Computing Unit Groups of Orders

Transcription

1 Computing Unit Groups of Orders Gabriele Nebe, Oliver Braun, Sebastian Schönnenbeck Lehrstuhl D für Mathematik Bad Boll, March 5, 2014

2 The classical Voronoi Algorithm Around 1900 Korkine, Zolotareff, and Voronoi developed a reduction theory for quadratic forms. The aim was to classify the densest lattice sphere packings in n-dimensional Euclidean space. lattice L = Z 1 n, Euclidean structure on L given by some some positive definite F R n n sym, (x, y) = xf y tr. Voronoi described an algorithm to find all local maxima of the density function on the space of all n-dimensional positive definite F. They are perfect forms (as will be defined below). There are only finitely many perfect forms up to the action of GL n(z), the unit group of the order Z n n. Later, Voronoi s algorithm has been used to compute generators and relations for GL n(z) but also its integral homology groups. It has been generalised to other situations: compute integral normalizer, the automorphism group of hyperbolic lattices and more general unit groups of orders.

3 Unit groups of orders A separable Q-algebra, so A = s rings over division algebras. i=1 Dn i n i i, is a direct sum of matrix An order Λ in A is a subring that is finitely generated as a Z-module and such that Λ Q = A. Its unit group is Λ := {u Λ v Λ, uv = 1}. Know in general: Λ is finitely generated. Example: A = K a number field, Λ = O K, its ring of integers. Then Dirichlet s unit theorem says that Λ = µk Z r+s 1. Example: Λ = 1, i, j, ij Z with i 2 = j 2 = (ij) 2 = 1. Then Λ = i, j the quaternion group of order 8. Example: A = QG for some finite group G, Λ = ZG. Example: A a division algebra with dim Z(A) (A) = d 2 > 4. Not much known about the structure of Λ. Voronoi s algorithm may be used to compute generators and relations for Λ and to solve the word problem. Seems to be practical for small A and for d = 3.

4 The classical Voronoi Algorithm Korkine, Zolotareff, Voronoi, Definition V := {X R n n X = X tr } space of symmetric matrices σ : V V R, σ(a, B) := trace(ab) Euclidean inner product on V. for F V, x R 1 n define F [x] := xf x tr = σ(f, x tr x) V >0 := {F V F positive definite } for F V >0 define the minimum µ(f ) := min{f [x] : 0 x Z 1 n } and M(F ) := {x Z 1 n F [x] = µ(f )} Vor(F ) := { x M(F ) axxtr x a x 0} the Voronoi domain F is called perfect dim(vor(f )) = dim(v) = n(n+1) 2. Remark GL n(z) acts on V >0 by (F, g) g 1 F g tr. Then M(g 1 F g tr ) = {xg x M(F )} Vor(g 1 F g tr ) = g tr Vor(F )g

5 The classical Voronoi Algorithm Korkine, Zolotareff, Voronoi, Definition V := {X R n n X = X tr } space of symmetric matrices σ : V V R, σ(a, B) := trace(ab) Euclidean inner product on V. for F V, x R 1 n define F [x] := xf x tr = σ(f, x tr x) V >0 := {F V F positive definite } for F V >0 define the minimum µ(f ) := min{f [x] : 0 x Z 1 n } and M(F ) := {x Z 1 n F [x] = µ(f )} Vor(F ) := { x M(F ) axxtr x a x 0} the Voronoi domain F is called perfect dim(vor(f )) = dim(v) = n(n+1) 2. Theorem (Voronoi) (a) T := {Vor(F ) F V >0, perfect } forms a face to face tesselation of V 0. (b) GL n(z) acts on T with finitely many orbits that may be computed algorithmically.

6 Example, generators for GL 2 (Z) n = 2, dim(v) = 3, dim(v >0 /R >0) = 2 compute in affine section of the projective space A 0 = {F V 0 trace(f ) = 1} ( 2 1 F 0 = 1 2 A 0 Vor(F 0) = conv(a = c ), µ(f 0) = 2, M(F 0) = {±(1, 0), ±(0, 1), ±(1, 1)} ( ) 10, b = 00 ( ) 00, c = ( ) 11 ) 11 Vor(F0) a b

7 Example, generators for GL 2 (Z) n = 2, dim(v) = 3, dim(v >0 /R >0) = 2 compute in affine section of the projective space A 0 = {F V 0 trace(f ) = 1} ( 2 1 F 0 = 1 2 A 0 Vor(F 0) = conv(a = c ), µ(f 0) = 2, M(F 0) = {±(1, 0), ±(0, 1), ±(1, 1)} ( ) 10, b = 00 ( ) 00, c = ( ) 11 ) 11 Vor(F0) a b Vor(F1) c

8 Example, generators for GL 2 (Z) Compute neighbor: F 1 V >0 so that Vor(F 1) = conv(a, b, c ). linear equation on F 1: trace(f ( 1a) ) = trace(f 1b) = 2 and trace(f 1c) > 2, 01 so F 1 = F 0 + sx where X = generates a, b. 10 ( ) 21 For s = 2 the matrix F 1 = has again 6 minimal vectors 12 M(F 1) = {±(1, 0), ±(0, 1), ±(1, 1)} ( ) 1 1 A 0 Vor(F 1) = conv(a, b, c := 1 ) c Vor(F0) a b Vor(F1) c

9 Example, generators for GL 2 ((Z) ) ( ) Stab GL2 (Z)(F 0) = g =, h = (a, b) g = (b, c), (b, c) g = (c, a) Compute isometry t = diag(1, 1) GL 2(Z), so t 1 F 0t tr = F 1. Then GL 2(Z) = g, h, t. c Vor(F0) g a t b Vor(F1)

10 GL 2 (Z) = g, h, t. ( ) ( ) Stab GL2 (Z)(F 0) = g =, h = (a, b) g = (b, c), (b, c) g = (c, a) Compute isometry t = diag(1, 1) GL 2(Z), so t 1 F 0t tr = F 1. c Vor(F1g 2 ) Vor(F1g) Vor(F0) g a t b Vor(F1) c

11 Variations of Voronoi s algorithm Many authors used this algorithm to compute integral homology groups of SL n(z) and related groups, as developed C. Soulé in Max Köcher developed a general Voronoi Theory for pairs of dual cones in the 1950s. σ : V 1 V 2 R non degenerate and positive on the cones V 1 >0 V 2 >0. discrete admissible set D V 2 >0 used to define minimal vectors and perfection for F V 1 >0 and Vor D(F ) V >0 2. J. Opgenorth (2001) used Köcher s theory to compute the integral normalizer N GLn(Z)(G) for a finite unimodular group G. M. Mertens (Masterthesis, 2012) applied Köcher s theory to compute automorphism groups of hyperbolic lattices. This talk will explain how to apply it to obtain generators and relations for unit group of orders in semi-simple rational algebras and an algorithm to solve the word problem in these generators.

12 Orders in semi-simple rational algebras. The positive cone K some rational division algebra, A = K n n A R := A Q R semi-simple real algebra so A R is isomorphic to a direct sum of matrix rings over of H, R or C. A R carries a canonical involution (depending on the choice of the isomorphism) that we use to define symmetric elements: V := Sym(A R ) := { F A R F = F } σ(f 1, F 2) := trace(f 1F 2) defines a Euclidean inner product on V. In general the involution will not fix the set A. The simple A-module. Let V = K 1 n denote the simple right A-module, V R = V Q R. For x V we have x x V. F V is called positive if F [x] := σ(f, x x) > 0 for all 0 x V R.

13 Minimal vectors. The discrete admissible set O maximal order in K, L some O-lattice in the simple A-module V Λ := End O(L) is a maximal order in A with unit group Λ := GL(L) = {a A al = L}. L-minimal vectors Let F V >0. µ(f ) := µ L(F ) = min{f [l] 0 l L} the L-minimum of F. M L(F ) := {l L F [l] = µ L(F )} the finite set of L-minimal vectors. Vor L(F ) := { x M L (F ) axx x a x 0} V 0 Voronoi domain of F. F is called L-perfect dim(vor L(F )) = dim(v). Theorem T := {Vor L(F ) F V >0, L-perfect } forms a face to face tesselation of V 0. Λ acts on T with finitely many orbits.

14 Generators for Λ Compute R := {F 1,..., F s} set of representatives of Λ -orbits on the L-perfect forms, such that their Voronoi-graph is connected. For all neighbors F of one of these F i (so Vor(F ) Vor(F i) has codimension 1) compute some g F Λ such that g F F R. Then Λ = Aut(F i), g F F i R, F neighbor of some F j R. 1 a b e f 1 2 c d so here Λ = Aut(F 1), Aut(F 2), Aut(F 3), a, b, c, d, e, f.

15 Example Q 2,3. Take the rational quaternion algebra ramified at 2 and 3, Q 2,3 = i, j i 2 = 2, j 2 = 3, ij = ji = diag( 2, ( 0 1 2), 3 0 Maximal order Λ = 1, i, 1 (1 + i + ij), 1 (j + ij) 2 2 V = A = Q 2,3, A R = R 2 2, L = Λ Embed A into A R using the maximal subfield Q[ 2]. Get three perfect forms: ( F 1 = ), F 2 = F 3 = diag( , ) ( ) )

16 The tesselation for Q 2,3 Q[ 2] 2 2.

17 Λ / ±1 = a, b, t a 3, b 2, atbt a t b 1 t

18 Easy solution of the word problem PX a t P t b

19 Easy solution of the word problem PX a Pa P t t b

20 Easy solution of the word problem Pa 1 X PX a Pa P t t b

21 Easy solution of the word problem Pa 1 X PX a P t t b

22 The tesselation for Q 2,3 Q[ 3] 2 2.

23 Conclusion Algorithm works quite well for indefinite quaternion algebras over the rationals Obtain presentation and algorithm to solve the word problem For Q 19,37 our algorithm computes the presentation within 5 minutes (288 perfect forms, 88 generators) whereas the Magma implementation FuchsianGroup does not return a result after four hours Reasonably fast for quaternion algebras with imaginary quadratic center or matrix rings of degree 2 over imaginary quadratic fields For the rational division algebra of degree 3 ramified at 2 and 3 compute presentation of Λ, 431 perfect forms, 50 generators in about 10 minutes. Quaternion algebra with center Q[ζ 5]: > perfect forms. Masterthesis by Oliver Braun: The tesselation T can be used to compute the maximal finite subgroups of Λ. Masterthesis by Sebastian Schönnenbeck: Compute integral homology of Λ.

24 Calculating maximal finite subgroups Minimal classes Definition Let A, B V >0. A and B are minimally equivalent if M L(A) = M L(B). C := Cl L(A) = {X V >0 M L(X) = M L(A)} is the minimal class of A. In this case M L(C) := M L(A). Call C well-rounded if M L(C) contains a K-basis of V = K 1 n. Remark A V >0 is L-perfect if and only if Cl L(A) = {αa α R >0}.

25 Calculating maximal finite subgroups Minimal classes Definition Let A, B V >0. A and B are minimally equivalent if M L(A) = M L(B). C := Cl L(A) = {X V >0 M L(X) = M L(A)} is the minimal class of A. In this case M L(C) := M L(A). Call C well-rounded if M L(C) contains a K-basis of V = K 1 n. Remark A V >0 is L-perfect if and only if Cl L(A) = {αa α R >0}. Remark dim R (V) dim R ( x x x M L(A) ), the perfection corank, is constant on Cl L(A).

26 Calculating minimal classes Theorem Let A V >0 be L-perfect. Any codimension-k-face of Vor L(A) corresponds to a minimal class of perfection corank k, represented by A + 1 2k k ρ ir i = 1 k i=1 k i=1 ( A + ρi 2 Ri ) V >0 with facet vectors R i and ρ i R >0 such that A + ρ ir i is a perfect neighbour of A (and the codimension-k-face in question is the intersection is the intersection of the facets with facet vectors R i).

27 Calculating minimal classes Theorem Let A V >0 be L-perfect. Any codimension-k-face of Vor L(A) corresponds to a minimal class of perfection corank k, represented by A + 1 2k k ρ ir i = 1 k i=1 k i=1 ( A + ρi 2 Ri ) V >0 with facet vectors R i and ρ i R >0 such that A + ρ ir i is a perfect neighbour of A (and the codimension-k-face in question is the intersection is the intersection of the facets with facet vectors R i). Example: ( The minimal ) classes in dimension 2 over Z 2 1 F 0 =, M(F 1 2 0) = {±(1, 0), ± (0, 1), ± (1, 1)} (( ) ( ) ( )) A 0 Vor(F 0) = conv,, ( ) ( ) ( ) Facet vectors R 1 =, R =, R = 1 2 All perfect neighbours of F 0 are given by F 0 + 2R i, so ρ i = 2 for all 1 i 3.

28 Now consider the dual of the tesselation of T = { Vor(F ) F V >0 perfect }.

29 Now consider the dual of the tesselation of T = { Vor(F ) F V >0 perfect }. F 1 F 2 F 0 F 3

30 Now consider the dual of the tesselation of T = { Vor(F ) F V >0 perfect }. F 1 F 2 F 0 F 3 The well-rounded classes have perfection corank 0 or 1. The corank 0 class is the perfect class of F 0. The corank 1 classes are represented by ( ) 2 0 F 0 + R 1 = 0 2 ( ) 4 2 F 0 + R 2 = 2 2 ( ) 2 2 F 0 + R 3 = 2 4 The minimal classes represented by these three matrices are in the same orbit under GL 2(Z). This is easily checked for well-rounded minimal classes using a theorem by A.-M. Bergé. The corank 2 class is represented by 1 2 (2F0 + R1 + R2) = ( ).

31 Maximal finite subgroups Theorem (Coulangeon, Nebe (2013)) G GL(L) maximal finite = G = Aut L(C), where C is a well-rounded minimal class, such that dim(c F(G)) = 1. F(G) := {A V A[g] = A g G} Remark This theorem yields a finite set of finite subgroups of GL(L), containing a set of representatives of conjugacy classes of maximal finite subgroups of GL(L).

32 Maximal finite subgroups Theorem (Coulangeon, Nebe (2013)) G GL(L) maximal finite = G = Aut L(C), where C is a well-rounded minimal class, such that dim(c F(G)) = 1. F(G) := {A V A[g] = A g G} Remark This theorem yields a finite set of finite subgroups of GL(L), containing a set of representatives of conjugacy classes of maximal finite subgroups of GL(L). There are algorithmic methods to check if a finite subgroup is maximal finite and whether two maximal finite subgroups are conjugate.

33 Maximal finite subgroups Theorem (Coulangeon, Nebe (2013)) G GL(L) maximal finite = G = Aut L(C), where C is a well-rounded minimal class, such that dim(c F(G)) = 1. F(G) := {A V A[g] = A g G} Remark This theorem yields a finite set of finite subgroups of GL(L), containing a set of representatives of conjugacy classes of maximal finite subgroups of GL(L). There are algorithmic methods to check if a finite subgroup is maximal finite and whether two maximal finite subgroups are conjugate. Therefore in the previous example, we obtain two conjugacy classes of maximal finite subgroups: The stabilizer of the perfect form F 0, which is isomorphic to D 12. The stabilizer of the corank 1 class, which is isomorphic to D 8. These groups are indeed maximal finite.

34 Example: Q( 6) O = Z[ 6], L 0 = O O, L 1 = O p, where p (2). Well-rounded minimal classes for Q( 6) L = L 0 L = L 1 C G = Aut L(C) max. C G = Aut L(C) max. P 1 SL(2, 3) yes P 1 Q 8 yes C 1 D 12 yes P 2 C 3 C 4 yes C 2 D 12 yes C 1 D 8 yes C 3 C 4 no C 2 C 4 no C 4 D 8 yes C 3 C 4 no D 1 D 8 yes C 4 D 12 yes D 2 D 8 yes D 1 C 2 C 2 yes D 3 C 2 C 2 yes D 2 C 2 C 2 yes = GL(L 0) = GL(L1)

35 Resolutions for Unit Groups of Orders Setup: As before A = K n n for some rational division algebra K, O a maximal order in K and Λ = End O(L) for some O-lattice L.

36 Resolutions for Unit Groups of Orders Setup: As before A = K n n for some rational division algebra K, O a maximal order in K and Λ = End O(L) for some O-lattice L. Target Compute a ZΛ -free resolution of Z (which may then be used to compute e.g. the integral homology of Λ ).

37 Resolutions for Unit Groups of Orders Setup: As before A = K n n for some rational division algebra K, O a maximal order in K and Λ = End O(L) for some O-lattice L. Target Compute a ZΛ -free resolution of Z (which may then be used to compute e.g. the integral homology of Λ ). Basic idea Find a cell complex with a suitable Λ -action and employ its cellular chain complex.

38 Resolutions for Unit Groups of Orders Setup: As before A = K n n for some rational division algebra K, O a maximal order in K and Λ = End O(L) for some O-lattice L. Target Compute a ZΛ -free resolution of Z (which may then be used to compute e.g. the integral homology of Λ ). Basic idea Find a cell complex with a suitable Λ -action and employ its cellular chain complex. Reminder A R := A Q R carries a canonical involution, V := { F A R F = F }.

39 Resolutions for Unit Groups of Orders Setup: As before A = K n n for some rational division algebra K, O a maximal order in K and Λ = End O(L) for some O-lattice L. Target Compute a ZΛ -free resolution of Z (which may then be used to compute e.g. the integral homology of Λ ). Basic idea Find a cell complex with a suitable Λ -action and employ its cellular chain complex. Reminder A R := A Q R carries a canonical involution, V := { F A R F = F }. Λ acts on V >0 via (g, F ) gf g.

40 The cell decomposition of V >0 Minimal classes For F V >0 define Cl L(F ) := {F V >0 M L(F ) = M L(F )} the minimal class corresponding to F.

41 The cell decomposition of V >0 Minimal classes For F V >0 define Cl L(F ) := {F V >0 M L(F ) = M L(F )} the minimal class corresponding to F. V >0 decomposes into the disjoint union of all minimal classes.

42 The cell decomposition of V >0 Minimal classes For F V >0 define Cl L(F ) := {F V >0 M L(F ) = M L(F )} the minimal class corresponding to F. V >0 decomposes into the disjoint union of all minimal classes. Properties of this decomposition: Partial ordering on the minimal classes: C C M L(C) M L(C ).

43 The cell decomposition of V >0 Minimal classes For F V >0 define Cl L(F ) := {F V >0 M L(F ) = M L(F )} the minimal class corresponding to F. V >0 decomposes into the disjoint union of all minimal classes. Properties of this decomposition: Partial ordering on the minimal classes: C C M L(C) M L(C ). Each minimal class is a convex set in V.

44 The cell decomposition of V >0 Minimal classes For F V >0 define Cl L(F ) := {F V >0 M L(F ) = M L(F )} the minimal class corresponding to F. V >0 decomposes into the disjoint union of all minimal classes. Properties of this decomposition: Partial ordering on the minimal classes: C C M L(C) M L(C ). Each minimal class is a convex set in V. The decomposition as well as the partial ordering are compatible with the Λ -action.

45 The cell decomposition of V >0 Minimal classes For F V >0 define Cl L(F ) := {F V >0 M L(F ) = M L(F )} the minimal class corresponding to F. V >0 decomposes into the disjoint union of all minimal classes. Properties of this decomposition: Partial ordering on the minimal classes: C C M L(C) M L(C ). Each minimal class is a convex set in V. The decomposition as well as the partial ordering are compatible with the Λ -action. We have C = C C C.

46 The cell decomposition of V >0 Minimal classes For F V >0 define Cl L(F ) := {F V >0 M L(F ) = M L(F )} the minimal class corresponding to F. V >0 decomposes into the disjoint union of all minimal classes. Properties of this decomposition: Partial ordering on the minimal classes: C C M L(C) M L(C ). Each minimal class is a convex set in V. The decomposition as well as the partial ordering are compatible with the Λ -action. We have C = C C C. The cellular chain complex The decomposition yields an acyclic chain complex C, where C n is the free Abelian group on the minimal classes in dimension n. C n becomes a Λ -module by means of the Λ -action on V.

47 Assembling the information Problem 1: The modules C n are not necessarily free.

48 Assembling the information Problem 1: The modules C n are not necessarily free. Perturbations - C.T.C. Wall (1961) There is an algorithm which takes as input the cellular chain complex and free resolutions of Z for the occuring stabilizers of cells and outputs a free resolution of Z for Λ.

49 Assembling the information Problem 1: The modules C n are not necessarily free. Perturbations - C.T.C. Wall (1961) There is an algorithm which takes as input the cellular chain complex and free resolutions of Z for the occuring stabilizers of cells and outputs a free resolution of Z for Λ. Problem 2: Some cells have infinite stabilisers.

50 Assembling the information Problem 1: The modules C n are not necessarily free. Perturbations - C.T.C. Wall (1961) There is an algorithm which takes as input the cellular chain complex and free resolutions of Z for the occuring stabilizers of cells and outputs a free resolution of Z for Λ. Problem 2: Some cells have infinite stabilisers. Solution: Consider only a certain retract of V >0.

51 Assembling the information Problem 1: The modules C n are not necessarily free. Perturbations - C.T.C. Wall (1961) There is an algorithm which takes as input the cellular chain complex and free resolutions of Z for the occuring stabilizers of cells and outputs a free resolution of Z for Λ. Problem 2: Some cells have infinite stabilisers. Solution: Consider only a certain retract of V >0. The well-rounded retract F V >0 is called well-rounded, if M L(F ) contains a K-Basis of K n.

52 Assembling the information Problem 1: The modules C n are not necessarily free. Perturbations - C.T.C. Wall (1961) There is an algorithm which takes as input the cellular chain complex and free resolutions of Z for the occuring stabilizers of cells and outputs a free resolution of Z for Λ. Problem 2: Some cells have infinite stabilisers. Solution: Consider only a certain retract of V >0. The well-rounded retract F V >0 is called well-rounded, if M L(F ) contains a K-Basis of K n. V >0,wr =1 := {F V >0 F well-rounded, µ L(F ) = 1}.

53 The well-rounded retract Properties of the well-rounded retract In V >0,wr =1 we have:

54 The well-rounded retract Properties of the well-rounded retract In V >0,wr =1 we have: There are only finitely many Λ -orbits in any dimension and every occuring stabiliser is finite.

55 The well-rounded retract Properties of the well-rounded retract In V >0,wr =1 we have: There are only finitely many Λ -orbits in any dimension and every occuring stabiliser is finite. The topological closure of each cell is a polytope.

56 The well-rounded retract Properties of the well-rounded retract In V >0,wr =1 we have: There are only finitely many Λ -orbits in any dimension and every occuring stabiliser is finite. The topological closure of each cell is a polytope. V >0,wr =1 is a retract of V >0, especially we have that the cellular chain complex is again acyclic and H 0 = Z (A. Ash, 1984).

57 The well-rounded retract Properties of the well-rounded retract In V >0,wr =1 we have: There are only finitely many Λ -orbits in any dimension and every occuring stabiliser is finite. The topological closure of each cell is a polytope. V >0,wr =1 is a retract of V >0, especially we have that the cellular chain complex is again acyclic and H 0 = Z (A. Ash, 1984). Summary

58 The well-rounded retract Properties of the well-rounded retract In V >0,wr =1 we have: There are only finitely many Λ -orbits in any dimension and every occuring stabiliser is finite. The topological closure of each cell is a polytope. V >0,wr =1 is a retract of V >0, especially we have that the cellular chain complex is again acyclic and H 0 = Z (A. Ash, 1984). Summary The group Λ acts on the space of positive definite forms.

59 The well-rounded retract Properties of the well-rounded retract In V >0,wr =1 we have: There are only finitely many Λ -orbits in any dimension and every occuring stabiliser is finite. The topological closure of each cell is a polytope. V >0,wr =1 is a retract of V >0, especially we have that the cellular chain complex is again acyclic and H 0 = Z (A. Ash, 1984). Summary The group Λ acts on the space of positive definite forms. This space decomposes into cells in a Λ -compatible way.

60 The well-rounded retract Properties of the well-rounded retract In V >0,wr =1 we have: There are only finitely many Λ -orbits in any dimension and every occuring stabiliser is finite. The topological closure of each cell is a polytope. V >0,wr =1 is a retract of V >0, especially we have that the cellular chain complex is again acyclic and H 0 = Z (A. Ash, 1984). Summary The group Λ acts on the space of positive definite forms. This space decomposes into cells in a Λ -compatible way. There is a subspace such that each cell in it is a polytope and has finite stabiliser in Λ.

61 The well-rounded retract Properties of the well-rounded retract In V >0,wr =1 we have: There are only finitely many Λ -orbits in any dimension and every occuring stabiliser is finite. The topological closure of each cell is a polytope. V >0,wr =1 is a retract of V >0, especially we have that the cellular chain complex is again acyclic and H 0 = Z (A. Ash, 1984). Summary The group Λ acts on the space of positive definite forms. This space decomposes into cells in a Λ -compatible way. There is a subspace such that each cell in it is a polytope and has finite stabiliser in Λ. We may use this cellular decomposition and the finite stabilisers to construct a free ZΛ -resolution of Z.

62 Example 1: Linear Groups over Imaginary Quadratic Integers Q( 5)

63 Example 1: Linear Groups over Imaginary Quadratic Integers Q( 5) K := Q( 5), A := K 2 2, O := Z [ 5 ]. Λ i := End O (L i ) where L 1 := O 2 and L 2 := O where 2 = (2).

64 Example 1: Linear Groups over Imaginary Quadratic Integers Q( 5) K := Q( 5), A := K 2 2, O := Z [ 5 ]. Λ i := End O (L i ) where L 1 := O 2 and L 2 := O where 2 = (2). 1. G 1 := GL(L 1 ): C2 5 n = 1 C H n(g 1, Z) = 4 2 C 12 Z n = 2 C2 8 C 24 n = 3 C2 7 n = 4

65 Example 1: Linear Groups over Imaginary Quadratic Integers Q( 5) K := Q( 5), A := K 2 2, O := Z [ 5 ]. Λ i := End O (L i ) where L 1 := O 2 and L 2 := O where 2 = (2). 1. G 1 := GL(L 1 ): C2 5 n = 1 C H n(g 1, Z) = 4 2 C 12 Z n = 2 C2 8 C 24 n = 3 C2 7 n = 4 2. G 2 := GL(L 2 ): C2 3 n = 1 C H n(g 2, Z) = 2 2 C 12 Z n = 2 C2 8 C 24 n = 3 C2 7 n = 4

66 Example 1: Linear Groups over Imaginary Quadratic Integers Q( 5) K := Q( 5), A := K 2 2, O := Z [ 5 ]. Λ i := End O (L i ) where L 1 := O 2 and L 2 := O where 2 = (2). 1. G 1 := GL(L 1 ): C2 5 n = 1 C H n(g 1, Z) = 4 2 C 12 Z n = 2 C2 8 C 24 n = 3 C2 7 n = 4 2. G 2 := GL(L 2 ): C2 3 n = 1 C H n(g 2, Z) = 2 2 C 12 Z n = 2 C2 8 C 24 n = 3 C2 7 n = 4 Especially: G 1 G 2.

67 Example 2: The well-rounded retract and the Voronoi domains A = K = ( ) 2,3, O = Λ = 1, i, j, k Q Z

68 Example 2: The well-rounded retract and the Voronoi domains A = K = ( ) 2,3, O = Λ = 1, i, j, k Q Z

69 Example 2: The well-rounded retract and the Voronoi domains A = K = ( ) 2,3, O = Λ = 1, i, j, k Q Z

70 Example 2: The well-rounded retract and the Voronoi domains A = K = ( ) 2,3, O = Λ = 1, i, j, k Q Z

71 Example 2: The well-rounded retract and the Voronoi domains A = K = ( ) 2,3, O = Λ = 1, i, j, k Q Z

72 References A. Ash Small-dimensional classifying spaces for arithmetic subgroups of general linear groups Duke Mathematical Journal, 51, Renaud Coulangeon, Gabriele Nebe Maximal finite subgroups and minimal classes arxiv: [math.nt], to appear in l Ens. Math. C.T.C. Wall Resolutions for extensions of groups Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge Univ. Press, vol. 57, 1961