A model for E

PSL

2

(Z)

is given bythe modular tree[21℄. Thereisarigid ell omplexstru ture

T

1

onit,givenasfollows. By[21℄,themodulartree admitsastri tfundamental domainfor

PSL

2

(Z)

,oftheshape

Z/3Z

b

b

Z/2Z

with vertex stabilizers as indi ated and trivialedge stabilizer. We obtain the ell omplex

T

1

by tessellatingthemodulartreewiththe

PSL

2

(Z)

-imagesofthisfundamentaldomain. Obviously,

T

1

is rigid,andityieldstheBredon hain omplex

0 → R

C

(h1i)

d

−→ R

C

(Z/2Z) ⊕ R

C

(Z/3Z) → 0.

Themap

d

in theaboveBredon hain omplexisinje tive,andas

R

C

(Z/nZ) ∼

= Z

n

,wereado

H

Fin

1

(

E

PSL

2

(Z); R

C

) = 0,

H

Fin

0

(

E

PSL

2

(Z); R

C

) ∼

= Z

4

_.

Nowweequipthemodulartree withanalternativeequivariant ellstru ture

T

2

,indu edbythe non-stri tfundamentaldomain

Z/3Z

b

b

Z/3Z

wherethe(set-wise)edgestabilizeris

Z/2Z

,ippingtheedgeontoitself. It anbeseenasaramied double overofthefundamental domainfor

T

1

dis ussedabove. Asystemofrepresentative ellsfor

T

2

isgivenbytheedgeofdoublelength,andonevertexofstabilizertype

Z/3Z

. Thisyieldsa hain omplex

0 →

R

C

^

(Z/2Z) → R

C

(Z/3Z) → 0,

where the tilde ould be any onstru tion whi h takesthe non-trivial

Z/2Z

-a tion on the edge of double length into a ount (similar to the oe ients in the orientation module for group

homol-ogy omputed from non-rigid ell omplexes). But no matter how this onstru tion is done, from

R

C

(Z/3Z) ∼

= Z

3

, we an neverrea h

H

Fin

0

(

E

PSL

2

(Z); R

C

) ∼

= Z

4

_.

Hen e

T

2

is ourdesired

ounterex-ample.



Remark 16. We ould of ourse drop the ondition module-wise in the above proposition, and

investigatewhether thereisareasonable onstru tionwhi hmapstherepresentationringtoa

om-plexof modulesand yields aquasi-isomorphism from thetotal omplexto theBredon omplexfor

thesubdivided tree. Butwith su h a onstru tion, one would only super ially hide thefa t that

subgroupsand oe ientsin the omplexrepresentationring withoutsubdividingthe ell omplex

under onsiderationtomakeitrigid.

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