On the integral homology of PSL4(Z) and other arithmetic groups

0.15 MB·English

by  Mathieu Dutour Sikiric

ON THE INTEGRAL HOMOLOGY OF PSL (Z) AND OTHER
4
ARITHMETIC GROUPS
MATHIEU DUTOUR SIKIRIC´, GRAHAM ELLIS, AND ACHILL SCHU¨RMANN
1
1 Abstract. WedeterminetheintegralhomologyofPSL4(Z)indegrees≤5anddetermine
0
its p-partin higher degrees for the primes p≥5. Our method applies to other arithmetic
2
groups; as illustrations we include descriptions of the integral homology of PGL3(Z[i])
an and PGL3(Z[exp(2πi/3)]) in degrees ≤5.
J

4
2

1. Introduction
]
T
K We obtain the following partial description of the integral homology of the projective
. special linear group PSL4(Z) (writing H(p) to denote the p-primary part of an Abelian
h
group H).
t
a
m Theorem 1.
[
0 n = 1

1  (Z )3 n = 2
2
v (i) H (PSL (Z),Z) =  Z⊕(Z )2 ⊕(Z )2 ⊕Z , n = 3
6 n 4  4 3 5
0  (Z )4 ⊕Z , n = 4
2 5
5 (Z )13, n = 5.
4  2

1.  (Z )2 n ≡ 3 mod 4 (n ≥ 6)
0 (ii) Hn(PSL4(Z),Z)(5) = (cid:26) 0 5 n ≡ 0,1,2 mod 4 (n ≥ 6)
1
1 (iii) H (PSL (Z),Z) = 0 for all primes p ≥ 7 and n ≥ 0.
: n 4 (p)
v
i (iv) H (PSL (Z),Z) is finite for all positive n 6= 3.
X n 4
r The proof of Theorem 1(i) involves four steps. The first, described in Section 2, uses
a
computer calculations of perfect forms to explicitly determine a CW-structure on a homo-
topy retract X of the space S4 of 4×4 positive definite symmetric matrices. The retract
>0
X is 6-dimensional, contractible and admits a cellular action of G = PSL (Z) in which
4
each cell e has a finite stabilizer group Ge. (We remark that an analogous 3-dimensional
retract of S3 , due to Avner Ash, was used by Christophe Soul´e [18] in his calculation of
>0
the integral cohomology of SL (Z). Furthermore, versions of our retract X for PSL (Z)
3 4
have been determined by Ash et al. in [2] and by Elbaz-Vincent et al. in [6, 5], who used
it to compute cohomology rationally and integrally at large primes only.) We will refer
FirstauthorhasbeensupportedbyMarieCuriefellowshipMTKD-CT-2006-042685andbytheCroatian
Ministry of Science, Education and Sport under contract 098-0982705-2707.
1
2 MATHIEU DUTOURSIKIRIC´,GRAHAMELLIS, ANDACHILL SCHU¨RMANN
to [15] and [11] for full details on perfect forms and limit our exposition to those facts
necessary for Theorem 1.
The next step is described inSection 3 anduses algorithms from[7, 9] todetermine small
free ZGe-resolutions of Z for each stabilizer group. Implementations of these algorithms
are available in the HAP [8] software package. However, Theorem 1 was obtained using an
implementation specially adapted by the first author.
The third step is described in Section 4. It uses a generalization of a lemma of C.T.C.
Wall to combine the resolutions for the stabilizer groups with the cellular chain complex
C (X) to form a free ZG-resolution R of Z. (For a full exposition of this technique we will
∗ ∗
refer to [10] where it was illustrated with the detailed hand calculation of free resolutions
for generalized triangle groups acting on hyperbolic 3-space.) The technique has been
automated in the software package [8] and such an automation was used to obtain six
terms of a free ZG-resolution R .
∗
The final step in the proof of Theorem 1(i) is routine: the Smith Normal Form algorithm
is used to compute the homology of the chain complex R∗ ⊗ZG Z.
Theorem 1(ii)-(iv) are standard applications of the Leray spectral sequence. Details are
given in Section 5. The following result, which gives bounds on the 3-primary part of the
homology, is also derived in Section 5.
Proposition 2. At the prime p = 3 the Leray spectral sequence
E1 = H (Gep,Zρ) ⇒ H (PSL (Z),Z)
p,q q (3) n 4 (3)
M[ep]
(where [ep] ranges over the G-orbits of p-dimensional ep cells in X and ρ is a suitable action
of Gep on Z) has first page
(Z )q/2 p = 0,q = 2+4k
3
 (Z )(5+q)/2 p = 0,q = 3+4k
3
 (Z )(q+2)/4 p = 1,q = 2+4k
 3
 (Z )(q+5)/4 p = 1,q = 3+4k
 3
 Z p = 3,q = 1+2k
 3
E1 =  Z p = 4,q = 1+4k (k ≥ 0).
p,q  3
 (Z )(q+2)/4 p = 4,q = 2+4k
3
(Z )(q+5)/4 p = 4,q = 3+4k
 3
 Z p = 5,q = 1+2k
 3
 Z p = 6,q = 1+4k
 3

 0 otherwise





The techniques underlying Theorem 1 can in principle be applied to other arithmetic
groups. For example, they yield the following descriptions of the low-dimensional integral
homology of the 3-dimensional projective general linear groups PGL (Z[i]), PGL (Z[ω])
3 3
over the Gaussian and Eisenstein integers (where ω = exp(2πi/3)).
ON THE INTEGRAL HOMOLOGY OF PSL4(Z) AND OTHER ARITHMETIC GROUPS 3
Theorem 3.
0 n = 1
 (Z )2 n = 2
2
H (PGL (Z[i]),Z) =  Z⊕Z ⊕Z ⊕Z , n = 3
n 3  4 8 3

 Z⊕Z , n = 4
2
(Z )5 ⊕(Z )2, n = 5.
 2 4



 Z n = 1
3
 Z n = 2
3
H (PGL (Z[ω]),Z) =  Z⊕Z ⊕(Z )3, n = 3
n 3  8 3

 Z⊕(Z )2, n = 4
3
(Z )2 ⊕(Z )3, n = 5.
 2 3



The method of Theorem 1(i) and Theorem 3 has also been used to obtain the first few
terms of an explicit resolution for the symplectic group Sp (Z). Although the integral
4
(co)homology has previously been calculated in this case by other means (see [4]), an
advantage of an explicit resolution is that in principle it can be used to compute the
cohomology and explicit cocycles for finite index subgroups such as congruence subgroups.
See the final section for details.
We will not give details for the proof of Theorem 3. Instead, in Section 6 we explain
how the results of the theorem can be computed using the HAP [8] software package. (We
remark that the integral homology of PSL (O) for various rings of quadratic integers O
2
have been calculated by J. Schwermer and K. Vogtmann in [17] and by A. Rahm and M.
Fuchs [13, 14].)
2. Perfect forms and a cell complex with PSL (Z)-action.
4
In the space Sn of positive definite, real symmetric n × n matrices, we consider a
>0
specific SL (Z) invariant polyhedral complex. Such complexes are classically studied in
n
the arithmetic theory of quadratic forms. One of them goes back to the work of Voronoi
[20] on perfect quadratic forms.
For A ∈ Sn an associated positive definite quadratic form on Rn is defined by A[x] =
>0
xtAx. By this correspondence we simply identify quadratic forms and symmetric matrices.
The arithmetical minimum of A ∈ Sn is defined by
>0
min(A) = min A[v].
06=v∈Zn
The finite set of its representatives is denoted by
Min(A) = {v ∈ Zn with A[v] = min(A)}.
We consider the set of forms having a fixed arithmetical minimum, say 1:
Sn = {A ∈ Sn with min(A) = 1}.
=1 >0
This set is the boundary of a locally finite polyhedral set known as Ryshkov polyhedron
(see [15] for details). In particular, Sn is a piecewise linear surface of dimension n+1 −1
=1 2
and the support of a cell complex obtained from the faces of the Ryshkov pol(cid:0)yhed(cid:1)ron.
4 MATHIEU DUTOURSIKIRIC´,GRAHAMELLIS, ANDACHILL SCHU¨RMANN
Moreover, Sn is contractible. Each v ∈ Zn determines one of the top-dimensional cells by
=1
the linear condition A[v] = 1. The 0-dimensional cells in Sn correspond to perfect qua-
=1
dratic forms. These are characterized as being uniquely determined by their arithmetical
minimum (here 1) and its representatives.
The group SL (Z) acts on Sn and Sn by A 7→ PAPt. Some higher dimensional cells
n >0 =1
of Sn (and in particular all top-dimensional ones) have infinite stabilizer groups. To avoid
=1
problems arising fromsuch stabilizers, we consider adeformationretract. Aquadraticform
A ∈ Sn is said to be well-rounded if there exist linearly independent v ,...,v ∈ Min(A).
>0 1 n
ThesetSn ofwellroundedformsinSn definesan n -dimensionalpolyhedralsubcomplex,
wr =1 2
in which all cells are bounded polyhedra and thus h(cid:0)a(cid:1)ve finite stabilizer groups. By a result
of Ash [1], Sn is a homotopy retract of Sn (and Sn ) and is thus contractible. In fact,
wr >0 =1
Sn is a minimal deformation retract as there is no proper closed subset of Sn which is
wr wr
SL (Z) invariant and contractible (see [12]).
n
The 0-dimensional cells of Sn (the perfect forms) can be enumerated up to SL (Z)
wr n
equivalence using Voronoi’s algorithm. In essence it is a graph traversal search on the
1-dimensional subcomplex of Sn (see [20, 11, 15] for more details). Having a complete list
=1
of inequivalent perfect forms one can obtain all other orbits of cells, as these all contain at
least one of the perfect forms.
In dimension 4 there are just two perfect forms up to SL (Z) equivalence (associated to
4
the root lattices A and D ). These yield a PSL (Z)-equivariant CW-decomposition of Sn
4 4 4 wr
involving the following cells and cell stabilizer groups:
Dim. 0: two cell orbits with stabilizers A and (A ×A ) : C .
5 4 4 2
Dim. 1: two cell orbits with stabilizers S and S ×S .
3 3 3
Dim. 2: two cell orbits, both with stabilizer C ×C .
2 2
Dim. 3: four cell orbits, two with stabilizer C ×C and two with stabilizer S .
2 2 4
Dim. 4: four cell orbits with stabilizers S , D , S and C ×S ×S .
3 8 4 2 3 3
Dim. 5: three cell orbits with stabilizers D , S and A .
24 4 5
Dim. 6: one cell orbit with stabilizer ((C ×C ×C ×C ) : C ) : C .
2 2 2 2 3 2
3. Resolutions for stabilizer subgroups
The method outlined in Section 4 for computing n terms of a free ZG-resolution for
G = PSL (Z) will require n − p terms of a free ZGe-resolution for each stabilizer group
4
of a p-dimensional cell in Sn . An algorithm for computing a reasonably small free ZGe-
wr
resolution RGe → Z for a generic finite group Ge was described in [7]. The implementation
∗
of this algorithm in [7] produces, for instance, a free ZA -resolution with 26 free generators
5
in dimension 6. The algorithm could be used to obtain 6−p terms for each of the stabilizer
groups in the above tessellation of Sn .
wr
However, for computational efficiency it is necessary to craft smaller resolutions for some
of the stabilizer groups Ge. One way to do this is to find a suitable polytope P on which Ge
acts with small cell stabilizers. Then, using Lemma 4 below, we can combine resolutions
for those subgroups of Ge stabilizing cells of P with the cellular chain complex C (P) in
∗
order to produce a free ZGe-resolution RGe → Z. The idea was illustrated in [10] with
∗
ON THE INTEGRAL HOMOLOGY OF PSL4(Z) AND OTHER ARITHMETIC GROUPS 5
the hand calculation of a free ZA -resolution involving n+1 free generators in degree n.
4
The same technique can be used, for instance, to construct a free ZA -resolution involving
5
n+1 free generators in degree n; in this case we take P to be the icosahedron and let A
5
act on P as a subgroup of the symmetry group C ×A of the icosahedron.
2 5
The first author used this polytopal method to craft small resolutions for the larger
stabilizer groups in the tessellation of Sn .
wr
Both, the algorithm in [7] and the polytopal method [9, 10] can return a free ZGe-
resolution RGe endowed with a contracting homotopy. Latter is encoded as a sequence of
∗
Z-linear homomorphisms h : RGe → RGe satisfying
n n n+1
h d +d h = 1 (n ≥ 0,h = 0).
n−1 n n+1 n −1
This contracting homotopy is required by Lemma 4 below.
4. Perturbation theory
It was observed in [9] that a method of Wall [21] for constructing free resolutions for
group extensions can be extended to a method for constructing free resolutions for groups
acting on contractible cellular spaces with nice stabilizer subgroups. We recall the method.
Suppose a chain complex
C : ··· → C → C → ··· → C → Z,
∗ p p−1 0
of ZG-modules is given, where the modules C are not necessarily ZG-free. Suppose also
p
that for each p ≥ 0 we are given a free ZG-resolution A of the module C
p,∗ p
A : → A → A → ··· → A → C
p,∗ p,q p,q−1 p,0 p
endowed with a contracting homotopy h : A → A (q ≥ 0).
q p,q p,q+1
The following lemma explains how to construct a chain complex R of free ZG-modules
∗
R = A and a surjective ZG-chain map
n p+q=n p,q
L
φ : R → C
∗ ∗ ∗
∼=
which induces homology isomorphisms H (φ ): H (R ) → H (C ) for n ≥ 0. We state a
n ∗ n ∗ n ∗
slightly more general version of the lemma than was given in [9]. However, the proof is the
same and boils down to the proof of the special case given in [21].
Lemma 4. [21, 9]
(i) Let A (p,q ≥ 0) be a bigraded family of free ZG-modules. Suppose that there
p,q
are ZG-module homomorphisms d : A → A such that (A ,d ) is an acyclic
0 p,q p,q−1 p,∗ 0
chain complex for each p. Set C = H (A ,d ) and suppose further that there
p 0 p,∗ 0
are ZG-homomorphisms δ: C → C for which (C ,δ) is a chain complex. Then
p p−1 ∗
there exist ZG-homomorphisms d : A → A (k ≥ 1,p > k) such that
k p,q p−1,q+k−1
d = d +d +··· : R = A → R = A
0 1 n p,q n−1 p,q
pM+q=n p+Mq=n−1
is a differential on a chain complex R of free ZG-modules.
∗
6 MATHIEU DUTOURSIKIRIC´,GRAHAMELLIS, ANDACHILL SCHU¨RMANN
(ii) The canonical chain maps φ : A → H (A ) constitute a chain map φ : R → C
p p,∗ 0 p,∗ ∗ ∗ ∗
which is an homology isomorphism.
(iii) Suppose that there exist Z-homomorphismsh : A → A such that d h d (x) =
0 p,q p,q+1 0 0 0
d (x) for all x ∈ A . Then we can construct d by first lifting δ to d : A →
0 p,q+1 k 1 p,0
A and recursively defining d = −h ( k d d ) on free generators of the
p−1,0 k 0 i=1 i k−i
module Ap,q. P
(iv) Supposethat C is acyclic, that H (S ) ∼= Z and that each C is a free Z-module. We
∗ 0 ∗ p
can construct Z-module homomorphisms h: R → R satisfying dhd(x) = d(x)
n n+1
by setting h(a ) = h (a ) − hd+h (a ) + ǫ(a ) for free generators a of the
p,q 0 p,q 0 p,q p,q p,q
module A . Here d+ = p d and, for q ≥ 1, ǫ = 0. For q = 0 we define
p,q i=1 i
ǫ = h1 − h0d+h1 + hd+h1P+ hd+h0d+h1 where h1: Ap,0 → Ap+1,0 is a Z-linear
homomorphism induced by a contracting homotopy on C .
∗
For G = PSL (Z) and X = Sn the cellular chain complex C (X) can be viewed as a
4 wr ∗
complex of ZG-modules. Moreover,
Cp(X) ∼= ZG⊗ZGep Z
M[ep]
where [ep] ranges over the orbits of p-dimensional cells. A free ZG-resolution A of C (X)
p,∗ p
can thus be obtained as
Ap,∗ ∼= R∗Gep ⊗ZGep Z
M[ep]
where RGep is any free ZGep-resolution of Z. Moreover, contracting homotopies on RGep
∗ ∗
induce a contracting homotopy on A . Lemma 4 thus gives an automated procedure to
p,∗
combine the chain complex C (X) with resolutions for the stabilizer groups to produce a
∗
free ZG-resolution RG of Z.
∗
The first six terms of such a resolution RG were computed and, via the Smith Normal
∗
Form algorithm, used to prove Theorem 1(i).
5. The Leray spectral sequence
Details on the Leray spectral sequence can be found in [3]. For G = PSL (Z) and
4
X = Sn it has the form
wr
E1 = H (Gep,Zρ) ⇒ H (PSL (Z),Z)
p,q q n 4
M[ep]
where [ep] ranges over the G-orbits of p-dimensional cells ep in the tessellation of X and
the integer coefficients are twisted by an action ρ: Gep → Aut(Z) = {±1}.
Each stabilizer group is finite. The homology of any finite group is finite in degrees
≥ 1. So since X is 6-dimensional the spectral sequence immediately implies that H (G,Z)
n
is finite for n ≥ 6. A computer calculation shows that the homomorphism E = Z →
6,0
E = Z3 is injective, and H (G,Z) is thus finite. Theorem 1(iv) therefore follows from
5,0 6
Theorem 1(i).
ON THE INTEGRAL HOMOLOGY OF PSL4(Z) AND OTHER ARITHMETIC GROUPS 7
No prime p ≥ 7 divides the order of any of the stabilizer subgroups Ge and thus no p ≥ 7
divides the order of H (Ge,Zρ). So Theorem 1(iii) follows from the spectral sequence.
q
The only 3-groups and 5-groups arising as subgroups of stabilizer groups are C , C ×C
3 3 3
and C . These prime-power groups can only act trivially on Z. Using Cartan and Eilen-
5
berg’s identification of the p-part Hq(Ge,Zρ) of the cohomology of Ge with the G-stable
(p)
elements in the cohomology Hq(Syl (Ge),Z) of the Sylow p-subgroup, it is straightforward
p
to determine H (Ge,Zρ) = Hq(Ge,Zρ) for p = 3,5. The 3-part is presented in
q−1 (p) (p)
Proposition 2. The 5-part is given in the following.
Proposition 5. At the prime p = 5 the Leray spectral sequence E1 ⇒ H (PSL (Z),Z)
p,q n 4 (5)
has first page
Z p = 0,q = 3+4k
5
E1 =  Z p = 5,q = 3+4k (k ≥ 0).
p,q 5
 0 otherwise
Proposition 5 directly implies that
0 or (Z )2 if n ≡ 3 mod 4 (n ≥ 6)
H (PSL (Z),Z) = 5
n 4 (5) (cid:26) 0 if n ≡ 0,1,2 mod 4 (n ≥ 6)
The alternating group A is the stabilizer group of a 0-cell of X. A computer cal-
5
culation shows that the inclusion A ֒→ PSL (Z) induces a surjection in cohomology
5 4
H4(PSL (Z),Z) → H4(A ,Z) . The ring H∗(A ,Z) ∼= Z [x] is generated by a sin-
4 (5) 5 (5) 5 (5) 5
gle class x in degree 4. Hence the ring homomorphism H∗(PSL (Z),Z) → H∗(A ,Z)
4 (5) 5 (5)
is surjective and consequently H (PSL (Z),Z) ∼= Hn+1(PSL (Z),Z) is non-trivial for
n 4 (5) 4 (5)
n ≡ 3 mod 4 (n ≥ 7). This proves Theorem 1(ii).
6. The groups PGL (Z[i]) and PGL (Z[ω])
3 3
The above techniques apply to other arithmetic groups. In particular, the first author
using theconstruction of[16]forequivariant perfect formshascomputed two 8-dimensional
contractible CW-spaces on which the groups PGL (Z[i]) and PGL (Z[ω]) act cellularly
3 3
with finite stabilizer groups. These spaces, together with the above 6-dimensional space
for PSL (Z), have been stored in the HAP package [8] for the GAP computer algebra
4
system.
Thefollowingshort GAPsession illustrateshowthespaceforPGL (Z[i]) canbeaccessed
3
andusedtocompute: (i)fivetermsofafreeresolutionand(ii)thefourthintegralhomology
group.
gap> C:=ContractibleGcomplex("PGL(3,Z[i])");;
gap> R:=FreeGResolution(C,5);;
gap> Homology(TensorWithIntegers(R),4);
[ 2, 0 ]
The resolution can also be used to determine the homology of finite index subgroups of
PGL (Z[i]), though GAP’s standard implementation of the Smith Normal Form algorithm
3
does not work well when the index is large.
8 MATHIEU DUTOURSIKIRIC´,GRAHAMELLIS, ANDACHILL SCHU¨RMANN
For the group Sp (Z) the 4-dimensional CW-complex of [19], which comes also from the
4
perfect forms in dimension 4, has been stored in the HAP package [8]. So, for instance,
the following GAP session computes the homology H (Sp (Z),Z) = (Z )2 ⊕Z ⊕Z in
3 4 2 12 240
agreement with calculations of [4].
gap> C:=ContractibleGcomplex("Sp(4,Z)");;
gap> R:=FreeGResolution(C,4);;
gap> Homology(TensorWithIntegers(R),3);
[ 2, 2, 12, 240 ]
7. Acknowledgment
We thank Mark McConnell and Paul Gunnells for help in using the cellular decomposi-
tion of [19].
References
[1] A. Ash, Small-dimensional classifying spaces for arithmetic subgroups of general linear groups, Duke
Math. J., 51 (1984), no. 2, 459–468.
[2] A. Ash, P.E. Gunnells and M. McConnell, Cohomology of congruence subgroups of SL(4,Z), J.
Number Theory 94 (2002), no. 1, 181–212.
[3] K. S. Brown, Cohomology of groups, Graduate texts in mathematics, vol. 87, Springer-Verlag 1982.
[4] A. Brownstein and R. Lee, Cohomology of the symplectic group Sp(4,Z). II. Computations at the
prime 2, Michigan Math. J., 41 (1994), no. 1, 181–208.
[5] J.-G.Dumas, P.Elbaz-Vincent,P.GiorgiandA. Urban´ska,Parallelcomputationofthe rankof large
sparse matrices from algebraic K-theory, PASCO’07, 43–52, ACM, New York, 2007.
[6] P. Elbaz-Vincent, H. Gangl and C. Soul´e, Quelques calculs de la cohomologie de GLN(Z) et de la
K-th´eorie de Z, C. R. Acad. Sci. Paris, Ser I, 335 (2002), no 4, 321–324.
[7] G. Ellis, Computing group resolutions, J. Symbolic Comput. 38 (2004), no 3, 1077–1118.
[8] G. Ellis, HAP - Homological Algebra Programming, a packagefor the GAP computer algebrasystem.
http://www.gap-system.org
[9] G. Ellis, J. Harris and E. Sk¨oldberg, Polytopal resolutions for finite groups, J. Reine Angew. Math.
598 (2006), 131–137.
[10] G. Ellis and G. Williams, On the cohomology of generalized triangle groups, Comment. Math. Helv.
80 (2005), no 3, 571–591.
[11] J. Martinet, Perfect lattices in Euclidean spaces, Springer, 2003.
[12] A. Pettet and J. Souto, Minimality of the well rounded retract, Geom. Topol., 12 (2008), no 3,
1543–1556.
[13] A.D. Rahm, (Co)homologies and K-theory of Bianchi groups using computational geometric models,
PhD thesis, Institut Fourier, Universit´e de Grenoble and Universit¨at G¨ottingen (2010), http://
tel.archives-ouvertes.fr/tel-00526976/en/.
[14] A.D. Rahm and M. Fuchs, The integral homology of PSL2 of imaginary quadratic inte-
gers with non-trivial class group, to appear in the Journal of Pure and Applied Algebra.
http://hal.archives-ouvertes.fr/hal-00370722/fr/, (2009).
[15] A.Schu¨rmann,Computationalgeometryofpositivedefinitequadraticforms,UniversityLectureNotes,
AMS, 2009.
[16] A. Schu¨rmann,Enumeratingperfect forms,in Quadraticforms — algebra,arithmetic, andgeometry,
359–377,Contemp. Math., 493, Amer. Math. Soc., Providence,RI, 2009.
[17] J. Schwermer and K. Vogtmann, The integral homology of SL2 and PSL2 of Euclidean imaginary
quadratic integers, Comment. Math. Helv. 58 (1983), no 4, 573–598.
ON THE INTEGRAL HOMOLOGY OF PSL4(Z) AND OTHER ARITHMETIC GROUPS 9
[18] C. Soul´e, The cohomology of SL3(Z), Topology 17 (1978), no 1, 1–22.
[19] R. MacPherson and M. McConnell, Explicit reduction theory for Siegel modular threefolds, Invent.
Math. 111 (1993), 575–625.
[20] G. Voronoi, Nouvelles applications des param`etres continues `a la th´eorie des formes quadratiques
1: Sur quelques propri´et´es des formes quadratiques positives parfaites, J. Reine Angew. Math. 133
(1908), 97–178.
[21] C.T.C. Wall. Resolutions of extensions of groups. Proc. Cambridge Philos. Soc. 57 (1961), 251–255.
M. Dutour Sikiric´, Rudjer Boskovic´ Institute, Bijenicka 54, 10000 Zagreb, Croatia
E-mail address: [email protected]
G. Ellis, Mathematics Department, National University of Ireland, Galway
E-mail address: [email protected]
A. Schu¨rmann, Institute of Mathematics, University of Rostock, 18051 Rostock, Ger-
many
E-mail address: [email protected]