![[Math.NT] 4 Jan 2006 Rtmtcsubgroups](http://data.docslib.org/img/e520377b81b15eb22ba57290cc23d3a6-1.webp)
arXiv:math/0601071v1 [math.NT] 4 Jan 2006 rtmtcsubgroups. arithmetic package. iersmercsae.AhadMCnel[]etn 3 oteBor coho of geod the computation of to the combination in [3] used a 17–20]. extend 15, been 13, to has [7] retraction retract McConnell well-rounded the The and relate Ash and compactification spaces. symmetric linear ooooyo Γ of cohomology s ecie the describes Ash as aPesnadMCnel[4 osrce pn nteSee up Siegel the Sp in spine group e a the non-linear constructed for no [14] were McConnell There and spaces. MacPherson symmetric linear in spines defining ooooia ieso fΓi .Teei nfc -iesoa defo 3-dimensional a fact in is There 3. is Γ retract of dimension cohomological steodr ftetrineeet fΓaeivril in invertible are Γ of elements torsion the of orders the as u h smrhs fchmlg ssilvldb sn utbesheaf suitable a using by valid still is cohomology of isomorphism the but maximal rtmtcsbru.BrladSre[]so httediscrepan the that show [9] Serre of and dimension Borel subgroup. arithmetic Γ h oal ymti pc Γ space symmetric locally the ooooyo sioopi otechmlg ftelclysymmetr locally the of cohomology the to isomorphic i.e. is Γ of cohomology eteascae o-opc ymti pc.LtΓb narith an be Γ points Let rational space. the symmetric of non-compact associated the be \ D 2000 e od n phrases. and words Key h rgnlmnsrp a rprdwt the with prepared was manuscript original The pnshv encntutdfrmn rus[,8 1 31,1 13–15, 11, 8, [1, groups many for constructed been have Spines h elruddrtatpoe h xsec n ie method a gives and existence the proves retract well-rounded The The Let spines H hnΓhstrin h orc ramn novstelanguage the involves treatment correct the torsion, has Γ When . ∗ G (Γ ita ooooia dimension cohomological virtual ahmtc ujc Classification. Subject Mathematics D opc ye spine A type. compact Abstract. osrc pn o h aeo SU(2 = Γ Γ of case of the dimension for cohomological spine virtual a the construct to equal dimension opt h ooooyo Γ of cohomology the compute . etera onso the of points real the be 0 Q E , ⊂ slttrsin torus -split ) D = ∼ D XLCTRDCINFOR REDUCTION EXPLICIT hti nain ne h cino 2] uhsae r known are spaces Such [21]. Γ of action the under invariant is that H \ 4 n h vcd( the and D ( ∗ Z e Γ Let elruddretract well-rounded (Γ ). aihsi ere above degrees in vanishes \ G D \ pn,Pcr oua ru,lclysmercsae coho space, symmetric locally group, modular Picard spine, ( D ; Q G E .Lt( Let ). ea rtmtcqoin fasmercsaeo non- of space symmetric a of quotient arithmetic an be ,where ), hsi u case, our in Thus . D \ 0 G D saΓeuvratdfrainrtato of retraction deformation Γ-equivariant a is sgvnb the by given is ) 1. Q sa is \ ,ρ E, rn ieragbacgopSU(2 group algebraic linear 1 -rank D Introduction A YASAKI DAN E ihvroslclcoefficients. local various with K eoe h oa ytmdfie y( by defined system local the denotes eaΓmdl over Γ-module a be ) rmr 15;Scnay53C35. Secondary 11F57; Primary ehdfrcntutn pn o all for spine a constructing for method a , (Γ vd of (vcd) 1 , )since 1) , 1; p A hr Γ where , M D Z G [ S i Q ) h pn ste sdto used then is spine The ]). s4dmninl n h virtual the and 4-dimensional, is stesals integer smallest the is -L D rn of -rank A T scnrcil,adtegroup the and contractible, is E SU(2 ar ytmadteX the and system macro X ⊂ R G , G 1; R ( eexplicitly We . Q h ieso fa of dimension the , fΓi torsion-free, is Γ If . Z saytorsion-free any is ) [ i ]) ei subgroup metic ybtenthe between cy ooy[,4–6, [2, mology D ,2] n[3], In 20]. 8, e half-space per , ape until xamples ) n let and 1), scactions. esic with fexplicitly of forbifolds, of p uhthat such E cspace, ic ooyof mology ,ρ E, rmation el-Serre slong as Y on ) -pic D 2 DAN YASAKI
In this paper, we provide another non-linear example by using the method of [21] to compute a spine for SU(2, 1; Z[i]). Sections 2 and 3 set notation and define the exhaustion functions that are used to describe the pieces of the spine. In Section 4, we classify certain of configurations of isotropic line in C3. We show the spine has the structure of a cell complex with cells related to these configurations in Section 6. Explicit Γ-representatives of cells are fixed, and their stabilizers are computed in Section 8. After subdivision, we obtain a regular cell complex for D0 on which Γ acts cellularly. In Section 10, we recall some facts about orbifolds and develop machinery to investigate the cohomology of Γ. The results of Section 10 hold in more generality, and may be of independent interest. We apply these methods in Section 11 to Γ = SU(2, 1; Z[i]) to compute the cohomology of Γ with coefficients in various Γ-modules. I would like to thank my thesis advisor, Les Saper for his insight into this work. I would also like to thank Paul Gunnells for helpful conversations.
2. Preliminaries Let G be the identity component of the real points of the algebraic group G = SU(2, 1), realized explicitly as 0 0 i 0 0 i SU(2, 1) = g SL(3, C) g∗ 0 1 0 g = 0 1 0 . ∈ − − i 0 0 i 0 0 − − Alternatively, let be the (2,1)-quadratic form on C3 defined by Q 0 0 i (u, v)= u∗ 0 1 0 v. Q i −0 0 − Then G is the group of determinant 1 complex linear transformations of C3 that preserve . Let Γ be the arithmetic subgroup Γ = SU(2, 1) SL3(Z[i]). Let θ Qdenote the Cartan involution given by inverse conjugate∩ transpose and let K be the fixed points under θ. Let D = G/K be the associated Riemannian symmetric space of non-compact type. Let denote the set of (proper) rational parabolic subgroups of G. P Let P G be the rational parabolic subgroup of upper triangular matrices, and 0 ⊂ fix subgroups N0, A0, and M0: 2 2 yζ βζ− ζ r + i β /2 /y | | P = 0 ζ 2 iβζ/y ζ,β C, ζ =1, r R, y R , 0 − ∈ | | ∈ ∈ >0 0 0 ζ/y 1 β r + i β 2/2 | | N = 0 1 iβ β C, r R , 0 ∈ ∈ 00 1 y 0 0 A = 01 0 y R , 0 ∈ >0 0 0 1/y ζ 0 0 2 M = 0 ζ− 0 ζ C, ζ =1 . 0 ∈ | | 0 0 ζ
EXPLICIT REDUCTION FOR SU(2, 1; Z[i]) 3
P0 acts transitively on D, and every point z D can be written as p x0 for some p P . Using Langlands decomposition, there∈ exists u N ,a A ,· and m M ∈ 0 ∈ 0 ∈ 0 ∈ 0 such that p = uam. Since M0 K, z can be written as ua x0. Denote such a point z = (y,β,r). ⊂ · Zink showed that Γ has class number 1 [22]. Thus Γ G(Q)/P0(Q) consists of a single point, and all the parabolic subgroups of G are\ Γ-conjugate. The rational parabolic subgroups of G are parametrized by the maximal isotropic subspaces of C3 which they stabilize. These are 1-dimensional, and so to each P , there is 3 ∈t P 3 an associated reduced, isotropic vector vP Z[i] . (A vector (n,p,q) Z[i] is reduced if (n,p,q) generate Z[i] as an ideal.)∈ Similarly, given a reduced,∈ isotropic 3 vector v in Z[i] , there is an associated rational parabolic subgroup Pv. Notice, however, that vP is only well-defined up to scaling by Z[i]∗ = 1, i . Thus, the {± ± }γ vectors v and εv will be treated interchangeably for ε Z[i]∗. If P = Q for some γ Γ, then v = γv . ∈ ∈ P Q Unless explicitly mentioned otherwise, the vector vP will be written as vP = (n,p,q)t. The isotropic condition (v , v ) = 0 implies that Q P P (1) p 2 = 2 Im(nq). | | In particular, q = 0 for P = P0. Furthermore, since there are no isotropic 2-planes in C3, 6 6 (2) (v , v ) =0 for P = Q. Q P Q 6 6 Because these elements of Γ will be used frequently, set once and for all 0 0 1 1 1+ i i 1 i(1 + i) i w = 01− 0 , σ = 0 1 1+ i , σˇ = 0 1 i(1 + i) , 10 0 00 1 00− 1 1 0 1 i 0 0 τ = 0 1 0 ,ǫ = 0 1 0 , and ξ = τwτσwǫ3. 0 0 1 0− 0 i Note thatσ ˇ is contained in the group generated by ǫ,w,σ . In particular,σ ˇ = 1 { } wσǫwσ− w.
3. Construction of the spine In this section we briefly describe the construction of a Γ-invariant, 3-dimensional cell complex which is a deformation retract of D. This construction is described for the general Q-rank 1 case in [21]. We first define an exhaustion function fP for every rational parabolic subgroup P G. These exhaustion functions are then used to define a decomposition of D into⊆ sets D( ) for . I I⊂P 3.1. Exhaustion functions. Let z = (y,β,r) D and P a rational parabolic subgroup of G with associated isotropic vector∈ (n,p,q)t. Then the exhaustion function fP can be written as (3) f (z) f (z)= y 0 ≡ P0 y (4) fP (z)= 1/2 . n βp + i β 2/2 r q 2 + y2 p iβq 2 + y4 q 2 | − | | − | | − | | | 4 DAN YASAKI
The family of exhaustion functions defined above is Γ-invariant in the sense that 1 (5) fγ (z)= f (γ− z) for γ Γ. P P · ∈ 3.2. Admissible sets. For a parabolic P , define D(P ) D to be the set of z D such that f (z) f (z) for every Q P . More generally,⊂ for a subset ∈ , P ≥ Q ∈P\{ } I⊆P (6) E( )= z D f (z)= f (z) for every pair P,Q I { ∈ | P Q ∈ I} (7) D( )= D(P ) I P \∈I (8) D′( )= D( ) D( ′). I I \ I ′) I[I It follows that D′( ) D( ) E( ) and D( )= ˜ D′(˜). Let f denote the I ⊆ I ⊂ I I I I restriction to E( ) of f for P . I⊇I I P ∈ I ` Definition 3.1. Let , P , and z E( ). Then z is called a first contact for if f (z) is a globalI⊆P maximum∈ I of f on∈E( I). I I I I Definition 3.2. A subset is called admissible if D( ) is non-empty and I ⊂P I strongly admissible if D′( ) is non-empty. I Let D D denote the subset 0 ⊂ (9) D = D′( ). 0 I >1 |I|a The deformation is defined separately on each D(P ) for P . For z D(P ), we ∈P ∈ use the (negative) gradient flow of fP to flow z to a point on D0. This corresponds to using the geodesic action [9] of AP on z [21].
Let fD0 denote the function on D0 given by
(10) fD0 (z)= f (z) for z D( ). I ∈ I 3.3. First contact points. Given the explicit description of the exhaustion func- tions in coordinates, one can readily describe the set E( P0, P ). Writing z = (y,β,r) and using (3), { } 2 f (z) 2 2 (11) 0 = n βp + i β 2/2 r q + y2 p iβq + y4 q 2. f (z) − | | − − | | P Proposition 3.3. Let P be a rational parabolic subgroup of G with associated isotropic vector (n,p,q)t. Every z = (y,β,r) E( P , P ) satisfies ∈ { 0 } 1 p 2 1 n βp 2 y2 = iβ¯ + Re − r −2 q − s q 2 − q − | | 2 f0(z) Proof. Note that for z E( P0 , P ), = 1. Then the result follows from ∈ { } fP (z) (11) using the quadratic formula to solve fory2, and simplifying the result using the isotropic condition (1).
Proposition 3.3 allows us to easily calculate the first contact point for P0, P . The Γ-invariance of the exhaustion functions allows us to translate this for{ general} P,Q . { } EXPLICIT REDUCTION FOR SU(2, 1; Z[i]) 5
Proposition 3.4. Let = P,Q . Let z be a first contact point for . Then I { }⊂P I 1 fP (z)= fQ(z)= . (v , v ) |Q P Q | In particular, the first contact for P , P pis z = 1/ q ,i(p/q), Re(n/q) . { 0 } | | p 4. Configurations of vectors Definition 4.1. Let be a subset of vectors in C3. Then is said to be c-bounded if J J (u, v) 2 c for every u and v in . |Q | ≤ J Set once and for all the following sets i of isotropic vectors in C3. Jj 1 0 1 i 2 = 0 , 0 2 = 0 , 1+ i J1 J2 0 1 0 1+ i 1 i 1+ i 3 = 2 0 3 = 2 1+ i 3 = 2 1+ i J1 J1 ∪ J2 J1 ∪ J3 J1 ∪ 1 1 1 i 1 − 4 = 3 1+ i 4 = 3 1+ i J1 J1 ∪ J2 J3 ∪ − 1 1+ i 1+ i 5 = 4 1+ i J J1 ∪ 1 1 0 1 1+ i 1+i i 2i i − − 8 = 0 , 0 , 1+ i , 1+ i , 1 i , 1+ i , 2 , 2 . J − 0 1 1+ i 1 1 1+ i 1 2 8 One easily checks that these sets are not Γ-conjugate, and is 4-bounded, while the other sets are 2-bounded. J Proposition 4.2. A 2-bounded set of reduced, integral, isotropic vectors is Γ- conjugate to exactly one of 2, 2, 3, 3, 3, 4, 4, or 5. J1 J2 J1 J2 J3 J1 J2 J Proof. Let denote such a subset of order 2. Since G has class number 1, we can assume thatJ one of the vectors of is (1, 0, 0)t. Let v = (n,p,q)t be the other vector of . Let σ, σ,ˇ τ, and ǫ be definedJ as in Section 2. These elements preserve t J (1, 0, 0) (up to scaling by Z[i]∗). By applying powers of σ andσ ˇ to v, we can add any Z-linear combination of (1 + i)q and (1 + i)iq to p to force it into the square in the complex plane with vertices q, iq, q, and iq. By applying powers of iǫ to v, one can force p to lie in the triangle with− q, iq,− and 0 as vertices while leaving− q fixed. Then by applying powers of τ to v, one can add a Z-scalar multiple of q to n so that n now has the form dq + ciq, where 1 < d 1 for some c R. Since − 2 ≤ 2 ∈ v is isotropic, p 2 = 2 Im(nq)=2c q 2. Since is 2-bounded, q 2 2, so that in | | 2| | 2 J | | ≤ particular, is Γ-equivalent to 1 or 2 . J J J 2 2 For > 2, we can arrange that 1 or 2 . The 2-bounded condition in each|J of | these two cases has onlyJ finitely ⊃ J manyJ ⊃solutions. J The proposition then follows from listing the solutions and identifying Γ-conjugate sets. 6 DAN YASAKI
5. Reduction theory
Proposition 5.1. Every point z D is conjugate under ΓP0 to a point (y,β,r), 1 1 ∈ where 2 < r 2 and β lies in the square in the complex plane with vertices 1+i − ≤1+i 0, 2 , i, and − 2 .
Proof. Consider the elements σ, σ,ˇ τ, ǫ ΓP0 defined in Section 2. The action of σ, σ,ˇ τ, ǫ leaves D(P ) {D stable,}⊂ and is given explicitly by { } 0 ∩ 0 σ (y,β,r) = (y,β +(1+ i), r Re(β) + Im(β)), · − σˇ (y,β,r) = (y,β + i(1 + i), r Re(β) Im(β)), (12) · − − τ (y,β,r) = (y,β,r + 1), and · ǫ (y,β,r) = (y, iβ,r). · − Thus, by applying powers of σ andσ ˇ, β can be put in the square in the complex plane with vertices 1, i, 1, and i. By applying a power of ǫ, β can be put in the − − 1+i 1+i square in the complex plane with vertices 0, 2 , i, and − 2 . Then by applying 1 1 powers of τ, it can be arranged that 2 < r 2 without changing the value of β. − ≤
Proposition 5.2. For each z D , ∈ 0 1
Note that on F , f0(z) fP (z) for all P . By Proposition 3.3, this is equivalent to the condition that ≥ ∈P
2 2 2 2 1 p 1 n βp (14) f0(z) = y iβ + Re − r . ≥−2 q − s q 2 − q − | | w Consider the rational parabolic subgroups P0, P , and Q corresponding to the t t t vectors vw = (0, 0, 1) , vP = (i, 1+ i, 1+ i) and vQ = ( 1, 1+ i, 1+ i) . Divide F into the following regions: − 9 3 1 F = x = (y,β,r) F β > , < r P ∈ | | 10 10 ≤ 2 n 9 1 o3 F = x = (y,β,r) F β > , r< Q ∈ | | 10 −2 ≤ −10 n o Fw = F F1 F2 . \{ ∪ } EXPLICIT REDUCTION FOR SU(2, 1; Z[i]) 7
Since β is constrained to lie in the square indicated in Proposition 5.1, one may calculate that on FP and FQ (10 √62) 10 √62 (15) − < Re(β) < − − 20 20 81 10√62 (16) β i 2 < − . | − | 100 Comparing f0 to fP on FP , f0 to fQ on FQ, and f0 to fw on Fw, (14) gives the desired bound on each piece. Since F = F F F , this proves the result. P ∪ Q ∪ w 6. Admissible sets
Since vP and vQ are integral vectors, (vP , vQ) Z[i]. Proposition 3.4 and Proposition 5.2 imply the following. Q ∈ Proposition 6.1. Let be an admissible set. Then I (v , v ) 2 4 for every P and Q in . |Q P Q | ≤ I In particular, the set of vectors associated to an admissible set is 4-bounded. Proposition 6.2. Let = P,Q be an admissible subset of . If (v , v ) 2 = I { } P |Q P Q | 4, then there exists a strongly admissible set ˜ of order 8 such that D( ) = I⊂P I D(˜)= z , where z is the first contact for . Let ˜ be the set of isotropic vectors I { } I J associated to ˜. Then ˜ is 4-bounded and is Γ-equivalent to 8. A set of isotropic vectors that isI not 2-bounded,J but is associated to a stronglyJ admissible set, is Γ- equivalent to 8. J Proof. Suppose = P,Q is admissible and (v , v ) 2 = 4. Since G has class I { } |Q P Q | number 1, is Γ-conjugate to a set of the form P0, P . By Γ-action, one only I t { } t needs to consider the cases when vP = (i, 2, 2) and vP = (1, 0, 2) . In the case that v = (1, 0, 2)t we claim that P , P is not admissible, and { 0 v} hence D( P0, Pv ) is empty. To see this, it suffices to show that fQ(x) >f0(x) on E( P , P { ) for some} Q. Proposition 3.3 implies that E( P , P ) is the set where { 0 v} { 0 v} β 2 (17) y2 = | | + r r2. − 2 − In particular, 0
Consider the rational parabolic subgroups Q1,Q2,Q3, and Q4 with associated isotropic vectors (0, 0, 1)t, (1+i, 1 i, 1)t, (2i, 2, 1)t, and ( 1+i, 1+i, 1)t respectively. Explicit computation shows that− on S, −
f (x) f (x) f (x)= 0 f (x)= 0 Q1 β 2 Q2 β ( 1+ i) 2 | | | − − | f (x) f (x) f (x)= 0 f (x)= 0 . Q3 β 2i 2 Q4 β (1 + i) 2 | − | | − |
2 Since β i < 1 and fQj (x) f0(x) on D( ), we see that β is forced to equal i. Thus | − | ≤ I
1 D( )= D( P0, Pv, P1, P2,Q1,Q2,Q3,Q4 )= , i, 0 . I { } √2
One checks that 8 is not properly contained in any 4-bounded set to complete the proof. J
Corollary 6.3. The set of isotropic vectors associated to a strongly admissible set of order less than eight is 2-bounded. Furthermore, there are no strongly admissible sets of order greater than eight.
Proof. Propositions 6.1 and 6.2 imply the first statement. The second follows from explicit calculation that shows there are no 4-bounded sets that properly contain 8. J
Proposition 6.4. Let i denote the set of rational parabolic subgroups associated Ij i as defined in Section 4. Then Jj
2 = P , wP 2 = P , ξP I1 { 0 0} I2 { 0 0} 3 2 τw 3 2 σw 3 2 τσw 1 = 1 P0 2 = 1 P0 3 = 1 P0 I I ∪{ } I I ∪{ } I−1 I ∪{ } 4 = 3 σwP 4 = 3 w τσwˇ P I1 I1 ∪{ 0} I2 I3 ∪{ 0} 5 4 τσw = 1 P0 I−1 I ∪{ } 2 4 8 = P , wP , wτσwP , τ σwP , τσwˇ P , τwτσwP , τ σσwˇ P , ǫwξ wP I { 0 0 0 0 0 0 0 0}
Using (3) and Propositions 3.4 and 6.4, one calculates the following first contact points z( ). I EXPLICIT REDUCTION FOR SU(2, 1; Z[i]) 9
1+√5 Proposition 6.5. Let φ = 2 . Then 1 1 z( 2)=(1, 0, 0) z( 2)= , i, I1 I2 √4 2 2 4 3 1 √3 1+ i z( 3)= , 0, z( 3)= , , 0 I1 4 2 I2 2 2 r 1 1 1 z( 3)= φ2 φ 2, (φ2 φ + i(1 φ + φ)), (1 φ + φ) I3 √2 − 2 − − 2 − q p p p p 3+ √3+ √2+ √6 1+ √3 √2 1 z( 4)= − , − (1 + √3i), I1 2 4 2 p 3+ i z( 4)= φ 1, (1 √2), 0 I2 − 2 − 1+2√p3 1+ i 1 1 z( 5)= − , , z( 8)= , i, 0 . I 2 2 2 I √2 p Theorem 6.6. Up to Γ-conjugacy, the strongly admissible sets are exactly 2, 2, 3, 3, 3, 4, 4, 5, 8. I1 I2 I1 I2 I3 I1 I2 I I Proof. Propositions 4.2 and 6.2 and Corollary 6.3 imply that every strongly admis- sible set must be one of the ones listed. Thus, it suffices to show that for each set listed, D′( ) is non-empty. In particular, it suffices to show that D′( ) contains Iits first contactI point. One uses (3) and Proposition 6.5 to show that forIP , 2 ∈P\I fP (z( )) 7. Pieces of the spine Theorem 7.1. Let (y,β,r) denote a point in D. Then 2 2 2 β (i) E( )= y = | | + √1 r2 I1 − 2 − n 2 o 2 (ii) E( 2)= y2 = 1 1 iβ¯ + 1 1 Re(β) r I2 − 2 − 2 − 2 − − q (iii) D( 2) E( 2) 1 (xi) D( 8)= 1 , i, 0 I √2 n o Proof. Proposition 3.3 implies (i) and (ii). Similarly, (v)-(xi) follow from computer calculations and repeated uses of Proposition 3.3. w To show the bounds in (iii), let P = P0 and consider the rational parabolic −1 τ τ t subgroups Q = P0 and R = P0 with associated isotropic vector vQ = (1, 0, 1) and v = ( 1, 0, 1)t respectively. Then (3) implies that R − y (19) fP (z)= ( iβ 2/2 r 2 + y2 β 2 + y4)1/2 | | − | y| | (20) fQ(z)= ( 1+ i β 2/2 r 2 + y2 β 2 + y4)1/2 | | | − | y | | (21) fR(z)= ( 1+ i β 2/2 r 2 + y2 β 2 + y4)1/2 |− | | − | | | 2 For z D( ), f (z) f (z) and fw (z) f (z) and hence (19), (20), and (21) ∈ I1 P ≥ Q P0 ≥ Q imply that 1 r 1 . Since f (z)= y, Proposition 5.2 implies that 1 8. Stabilizers 2 2 3 2 3 Proposition 8.1. The stabilizer in Γ of D′( 1 ) is e,ǫ,ǫ ,ǫ ,w,ǫw,ǫ w,ǫ w and is isomorphic to Z/4Z Z/2Z via the morphismI which{ sends ǫ to the generator} of Z/4Z and ǫw to the generator× of Z/2Z. 2 2 2 Proof. Note that Stab D′( ) = Stab . One can calculate that w Γ 0 Γ I1 Γ I1 ∈ P and hence w Stab 2 and acts non-trivially on the ordered pair (P , wP ). ∈ Γ I 1 0 0 Thus, Stab 2 = L L w, where γ L if and only if γP = P and γwP = wP . Γ 1 0 0 0 0 Since a parabolicI subgroup∪ · is its own normalizer,∈ one computes that L =Γ P wP = e,ǫ,ǫ2,ǫ3 , ∩ 0 ∩ 0 { } and hence, 2 2 3 2 3 Stab D′( ) = e,ǫ,ǫ ,ǫ ,w,ǫw,ǫ w,ǫ w . Γ I1 { } It is easily checked that the map given in the proposition is an isomorphism. EXPLICIT REDUCTION FOR SU(2, 1; Z[i]) 11 2 Proposition 8.2. The stabilizer in Γ of D′( 2 ) is the cyclic group of order eight generated by ξ. I 2 2 2 ξ Proof. Note that StabΓ D′( 2 ) = StabΓ 2 . Recall that 2 = P0, P0 . One 2 I1 I I 2 { } can calculate that ξ = ǫτσ− Γ 0 and hence ξ is in Stab and acts non- ∈ P Γ I2 trivially on the ordered pair (P , ξP ). Thus, Stab 2 = L L ξ, where γ L 0 0 Γ I2 ∪ · ∈ if and only if γP = P and γξP = ξP . Since a parabolic subgroup is its own 0 0 0 0 normalizer, one computes that L =Γ P ξP = e, ξ2, ξ4, ξ6 , ∩ 0 ∩ 0 { } 2 and hence Stab D′( ) is the cyclic group of order eight generated by ξ. Γ I2 3 Proposition 8.3. The stabilizer in Γ of D′( 1 ) is the cyclic group of order twelve generated by τǫw. I 3 3 Proof. Note that Stab D′( ) = Stab . One can easily check that Γ I1 Γ I1 −1 τ P , wP = P , τwP and wτ w P , wP = wP , τwP . { 0 0} { 0 0} { 0 0} { 0 0} Thus Stab 3 =Γ Γ Γ , where Γ I1 1 ∪ 2 ∪ 3 2 γτw τw Γ1 = γ StabΓ P0 = P0 , ∈ I1 2 γτw w Γ2 = γ τ Stab Γ 1 P0 = P 0 , and ∈ · I 1 2 γτw Γ3 = γ wτ − w Stab Γ 1 P0 = P0 . ∈ · I One calculates using Proposition 8.1 that Γ = e,ǫ,ǫ2,ǫ3 , 1 { } Γ = τw,τǫw,τǫ2w,τǫ3w , and 2 { } 1 1 1 2 1 3 Γ = wτ − , wτ − ǫ,wτ − ǫ , wτ − ǫ . 3 { } Explicit matrix multiplication shows that Γ1 Γ2 Γ3 is the cyclic group of order twelve generated by τǫw. ∪ ∪ 3 Proposition 8.4. The stabilizer in Γ of D′( ) is I2 3 1 1 3 3 2 Stab D′( ) = e,ǫw,wσˇ− w, wσˇ− ǫ ,σǫ w,σǫ Γ I2 { } S and is isomorphic to 3. 3 3 3 Proof. Note that StabΓ D′( 2 ) = StabΓ 2 . Fix an ordering of 2 . Since a parabolic subgroup is its ownI normalizer, theI group which preserves theI ordering w σw w is Γ P0 P0 P0. The proof of Proposition 8.1 shows that Γ P0 P0 = ǫ ∩ ∩ ∩ σw 3∩ ∩ h i and one can easily check that ǫ P0 = e . Thus StabΓ 2 is isomorphic h i ∩ { } 1 1 3 I3 2 to a subgroup of S3. One checks that e,ǫw,wσˇ− w, wσˇ− ǫ ,σǫ w,σǫ is a set 3 { }3 of six distinct elements in StabΓ 2 . Thus this set is exactly StabΓ 2 and is isomorphic to S . I I 3 3 Proposition 8.5. The stabilizer in Γ of D′( ) is trivial. I3 2 2 2 Proof. Since (v0, vw) = 1, (v0, vτσw) = 1, and (vw, vτσw) = 2, if 3|Q γ 2| 2 |Qγ 2 τσw| |Q 3 | γ StabΓ 3 , then 1 = 1 or 1 = P0, P0 . Thus StabΓ 3 = Γ1 Γ2, where∈ I I I I { } I ∪ Γ = γ Stab 2 γτσwP = τσwP and 1 ∈ Γ I1 0 0 2 γτσw w Γ2 = γ τσ Stab Γ 1 P0 = P 0 . ∈ · I 12 DAN YASAKI One checks using Proposition 8.1 that Γ = e and Γ = . 1 { } 2 ∅ 4 Proposition 8.6. The stabilizer in Γ of D′( ) is trivial. I1 4 4 Proof. Note that StabΓ D′( 1 ) = StabΓ 1 . The set of isotropic vectors as- 3 4 I I 3 sociated to 1 1 span a two dimensional subspace of C while the isotropic I ⊂ I 4 3 vectors associated to any other order three subset of 1 span all of C . Thus if γ Stab 4 then γ 3 = 3 and γσwP = σwP ThereforeI one calculates from ∈ Γ I1 I1 I1 0 0 Proposition 8.3 that Stab 4 = Stab 3 σwP = e . Γ I1 Γ I1 ∩ 0 { } 4 Proposition 8.7. The stabilizer in Γ of D′( 2 ) is cyclic of order two generated by ǫw. I Proof. Explicit computation show that −1 ǫw 3 w w τσwˇ 3 = P0, P0, P0 , −1 I { −1 } στwτσ 3 τσw w τσwˇ 3 = P0, P0, P0 , and −1 I { −1 } στwτσ w 3 = wP , τσwP , w τσwˇ P . I3 { 0 0 0} Then Stab 4 =Γ Γ Γ Γ , where Γ I2 1 ∪ 2 ∪ 3 ∪ 4 −1 −1 Γ = γ Stab 3 γw τσwˇ P = w τσwˇ P , 1 ∈ Γ I3 0 0 n −1 o Γ = γ ǫw Stab 3 γw τσwˇ P = τσwP , 2 ∈ · Γ I3 0 0 −1 n 1 3 γw τσwˇ ow Γ = γ στwτσ− Stab P = P , and 3 ∈ · Γ I3 0 0 −1 n 1 3 γw τσwˇ o Γ = γ στwτσ− w Stab P = P . 4 ∈ · Γ I3 0 0 n o 3 By Proposition 8.5, StabΓ 3 is trivial. It is easy to check that Γ1 = e , Γ2 = ǫw , and Γ =Γ = . I { } { } 3 4 ∅ 5 Proposition 8.8. The stabilizer group of D′( ) is the cyclic group of order two generated by σǫ2. I 5 5 Proof. Note that Stab D′( ) = Stab . With the exception of P , for Γ I Γ I 0 every P 5, there exists a Q 5 such that (v , v ) 2 = 2. Therefore if ∈ I ∈ I |Q P Q | γ Stab 5 , then γP = P . This implies that ∈ Γ I 0 0 5 w τw σw τσw StabΓ =ΓP0 StabΓ ( P0, P0, P0, P0 ) I ∩ −1 {−1 −1 } wτ ǫwσ τ 4 =ΓP0 StabΓ 2 ∩ −1 −1 −1 I wτ ǫwσ τ =Γ 0 e,ǫw from Proposition 8.7. P ∩ { } One easily checks that this intersection is e,σǫ2 . { } EXPLICIT REDUCTION FOR SU(2, 1; Z[i]) 13 8 Proposition 8.9. The stabilizer in Γ of D′( ) is the group of order 32 generated by ξ2 and ǫw given below: I 8 2 4 6 2 4 6 2 2 2 2 4 Stab D′( ) = e, ξ , ξ , ξ ,ǫw,ǫwξ ,ǫwξ ,ǫwξ , ξ ǫw,ξ ǫwξ , ξ ǫwξ , Γ I { 2 6 4 4 2 4 4 4 6 6 6 2 ξ ǫwξ , ξ ǫw,ξ ǫwξ , ξ ǫwξ , ξ ǫwξ , ξ ǫw,ξ ǫwξ , ξ6ǫwξ4, ξ6ǫwξ6,ǫwξ2ǫw,ǫwξ4ǫw,ǫwξ6ǫw,ξ2ǫwξ2ǫw, ξ2ǫwξ4ǫw,ξ2ǫwξ6ǫw,ξ4ǫwξ2ǫw,ξ4ǫwξ4ǫw,ξ4ǫwξ6ǫw, ξ6ǫwξ2ǫw,ξ6ǫwξ4ǫw,ξ6ǫwξ6ǫw } It is isomorphic to the group of order 32 with Hall-Senior number 31 [12]. 8 8 2 8 Proof. Note that Stab D′( ) = Stab . Consider . Since the set Γ I Γ I I1 ⊂ I of isotropic vectors associated to 2 is 1-bounded, if γ Stab 8 , then the I1 ∈ Γ I set of isotropic vectors associated to γ 2 is 1-bounded. There are 16 such subsets 1 = γI 2 8 with this property. ThenI I I1 ⊂ I 8 StabΓ = Γ , I 8 I I⊂Ia 2 is Γ-conjugate to 1 I I where 2 γ 8 2 8 Γ = γ γ StabΓ 1 1 = . I ∈ I · I {I \ I } {I \I} One can compute that each Γ has exactly two elements. Explicit computation of each Γ gives the 32 elementsI listed. I 8 8 Fix an ordering [ ]. This induces a homomorphism ψ : StabΓ S8. The I 8 I → subgroup of Stab which fixes the ordering is ker(ψ)=Γ 8 P . One Γ I ∩ P calculates that this intersection is trivial. Thus Stab 8 is isomorphic∈I to its Γ I T image ψ(Stab 8 ) in S . In particular, non-trivial elements of ψ(Stab 8 ) Γ 8 Γ written in disjointI cycle notation consist of 2, 4, or 8-cycles. Thus the exponentI 8 8 of StabΓ is 8. One computes that the center of StabΓ is cyclic of order four. ThereI are only three groups of order 32 with exponent 8I and center which is cyclic of order four, namely 26, 31, and 32 in the lists given in [12]. These are distinguished by the numberG of conjugacyG G classes of maximal elementary abelian subgroups, that is, subgroups which are isomorphic to (Z/2Z)r for some r. , G26 31, and 32 have one, two, and three conjugacy classes of maximal elementary G G 8 abelian subgroups, respectively. One calculates that StabΓ has two conjugacy classes of maximal elementary abelian subgroups. I 9. Structure of the spine 9.1. Cell structure and action of the stabilizers. From the previous compu- tations, one can get a very explicit description of the cellular structure of the spine. For example, to find the strongly admissible sets of order three that are on the 2 boundary of 1 , one needs to find all rational parabolic subgroups Q such that 2 I 3 3 3 = 1 Q is Γ-conjugate to either 1 , 2 , or 3 and hence reduces to a simple I I ∪{ } I I I t t calculation of the inner-products of vQ with (1, 0, 0) and (0, 0, 1) . Table 1 summarizesQ the strongly admissible sets up to Γ-conjugacy and gives their stabilizers. The incidence table is given in Table 2, where the entry below the diagonal means that each column cell has that many row cells in its boundary, and the entry 14 DAN YASAKI Table 1. Stabilizer groups of strongly admissible sets S.A.S. Stabilizer Generators 2 1 Z/2Z Z/4Z ǫw,ǫ I2 × h i 2 Z/8Z ξ I3 h i 1 Z/12Z τǫw I3 h i2 2 S3 ǫw,σǫ I3 h i 3 trivial e I4 h i 1 trivial e I4 h i 2 Z/2Z ǫw I5 Z/2Z hσǫ2i I8 ǫw,ξ2 h i I h i G31 • • •\ • • g bbbbb + \\\\ *WWW• •gg •/ WW • • // • W \\\\\ • •b+bbbb +g*gg // ?/ -- + // ? - + + * // /?? - + // • ? + + // • / ?• • • - • + •* // •b b+a a a - - ] ] ]+ \ \/• 7 • Õ •/ • - •+ • 9 / /\\\\*]]]]]•• - aaaabbbb × 99Õ / *+ + a ×7• 9 / // + • × *Õ 9 • / * + + - ×? 9• // * + ? * / * + + -• • ? * • ___ • ? //• * b b b+ \ \ \ W • ?* /•gg + W• ?• W *• + • WW \\\\\••bbbbb ggg D( 2) D( 2) I1 I2 Figure 1. The sets associated to the two Γ-conjugacy classes of 2 strongly admissible sets of order two are shown here. D( 1 ) is homeomorphic to a polytope with dodecagon, hexagon and quadri-I lateral faces, while D( 2) has only quadrilateral faces. I2 Table 2. Incidence types 2 2 3 3 3 4 4 5 8 1 2 1 2 3 1 2 2 I I I I I I I I I 1 3 3 2 5 4 8 16 I2 ∗ ∗ 2 0 0 1 1 2 2 8 I3 ∗ ∗ 1 2 0 1 0 2 0 I3 ∗ ∗ ∗ 2 4 0 1 0 2 0 I3 ∗ ∗ ∗ 3 12 8 2 4 6 32 I4 ∗ ∗ ∗ 1 40 8 12 6 2 4 0 I4 16 8 0 0 2 ∗ ∗ 1 16 I2 ∗ ∗ 5 32 8 12 6 3 2 1 I8 4 2 0 0 1 0 1 ∗ ∗ I ∗ ∗ above the diagonal means the column cell appears in the boundary of this many row cells. The entries below the diagonal can be read off from Figures 2-4, while the entries above the diagonal can be easily computed from Proposition 9.1, since the Γ-conjugacy class of a strongly admissible can be distinguished by the pairwise 3 -inner products of its associated isotropic vectors, except to distinguish 1 and Q3, we must also compute the dimension of the span of their isotropic vectors.I I2 EXPLICIT REDUCTION FOR SU(2, 1; Z[i]) 15 Define a map Q : ordered subsets of Mat(Z) as follows: Given an or- { P} → dered subset [ ] = (P1,...,Pn) , let Q ([ ]) be the n n matrix whose (i, j)- I 2 ⊂ P I × component is v , v . Q Pi Pj Proposition 9.1. Let be strongly admissible set. Then I 0 1 2 if is Γ-conjugate to 1 , 1 0! I I 0 2 2 if is Γ-conjugate to 2 , 2 0 I I ! 0 1 1 3 3 1 0 1 if is Γ-conjugate to 1 or 2 , I I I 1 1 0 0 1 2 1 0 1 if is Γ-conjugate to 3, I I3 2 1 0 0 11 2 1 01 1 if is -conjugate to 4, Γ 1 1 10 1 I I 2 11 0 0 11 2 Q ([ ]) = 1 02 1 I if is Γ-conjugate to 4, 1 20 1 I I2 2 11 0 01112 10121 5 11011 if is Γ-conjugate to , I I 12101 21110 01211214 10122141 21011412 12104121 if is -conjugate to 8. Γ 12140121 I I 21411012 14122101 41211210 In Figures 2-4, a 0-cell that is Γ-conjugate to 5 will be denoted by , and a 0-cell that is Γ-conjugate to 8 will be denoted byI⊛. A 1-cell that is Γ-conjugate• 4 I 4 to 1 will be denoted with a solid line, and a 1-cell that is Γ-conjugate to 2 will beI denoted with a dotted line. I 4 The 1-cell D( 1 ) is shown in Figure 2. The boundary consists of two 0-cells that are both conjugateI to D( 5). By Proposition 8.6, the stabilizer of D( 4) is trivial. I I1 16 DAN YASAKI 5 1 5 1 8 1 5 (τǫw)− τσǫw− (τǫw)− I · I ⊛ · I · I • • • D( 4) D( 4) I1 I2 Figure 2. Strongly admissible sets of order 4 5 4 M 1 q MM I 5 qqq• • MM I 1 11 qq M1 • 1 • 1 I • •11 1 11 • • 11 1 1 1 11 3 l2 1 1 • z( 1 ) • 1 I 1 1 l1 ______1 · •11 z( · 3) • 1 11 I2 •11 • 1 11 11 4 1 1 ⊛ I2 MMM 5 4 q 11 • MM 1 qqq• • IM I qq • • • • D( 3) D( 3) D( 3) I1 I2 I3 Figure 3. Strongly admissible sets of order 3 4 The 1-cell D( 2 ) is shown in Figure 2. The boundary consists of two 0-cells (one that is conjugateI to D( 5) and one that is conjugate to D( 8)). By Proposition 8.7, 4 I I the stabilizer of D( 2 ) is isomorphic to Z/2Z and is generated by ǫw. This acts on the cell by fixing itI pointwise. 3 The 2-cell D( 1 ) is shown in Figure 3. Its boundary consists of twelve 1-cells I 4 (all of which are conjugate to D( 1 )) and twelve 0-cells (all of which are conjugate 5 I 3 to D( )). By Proposition 8.3 the stabilizer of D( 1 ) is isomorphic to Z/12Z and is generatedI by τǫw. It is easily checked that τǫw actsI on the figure by rotation by π 3 3 6 about the first contact z( 1 ) for D( 1 ). 3 I I The 2-cell D( 2 ) is shown in Figure 3. The boundary consists of six 1-cells (all of I 4 5 which are conjugate to D( 1 )) and six 0-cells (all of which are conjugate to D( )). 3 I 3 I The point z( 2 ) is the first contact for 2 . The lines lj represent the gradient flows I 3 I of the f0 function restricted to D( 2 ) from the first contact to two of the vertices of 3 I 3 S D( 2 ). By Proposition 8.4, the stabilizer of D( 2 ) is isomorphic to 3 generated I 2 2 I 2 by ǫw and σ ǫ . One can check that ǫw acts as reflection about l1 and σǫ acts as reflection about l2. 3 The 2-cell D( 3 ) is shown in Figure 3. The boundary consists of four 1-cells I 4 4 (two that are conjugate to D( 1 ) and two that are conjugate to D( 2 )) and four 0-cells (three that are conjugateI to D( 5) and one that is conjugate toI D( 8)). By 3I I Proposition 8.5, the stabilizer of D( 3 ) is trivial. I 2 See Figure 4 for a picture of the face relations for D( 1 ). The outside of the figure I 2 is the dodecagon bottom face seen in Figure 1. The boundary of D( 1 ) consists 3 I of twenty-two 2-cells (two that are conjugate to D( 1 ), four that are conjugate to 3 3 I D( 2 ), and sixteen that are conjugate to D( 3 )), fifty-six 1-cells (forty that are I 4 I 4 conjugate to D( 1 ) and sixteen that are conjugate to D( 2 )), and thirty-six 0-cells (thirty-two thatI are conjugate to D( 5) and four that areI conjugate to D( 8)). By I I EXPLICIT REDUCTION FOR SU(2, 1; Z[i]) 17 i 5UU iii Ù• •5 UUU 8 iiii ÙÙ 55 UUUU ii Ù U ) I⊛ • •77 ו •) ⊛ 77 ×× ⊛ ) 7 M×× ) a qq MM )) d qqq • • MM1 ) G • •1 I ) uu• GG 11 w• II) uu GG 1 www I) a • • d • 3 w • τw •3 • I1 • 8⊛·⊛ I3 ⊛ 8 1 · G I • • I )I)I ww•1 • GG u b c ••) II ww 11 GG uu • ) w 1 u •• ) • MM 5 4 q • ) • MM 1 qq• ) 3 IM I q7q 8 b c ) ⊛3 × 7 ⊛ ) 4I ×ו•• • 77 I ) 2 ×× 3 7 U I 5 2 ⊛ UU 5 Ù ii 8 ••UUUU••5 I Ù iiii UUU5 ÙiÙiii I ••D( 2) D( 2) I1 I2 Figure 4. Strongly admissible sets of order 2 2 Proposition 8.1 the stabilizer of D( 1 ) is isomorphic to Z/4Z Z/2Z generated by I × π ǫ and ǫw. It is easily calculated that ǫ acts by rotating the figure by 2 and ǫw acts by an inversion, sending the interior 12-gon to the exterior one. − 2 The other 3-cell D( 2 ) is shown in Figure 4. The labels a,b,c,d are added I { } 2 to show the identifications that need to be made. The boundary of D( 2 ) consists 3 I of eight 2-cells (all of which are conjugate to D( 3 )), sixteen 1-cells (eight that 4 I 4 are conjugate to D( 1 ) and eight that are conjugate to D( 2 )) and ten 0-cells (eight that are conjugateI to D( 5) and two that are conjugateI to D( 8)). By I 2 I Proposition 8.2, the stabilizer of D( 2 ) is isomorphic to Z/8Z generated by ξ. It is easily checked that ξ acts on the figureI by the composition of an inversion sending π the exterior point to the interior point and a rotation of 4 . 9.2. The subdivision. From the explicit description of the cells and Γ-action, it is straightforward to subdivide the spine so that the stabilizer of each cell fixes the cell pointwise. 2 First consider D( 1 ). It must be divided into eight 3-cells that are conjugate via Stab 2 . One ofI the 3-cells X is represented by the shaded region in Figure 4. Γ I1 It is easy to see that the boundary of X consists of two 2-cells, C1 and C2, that are 3 3 3 Γ-conjugate to D( 3 ), half of the 2-cell D( 2 ) B, one-fourth of the 2-cell D( 1 ), A, and three new faces,I E , E , and D thatI lie inside D( 2) and such that E Iand 1 2 I1 1 E2 are Γ-conjugate. Next we turn to subdividing the boundary faces of X. From the description of the action of the Stab 3 on D( 3), it follows that A must Γ I1 I1 be subdivided into three Γ-conjugate 2-cells A1, A2, and A3. Similarly, because of 3 3 the action of StabΓ 2 on D( 2 ), B must be subdivided into three Γ-conjugate 2-cells B , B , and BI. One canI compute that the stabilizers of E are trivial, and 1 2 3 i that the stabilizer of D fixes D pointwise. The 1-cells of X, both the ones that 4 are Γ-conjugate to D( j ) (j = 1, 2) and the new ones that are introduced for the subdivision, do not needI to be subdivided because the stabilizers are either trivial, or the stabilizer acts on the 1-cell by fixing it pointwise. This yields X in Figure 5. 18 DAN YASAKI n g g n &\\\\\\\\\ m bbbbbbbb •& \\\\\\\\\\\-Obbbbbbbbbbbb • && oo -OO & a4 ooo • - OOOa1 & E2 oo - OO E1 & ooo - OOO & o6 oo a3 - a2 OOo3 r f2 & Moo A3 - A1 O f1 & MM A2 - qq 11 & • MMM -- qqq • • 1 & MMo4 d3 o-2 qq 1 & e4 d4 M1 q1 q d2 e2 i2 1i3 & 1 1 G2 11 & C2 • 1 • 1 C1 1 ⊛& 1 1 ⊛ o2 1 o3 p2 9 c2 1 B2 1 p1 N 9 d5 1 b1 1d1 Õ i2 qq d2 NNi3 99e3 1 1 e1 ÕÕ qqq • • NN 9 B3 11 B1 11 Õ qq NN 9 Õ r qq e2 N r 9 o q c o Õ NN G1 d1 C1 F2 q 99 •5 b2 • 1 •1 ÕÕ • NN qq • 9 Õ NN qqq 99 ÕÕ i1 NN e1 ⊛qq h 9 Õ o 11 qp 99 D ÕÕ 1 • 11 1 f2 9 Õf1 1 F1 9 Õ i1 1 h 99 ÕÕ 1 9 Õ 11 99 ÕÕ 9 Õ r 99 ÕÕ • 9ÕÕ n• X Y Figure 5. Subdivided cells 2 Similarly, D( 2 ) must be subdivided into eight 3-cells that are conjugate via 2 I StabΓ 2 . One of the 3-cells Y can be viewed in Figure 4 by taking a face C1 that I 3 2 is Γ-conjugate to D( 3 ) and looking at the set of points of D( 2 ) that would flow I I 2 to C1 under the gradient flow for f 2 and adding the first contact point of 2 . − I2 I Then the boundary of Y consists of five 2-cells, C1, F1, F2, G1, and G2 such that F1 is Γ-conjugate to F2 and G1 is Γ-conjugate to G2. The calculations above show that the stabilizer of C1 is trivial, and it is easy to check that the stabilizers of F1, F2, G1, and G2 are trivial as well. Hence the 2-cells do not need to be subdivided. Similarly, one checks that the stabilizers of the 1-cells are either trivial or fix the 1-cell pointwise. This yields Y in Figure 5. In the figures, the labels that only differ by a subscript are conjugate under Γ. For each Γ-conjugacy class, we fix a representative and compute the stabilizers. The results are given in Table 3. 10. Cohomology of Γ D with local coefficients \ 0 In this section only, we generalize to the case where D = G/K is a non-compact symmetric space, where G is the group of real points of a semisimple linear algebraic group G defined over Q. Let Γ be an arithmetic subgroup of G(Q). Let D0 D denote a spine with CW-structure such that the Γ-stabilizer of a cell fixes the⊂ cell pointwise. In order to set notation, we recall the definition of orbifold and the sheaf asso- ciated to the local system. We then prove that the cohomology of Γ D with local coefficients is isomorphic the the cohomology of Γ D with local coefficients.\ \ 0 10.1. Orbifolds. The notion of an orbifold was first introduced by Satake in [16]. He called them V-manifolds. Let M be a Hausdorff topological space. A local uniformizing system (l.u.s.) U, ΓU , φ for an open set V M is a collection of the following objects: { } ⊂ EXPLICIT REDUCTION FOR SU(2, 1; Z[i]) 19 Table 3. Representative Cells and Their Stabilizers Cell Dimension Stabilizer Generators X 3 trivial e Y 3 trivial hei h i A1 2 trivial e B 2 trivial hei 1 h i C1 2 trivial e D 2 Z/2Z hǫwi E 2 trivial h e i 1 h i F1 2 trivial e G 2 trivial hei 1 h i a1 1 trivial e h i 1 b1 1 Z/2Z σǫwσ− c 1 Z/2Z h ǫw i 1 h i d1 1 trivial e e 1 Z/2Z hǫwi 1 h i f1 1 Z/2Z ǫw g 1 Z/4Z h ǫ i h 1 Z/4Z hξ2i h i i1 1 trivial e m 0 Z/12Z τǫwh i n 0 Z/4Z Z/2Z hǫw,wi × h i o1 0 Z/2Z ǫw h i2 p1 0 31 ǫw,ξ G h 2i q 0 S3 ǫw,σǫ r 0 Z/8 h ξ i h i (i) U: a connected open subset of Rn. (ii) G: a finite group of linear transformations of U to itself such that the set of fixed points of G is either all of U or at least codimension 2. (iii) φ: a continuous ΓU -invariant map U V such that the induced map Γ U V is a homeomorphism. → U \ → Let V V ′ M be two open sets and let U, Γ , φ and U ′, Γ ′ , φ′ be local ⊂ ⊂ { U } { U } uniformizing systems for V and V ′ respectively. An injection λ : U, Γ , φ { U } → U ′, Γ ′ , φ′ is a smooth injection λ : U U ′ such that for any γ Γ , there { U } → ∈ U exists a γ′ ΓU ′ such that λ γ = γ′ λ and φ = φ′ λ. Let L be∈ a family of local◦ uniformizing◦ systems for◦ open sets in M. Then an open set V M is said to be L-uniformized if there exists a local uniformizing system for V⊂in L. Definition 10.1. An orbifold is a Hausdorff space M and a family L of local uniformizing systems for open sets in M satisfying the following conditions. (i) Let V V ′ M be two open sets and let U, Γ , φ and U ′, Γ ′ , φ′ be ⊂ ⊂ { U } { U } local uniformizing systems for V and V ′ respectively. Then there exists an injection U, ΓU , φ U ′, ΓU ′ , φ′ . (ii) The L-uniformized{ }→{ open sets form} a basis of open sets for M. 20 DAN YASAKI Two families of local uniformizing systems L1 and L2 defining an orbifold M are said to be equivalent if L1 L2 satisfies the conditions above. Equivalent families define the same orbifold structure∪ on M. Thus when talking about family of local uniformizing systems for an orbifold M, we will mean a maximal family. A smooth function on M is given locally on L-uniformized sets V by ΓU -invariant smooth functions on U. Similarly, a smooth p-form on M is given locally on V by ΓU -invariant smooth forms on U. Note that a smooth manifold is an example of an orbifold where every group Γ for U, Γ , φ L is the trivial group. If M˜ is a smooth manifold and Γ is a U { U } ∈ properly discontinuous group of automorphisms of M˜ , then the quotient Γ M˜ has a canonical structure of an orbifold. \ 10.2. Γ D as an orbifold. Since Γ acts properly discontinuously on D, X =Γ D has a canonical\ structure of an orbifold. Let π denote the projection D X. \ Given a connected open set V X with local uniformizing system L, we→ can and ⊂ will identify L with a triple U, StabΓ (U) , π U where U is a connected component 1 { | } of U˜ = π− (V ). A smooth function on V is given by a StabΓ (U)-invariant smooth function on U. In particular, for an open subset O X, a smooth function on O 1 ⊂ is given by a Γ-invariant function on π− (O). Since D0 is a spine, there exists a Γ-equivariant deformation retractr ˜ : D D0. Then there is an induced deformation retract r : X X =Γ D . → → 0 \ 0 10.3. Locally constant orbifold (fiber) bundles over Γ D. Let E be a Γ- \ module with Γ-action given by ρ : Γ GL(E). Let D Γ E denote the quotient space D E/ (x, v) (γ x, ρ(γ)v) γ→ Γ . For a Γ-invariant× deformation retract D D,× we can{ similarly∼ · define D | ∈E.} This yields the following maps. 0 ⊆ 0 ×Γ / D0 E / D E r × × KK π˜0 rr KKKπ˜ rrr KK x rr r˜ KK % rx r s K% D0 Γ E D0 4 D D Γ E × r KK × π0 r KK π rr KK rr KK y rr r K % ry qr K% X =Γ D q 1 Γ D = X 0 \ 0 \ Suppose that E is an N-dimensional vector space. If Γ is torsion-free, then X is a smooth manifold and D E is a flat rank-N vector bundle over X. If Γ has ×Γ torsion, then X is an orbifold and D Γ E is called a flat orbifold bundle over X. Consider the presheaf on X defined× as follows. For every open set U X, E ⊂ 1 f is locally constant and −1 (U)= f : π− (U) E f(γ x)=ρ(γ)f(x) γ Γ, x π (U) E → · ∀ ∈ ∈ n o with the obvious restriction maps. Let E denote the sheafification of . Similarly E define the presheaf 0 on X0 and let E0 denote the sheafification of 0. Note that if Γ is torsion-free, E Eis the sheaf associated to the local system X definedE by (E,ρ). We will extend this terminology to X, respectively X0, when Γ is not torsion-free EXPLICIT REDUCTION FOR SU(2, 1; Z[i]) 21 and say that E, respectively E0, is the sheaf associated to the local system on X, respectively X0, defined by (E,ρ). Proposition 10.2. The sheaf E0 on X0 is isomorphic to the push-forward r E of the sheaf E on X. ∗ Proof. Let V0 X0 be a contractible open set. Let U0 be a connected component ˜ 1 ⊂ of V0 = π0− (V0). Since Γ acts properly discontinuously on D0, by shrinking V0 (and hence shrinking U ) if necessary, one can arrange that Γ γ Γ γ U U = 0 U0 ≡{ ∈ | · 0∩ 0 6 ∅} is a finite group. A section f E (V ) is a locally constant map f : V˜ E such ∈ 0 0 0 → that f(γ x)= ρ(γ)f(x) for all γ Γ. Since V0 is a connected open set, it follows · 1∈ that π (U )= V and Γ U = π− (V ), and hence f is determined by its value v 0 0 0 · 0 0 0 U0 on U . Furthermore, by the Γ-equivariance of f, v EΓU0 , the subspace of E 0 U0 ∈ fixed by ΓU0 . 1 Recall that r E(V0) = E(r− (V0)). Since π0 r˜ = r π, there is a connected ∗ 1 1 ◦ 1 ◦ component U of π− (r− (V0)) such that U =r ˜− (U0). Since Γ acts properly discontinuously on D, Γ γ Γ γ U U = is finite. Thus a section ψ U ≡ { ∈ | · ∩ 6 ∅} ∈ r E(V0) is determined by its value uU on U. Furthermore, by the Γ-equivariance ∗ ΓU of ψ, uU E , the subspace fixed by ΓU . Thus to∈ show that the sheaves are isomorphic, it suffices to show that for suf- ficiently small V0 X0, the groups ΓU0 and ΓU defined above are equal. It is clear that Γ Γ⊂ . To show the opposite inclusion, suppose γ Γ . Then by U0 ⊆ U ∈ U definition γ U U = . Let x γ U U. In particular, x U,sor ˜(x) U0. Fur- · ∩ 6 ∅ 1 ∈ · ∩ 1 ∈ ∈ thermore, since x γ U, γ− x U. Thusr ˜(γ− x) U and the Γ-equivariance ofr ˜ implies thatr ˜∈(x) · γ U ·and∈ hence γ Γ . · ∈ ∈ · 0 ∈ U0 10.4. Cohomology of subspaces. We recall without proof two results of sheaf cohomology. A reference for this section is [10]. Theorem 10.3 ( [10, Theorem 10.6]). Let be a sheaf on a paracompact space X, and let A X be a closed subspace. Let F be the set of all open neighborhoods of A. Then ⊂ N lim H∗(N; N ) ∼= H∗(A; A). N F| F| −→∈N Theorem 10.4 ( [10, Theorem 11.7]). Let X be a paracompact space. Let f : X p 1 → Y a closed map and a sheaf on X. Suppose that H (f − (y); f −1(y))=0 for p> 0 and all y Y .F Then the natural map F| ∈ f † : H∗(Y ; f ) H∗(X; ) ∗F → F is an isomorphism. Let r : X X denote the deformation retraction arising from the Γ-equivariant → 0 deformation retraction of the symmetric space D to the spine D0. Let (E,ρ) be a Γ-module and let E and E0 denote the associated local systems on X and X0 respectively. E E Theorem 10.5. H∗(X; ) ∼= H∗(X0; 0). Lemma 10.6. For every z D, ∈ StabΓ (z) = StabΓ (˜rt(z)) for t< 1 and Stab (z) Stab (˜r (z)) . Γ ⊆ Γ 1 22 DAN YASAKI Proof. Let γ be an element of Stab (z). Then γ r˜ (z)=˜r (γ z)=˜r (z). Hence Γ · t t · t γ StabΓ (˜rt(z)). Notice that for each z / D0, c(t)=˜rt(z), 0 t 1 is a reparam- eterization∈ of a geodesic, and Γ acts by isometries.∈ Thus every≤ γ≤ Stab (z) fixes ∈ Γ the geodesic throughr ˜1(z) and z. In particular, StabΓ (z′) = StabΓ (z) whenever z =r ˜t(z′) for some t< 1. 1 Proof of Theorem 10.5. We will show that H∗(N; E ) = H∗(r− (y); E −1 ) for |N ∼ |r (y) all contractible neighborhoods N of y X0. Let U D be a connected component 1 1 ∈ ⊂ of π− (r− (y)). Lety ˜ denote the unique lift of y in U. Note that U is a finite union of geodesic rays emanating fromy ˜. By Proposition 10.6, the stabilizer groups Stab (˜x) are isomorphic forx ˜ in a connected component of U y˜. Let N˜ U be Γ \ ⊆ a contractible open subset containingy ˜ and let N = π(N˜). Then it is clear that 1 1 H∗(r− (y); E −1 ) = H∗(N, E ). Apply Theorem 10.3 with X = r− (y), = |r (y) ∼ |N F E −1 , A = y , and the family of all contractible open sets containing y. Then |r (y) { } N 1 E −1 E E H∗(r− (y); r (y)) ∼= lim H∗(N; N ) ∼= H∗(y; y). | N | | −→∈N p 1 In particular, H (r− (y); E −1 )=0 for p> 0. |r (y) By Theorem 10.4, this implies that H∗(X; E) = H∗(X0; r E). By Proposi- ∼ ∗ tion 10.2, r E = E0 and the result follows. ∗ ∼ 10.5. Computing the cohomology from the cell structure. The cellular struc- ture allows us to compute the cohomology H∗(Γ D0; E) combinatorially once Γ- representatives of cells and the stabilizers of those\ cells have been computed. We first set up some notation. For each p, fix a set Rp of representatives of Γ-conjugacy classes of p-cells of D0. Let [τ] denote the representative of the class of τ. For each representative [τ], fix a distinguished maximal flag of cells F = τ < τ < < τ , where τ is an i-cell and τ = [τ]. [τ] 0 1 ··· p i p For each cell τ, let γ Γ be such that γ F terminates in τ. Note that γ is τ ∈ τ · [τ] τ well-defined up to StabΓ (τ). For each fixed p-cell, let Sσ denote the simplicial complex arising from the poset of cells in σ with the partial ordering derived from containment. In particular, the vertices of Sσ are the cells contained in σ and the k-simplices are the (k + 1)-flags σ0 < σ1 < < σk. Define a map nσ : p-simplices of Sσ 1 by the equation ∂S = ··· n (F )∂F . Multiply by { 1 if necessary so}→{± that n (}γ F ) = ( 1)p. σ F Sσ σ σ σ [σ] Then for each∈ σ, define a map − − P (23) sgnσ(τ)= nσ(γτ F[τ] < σ). Theorem 10.7. The cohomology H∗(X0; E) can be computed from the complex 0 EStabΓ(σ) EStabΓ(σ) EStabΓ(σ) 0, → → →···→ → σ R0 σ R1 σ R M∈ M∈ M∈ where the differential d : EStabΓ(σ) EStabΓ(σ) → σ Rp−1 σ Rp ∈M M∈ is given by (dv)σ = sgnσ(τ)ρ (γτ ) v[τ]. τ a (p 1)-cell ∂σ −X ∈ EXPLICIT REDUCTION FOR SU(2, 1; Z[i]) 23 Here γ , sgn, and [ ] are defined above and the vector v is the [τ]-component of τ · [τ] the vector v EStabΓ(σ). σ Rp−1 ∈ ∈ L 11. Cohomology Recall that ǫ, w, σ, τ, and ξ were explicitly defined elements of G given in Section 2. Theorem 11.1. Let E be a Γ-module with the action of Γ given by ρ :Γ GL(E). → Then H∗(Γ D; E) can be computed from the following cochain complex. \ 0 C0 C1 C2 C3 0, → → → → → where 2 2 0 ρ(τǫw) ρ(ǫ),ρ(w) ρ(ǫw) ρ(ǫw),ρ(ξ ) ρ(ǫw),ρ(σǫ ) ρ(ξ) C = E Eh i E Eh i Eh i E ⊕ −1 ⊕ ⊕ ⊕ ⊕2 C1 = E Eρ(σǫwσ ) Eρ(ǫw) E Eρ(ǫw) Eρ(ǫw) Eρ(ǫ) Eρ(ξ ) E ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ C2 = E E E Eρ(ǫw) E E E ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ C3 = E E. ⊕ Then for (κ ) C0, (λ ) C1, and (µ ) C2, the differentials are given by i ∈ i ∈ i ∈ κ + ρ(ξ2)κ − 1 3 ρ(ξ)κ3 κ5 κ −κ 3 − 5 κ + ρ(ξ)κ − 3 3 d (κ)= κ3 κ4 0 − κ2 + κ4 − κ1 κ2 − κ4 κ6 − κ3 κ6 − λ + ρ(τǫw)λ + ρ(ξ)λ − 1 1 4 λ2 λ3 λ4 − − 2 λ4 + ρ(ξ)λ4 + λ5 ρ(ξ )λ5 2 − 1 d1(λ)= ρ(σǫ )λ2 + λ3 λ5 + ρ(τσwτσ− )λ5 λ6 + ρ(ǫ)λ6 − − 2 − λ1 ρ(ξ )λ5 λ6 + λ7 − − λ5 λ8 + λ9 − − λ4 + λ9 ρ(ξ)λ9 − Aµ + Bµ µ ρ(τσǫw 1)µ µ + µ ρ(ǫ)µ d (µ)= 1 2 3 − 3 4 5 5 2 µ− µ−+ ρ(ξ2)µ + µ−+ ρ(ξ)µ − − 3 − 6 6 7 7 where 2 1 1 A = ρ(I)+ ρ(τǫw)+ ρ(τǫw) and B = ρ(I)+ ρ(σǫwσ− ) ρ(σǫwσ− ǫw). − − An application of Theorem 11.1 is the following. Corollary 11.2. The group Γ = SU(2, 1; Z[i]) is generated by ǫ,w,τ,σ . { } Lemma 11.3. Let H be a subgroup of a group G. If H = G, then there exists a representation (E,ρ) of G such that Eρ(H) = Eρ(G). 6 6 24 DAN YASAKI Proof. Consider the representation (E,ρ) of functions φ : G/H C with the left → regular action of G on φ. The characteristic function χeH of the identity coset eH is fixed by H, but not fixed by G for G = H. 6 Proof of Corollary 11.2. Let ρ : Γ GL(E) be a representation of Γ. Then the 0 E → E Γ 0 E cohomology H (X; ) is equal to the global sections (X) ∼= E . Since H (X; ) 0 E ǫ,w,τ,σ is the kernel of d0 given in Theorem 11.1, H (X; ) ∼= Eh i. It is well-known 0 E ρ(Γ) that H (X; ) ∼= E . The result then follows from Lemma 11.3. The following corollary follows immediately from the form of d2 given in Theo- rem 11.1. Corollary 11.4. Let E be a Γ-module with action given by ρ :Γ GL(E). Let E denote the associated sheaf. Then → rank(H3(Γ D; E)) rank(E) rank(Eρ(ǫw)). \ ≤ − Corollary 11.5. Let E be a finite dimensional complex representation of SU(2, 1). Let Eij denote the iω1+jω2 weight space of E, where ω1 and ω2 are the fundamental (complex) weights of sl3C, the complexification of su(2, 1). Then dim H3(Γ D; E) dim E + dim (E ker(I + ρ(w))) . \ ≤ ij ii ∩ i>j i Xj(4) X ≡ 3 Proof. The dimension of H (Γ D; E) = 2 dim(E) rank(d2). From the form of d2 given in Theorem 11.1, it follows\ that − (24) dim H3(Γ D; E) dim(E) rank(Φ), \ ≤ − where Φ : Eρ(ǫw) E E is the linear map given by Φ(v)= v + v ρ(ǫ)v . Let ⊕ → 1 2 − 2 E+ denote the (+1)-eigenspace of ρ(ǫ) and E denote the ( 1)-eigenspace of ρ(w). Then (24) implies that − − 3 (25) dim H (Γ D; E) dim(E+ E ). \ ≤ ∩ − The result follows from translating the eigenvalue condition into a condition on weights of E. First consider the trivial representation E = Z. Then H∗(Γ D; E) is isomor- phic to the singular cohomology of Γ D. Proposition 11.1, allows\ us to explicitly compute the cohomology. \ Theorem 11.6. Let Z denote the constant sheaf of integers on Γ D. Then \ Z k =0, 2, Hk(Γ D; Z)= \ 0 k =1 or k 3. ( ≥ Theorem 11.7. Let E = Z[i]3 with the natural action of Γ and let E denote the associated sheaf on Γ D. Then \ 0 k =0, 1, Hk(Γ D; E)= Z2 k =2, \ Z/2Z k =3. i Theorem 11.8. The dimensions h of the cohomology groups for Γ with coefficients in Symn(V ), 0 n 20, where V C3 is the standard representation is as tabulated in Table≤ 11.8≤. ≡ EXPLICIT REDUCTION FOR SU(2, 1; Z[i]) 25 Table 4. Cohomology for Symn(V ) n 12345678910 h0 0000000000 h1 0000000000 h2 1001312251 h3 0001000100 n 11 12 13 14 15 16 17 18 19 20 h0 0000000000 h1 01003425811 h2 2374549575 h3 0100010001 References [1] A. Ash, Deformation retracts with lowest possible dimension of arithmetic quotients of self- adjoint homogeneous cones, Math. Ann. 225 (1977), no. 1, 69–76. [2] , Cohomology of congruence subgroups SL(n, Z), Math. Ann. 249 (1980), no. 1, 55–73. [3] , Small-dimensional classifying spaces for arithmetic subgroups of general linear groups, Duke Math. J. 51 (1984), no. 2, 459–468. [4] A. Ash, D. Grayson, and P. Green, Computations of cuspidal cohomology of congruence subgroups of SL(3, Z), J. Number Theory 19 (1984), no. 3, 412–436. [5] A. Ash and M. McConnell, Doubly cuspidal cohomology for principal congruence subgroups of GL(3, Z), Math. Comp. 59 (1992), no. 200, 673–688. [6] , Experimental indications of three-dimensional Galois representations from the co- homology of SL(3, Z), Experiment. Math. 1 (1992), no. 3, 209–223. [7] , Cohomology at infinity and the well-rounded retract for general linear groups, Duke Math. J. 90 (1997), no. 3, 549–576. [8] C. Batut, Classification of quintic eutactic forms, Math. Comp. 70 (2001), no. 233, 395–417 (electronic). [9] A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491, Avec un appendice: Arrondissement des vari´et´es `acoins, par A. Douady et L. H´erault. [10] G. E. Bredon, Sheaf theory, second ed., Graduate Texts in Mathematics, vol. 170, Springer- Verlag, New York, 1997. [11] A. Brownstein, Homology of Hilbert Modular Groups, Ph.D. thesis, University of Michigan, 1987. [12] M. Hall, Jr. and J. K. Senior, The groups of order 2n (n ≤ 6), The Macmillan Co., New York, 1964. [13] R. Lee and R. H. Szczarba, On the torsion in K4(Z) and K5(Z), Duke Math. J. 45 (1978), no. 1, 101–129. [14] R. MacPherson and M. McConnell, Explicit reduction theory for Siegel modular threefolds, Invent. Math. 111 (1993), no. 3, 575–625. [15] E. R. Mendoza, Cohomology of PGL2 over imaginary quadratic integers, Bonner Mathema- tische Schriften [Bonn Mathematical Publications], 128, Universit¨at Bonn Mathematisches Institut, Bonn, 1979, Dissertation, Rheinische Friedrich-Wilhelms-Universit¨at, Bonn, 1979. [16] I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363. [17] J. Schwermer and K. Vogtmann, The integral homology of SL2 and PSL2 of Euclidean imag- inary quadratic integers, Comment. Math. Helv. 58 (1983), no. 4, 573–598. [18] C. Soul´e, The cohomology of SL3(Z), Topology 17 (1978), no. 1, 1–22. [19] B. van Geemen and J. Top, A non-selfdual automorphic representation of GL3 and a Galois representation, Invent. Math. 117 (1994), no. 3, 391–401. [20] K. Vogtmann, Rational homology of Bianchi groups, Math. Ann. 272 (1985), no. 3, 399–419. [21] D. Yasaki, On the existence of spines for Q-rank 1 groups, Preprint, 2005. 26 DAN YASAKI [22] T. Zink, Uber¨ die Anzahl der Spitzen einiger arithmetischer Untergruppen unit¨arer Gruppen, Math. Nachr. 89 (1979), 315–320. Department of Mathematics and Statistics, Lederle Graduate Research Tower, Uni- versity of Massachusetts, Amherst, MA 01003-9305 E-mail address: [email protected]