Resolutions for Unit Groups of Orders › homes › Sebastian.Schoennen... · Resolutions for Unit Groups of Orders SebastianSchönnenbeck LehrstuhlDfürMathematik RWTHAachenUniversity

From densest sphere packings to perfect formsFrom perfect forms to the homology of infinite groups

Computational ResultsReferences

Resolutions for Unit Groups of Orders

Sebastian Schönnenbeck

Lehrstuhl D für MathematikRWTH Aachen University

31st October 2013

Sebastian Schönnenbeck Resolutions for Unit Groups of Orders



Overview

1 From densest sphere packings to perfect formsThe sphere packing problemPerfect lattices

2 From perfect forms to the homology of infinite groupsSome definitions from homological algebraQuadratic forms and unit groups of ordersPerturbationsThe well-rounded retract

3 Computational ResultsLinear groups over imaginary quadratic number fieldsMaximal orders in quaternion algebrasFurther applications of the well-rounded complex

4 References




The sphere packing problemPerfect lattices

Overview




4 References





Question: Given n-dimensional spheres all of equal radius, whatpercentage of n-dim. (Euclidean) space can we cover if we arrangethem in a non-overlapping “regular“ way?

Example in Dimension 2:

Square packing, ∼ 0.785 Hexagonal packing, ∼ 0.907





Question: Given n-dimensional spheres all of equal radius, whatpercentage of n-dim. (Euclidean) space can we cover if we arrangethem in a non-overlapping “regular“ way?Example in Dimension 2:







Square packing, ∼ 0.785

Hexagonal packing, ∼ 0.907












Square packing, ∼ 0.785

Hexagonal packing, ∼ 0.907











What do we mean by ”regular“?

Definition: LatticeLet B = (b1, ..., bn) be a basis of Rn. The (Z-)lattice generated byB is the set

L := 〈B〉Z =

{n∑

i=1

λibi | λi ∈ Z, 1 ≤ i ≤ n

}

The sphere packing induced by L is the arrangement of spheres whosecentres are exactly the points in L and whose radii are as large as possible(i.e. half of the minimal distance between two points in L).In dimension 2: The square packing belongs to the lattice with basis(e1, e2), the hexagonal packing to the lattice with basis(e1, 1/2(e1 +

√3e2)).







L := 〈B〉Z =

{n∑

i=1

λibi | λi ∈ Z, 1 ≤ i ≤ n

}


√3e2)).







L := 〈B〉Z =

{n∑

i=1

λibi | λi ∈ Z, 1 ≤ i ≤ n

}

The sphere packing induced by L is the arrangement of spheres whosecentres are exactly the points in L and whose radii are as large as possible(i.e. half of the minimal distance between two points in L).

In dimension 2: The square packing belongs to the lattice with basis(e1, e2), the hexagonal packing to the lattice with basis(e1, 1/2(e1 +

√3e2)).







L := 〈B〉Z =

{n∑

i=1

λibi | λi ∈ Z, 1 ≤ i ≤ n

}


√3e2)).





Definition and RemarkLet L = 〈b1, ..., bn〉Z ⊂ Rn be a lattice.

GL,B := ((bi , bj))i ,j ∈ Rn×n is called the Gram matrix of Lwith respect to B .min(L) := min{(v , v) | 0 6= v ∈ L} is called the minimum ofL.The density of the sphere packing corresponding to L isδ(L) = Vn

2n ·√

min(L)ndet(GL,B) , where Vn denotes the volume of the

n-dim. unit ball. This is independent of the choice of B .

Remark: The value of δ(L) only depends on GL,B (in fact it evenonly depends on the class of GL,B up to rescaling).

Consequently itmakes sense to speak of the density corresponding to a positivedefinite matrix.






GL,B := ((bi , bj))i ,j ∈ Rn×n is called the Gram matrix of Lwith respect to B .

min(L) := min{(v , v) | 0 6= v ∈ L} is called the minimum ofL.The density of the sphere packing corresponding to L isδ(L) = Vn

2n ·√










GL,B := ((bi , bj))i ,j ∈ Rn×n is called the Gram matrix of Lwith respect to B .min(L) := min{(v , v) | 0 6= v ∈ L} is called the minimum ofL.

The density of the sphere packing corresponding to L isδ(L) = Vn

2n ·√











2n ·√











2n ·√











2n ·√



Remark: The value of δ(L) only depends on GL,B (in fact it evenonly depends on the class of GL,B up to rescaling). Consequently itmakes sense to speak of the density corresponding to a positivedefinite matrix.





Densities for the sphere packings of certain latticesLet Vn be the volume of the n-dimensional unit ball.

Zn the n-dimensional standard lattice (or equivalently theidentity matrix) has density Vn

2n .

Let b1, ..., bn ∈ Rn with

(bi , bj) =

2 i = j−1 j = i + 1 or i = j + 10 else

and An the corresponding lattice. Thenthe densitiy of An is Vn√

2n(n+1).

The A3 sphere packing inthe real world

The lattice called E8 has density V816 ≈ 0.254.

Let Λ be the 24-dimensional Leech-lattice. Then Λ has densityequal to V24 ≈ 0.0019.







2n .


(bi , bj) =

2 i = j−1 j = i + 1 or i = j + 10 else


2n(n+1).










2n .


(bi , bj) =

2 i = j−1 j = i + 1 or i = j + 10 else


2n(n+1).










2n .


(bi , bj) =

2 i = j−1 j = i + 1 or i = j + 10 else


2n(n+1).










2n .


(bi , bj) =

2 i = j−1 j = i + 1 or i = j + 10 else


2n(n+1).










2n .


(bi , bj) =

2 i = j−1 j = i + 1 or i = j + 10 else


2n(n+1).








How do we find lattices with high density?

DefinitionLet L be a lattice in Rn.

Min(L) is the set of the shortest non-zero vectors in L.

L ist called perfect if the set {px := xxT | x ∈ Min(L)}contains a basis for the space of symmetric n × n matrices.L is called eutactic if there are λx > 0 such thatIn =

∑x∈Min(L) λxpx .

Theorem: Voronoi (1908)

A lattice is a local maximum of the density function if andonly if it is perfect and eutactic.

There are only finitely many perfect lattices up to rescalingand there is an algorithm to enumerate them







Min(L) is the set of the shortest non-zero vectors in L.

L ist called perfect if the set {px := xxT | x ∈ Min(L)}contains a basis for the space of symmetric n × n matrices.L is called eutactic if there are λx > 0 such thatIn =











Min(L) is the set of the shortest non-zero vectors in L.L ist called perfect if the set {px := xxT | x ∈ Min(L)}contains a basis for the space of symmetric n × n matrices.

L is called eutactic if there are λx > 0 such thatIn =











Min(L) is the set of the shortest non-zero vectors in L.L ist called perfect if the set {px := xxT | x ∈ Min(L)}contains a basis for the space of symmetric n × n matrices.L is called eutactic if there are λx > 0 such thatIn =

























A lattice is a local maximum of the density function if andonly if it is perfect and eutactic.There are only finitely many perfect lattices up to rescalingand there is an algorithm to enumerate them





Remark: Voronoi’s algorithm is of practical use only up todimension 7 as one needs to compute the facets of certain(n2

)-dimensional cones.

What is known about perfect lattices/densest sphere packings?Some Examples:

The densest lattices are known in each dimension up to 8.We know probable candidates for the densest lattices in higherdimensions e.g. the Leech-lattice in dimension 24.

Densest known lattice sphere packings in some low dimensionsDim. 2 3 4 5 6 7 8 24Lattice A∗∗

2 A∗∗3 D∗

4 D∗5 E∗

6 E∗7 E∗

8 ΛDensity 0.907 0.740 0.617 0.465 0.373 0.295 0.254 0.002

*: Provably best lattice sphere packing in the given dimension.

**: Provably best sphere packing (lattice or non-lattice) in the given dimension.










2 A∗∗3 D∗

4 D∗5 E∗

6 E∗7 E∗

8 ΛDensity 0.907 0.740 0.617 0.465 0.373 0.295 0.254 0.002










The densest lattices are known in each dimension up to 8.

We know probable candidates for the densest lattices in higherdimensions e.g. the Leech-lattice in dimension 24.


2 A∗∗3 D∗

4 D∗5 E∗

6 E∗7 E∗

8 ΛDensity 0.907 0.740 0.617 0.465 0.373 0.295 0.254 0.002












2 A∗∗3 D∗

4 D∗5 E∗

6 E∗7 E∗

8 ΛDensity 0.907 0.740 0.617 0.465 0.373 0.295 0.254 0.002












2 A∗∗3 D∗

4 D∗5 E∗

6 E∗7 E∗

8 ΛDensity 0.907 0.740 0.617 0.465 0.373 0.295 0.254 0.002







What’s next?

Coulangeon, Watanabe and many more generalized the concept ofperfect lattices, e.g. for lattices over rings other than Z such asintegers over number fields or even more generally maximal ordersover seperable algebras(e.g. 2001, [3]; 2003, [8]).

R. Coulangeon and G. Nebe use one of these generalizations tocompute a system of represantatives of the conjugacy classes ofmaximal finite subgroups in certain infinite groups (2013, [2]).

M. D. Sikiric, G. Ellis and A. Schürmann use a generalization ofVoronoi’s algorithm to do homology computations for certaininfinite groups (2011, [4])

Basic idea: Use the action of certain groups on the space of quadraticforms (with the perfect forms as distinguished points) to obtainstructural information about these groups.





What’s next?









What’s next?









What’s next?









What’s next?








Some definitions from homological algebraQuadratic forms and unit groups of ordersPerturbationsThe well-rounded retract

Overview




4 References





Definition: Chain Complexes

A positive chain complex C = {Cn, ∂n | n ≥ 0} over the ring R is afamiliy of R-modules Cn, n ≥ 0, together with a familiy ofmorphisms ∂n : Cn → Cn−1, n ≥ 1, with the property ∂n∂n+1 = 0for all n.

. . . ∂n+1 // Cn∂n // Cn−1

∂n−1 // . . . ∂2 // C1∂1 // C0 // 0

DefinitionHn(C ) := ker(∂n)/ img(∂n+1) is called the nth homologymodule of C .

C is called acyclic, if Hn(C ) = 0 for all n ≥ 1.C is called projective (free), if Cn is projective (free) for alln ∈ Z.







. . . ∂n+1 // Cn∂n // Cn−1

∂n−1 // . . . ∂2 // C1∂1 // C0 // 0

DefinitionHn(C ) := ker(∂n)/ img(∂n+1) is called the nth homologymodule of C .

C is called acyclic, if Hn(C ) = 0 for all n ≥ 1.C is called projective (free), if Cn is projective (free) for alln ∈ Z.







. . . ∂n+1 // Cn∂n // Cn−1

∂n−1 // . . . ∂2 // C1∂1 // C0 // 0

DefinitionHn(C ) := ker(∂n)/ img(∂n+1) is called the nth homologymodule of C .C is called acyclic, if Hn(C ) = 0 for all n ≥ 1.

C is called projective (free), if Cn is projective (free) for alln ∈ Z.







. . . ∂n+1 // Cn∂n // Cn−1

∂n−1 // . . . ∂2 // C1∂1 // C0 // 0

DefinitionHn(C ) := ker(∂n)/ img(∂n+1) is called the nth homologymodule of C .C is called acyclic, if Hn(C ) = 0 for all n ≥ 1.C is called projective (free), if Cn is projective (free) for alln ∈ Z.





Definition: ResolutionA a left R-module. An acyclic and projective (free) chain complexof the form

P : . . . // Pn // . . . // P2 // P1 // P0 // 0

with H0(P) ∼= A is called a projective (free) resolution of A.

RemarkA projective (free) resolution of A may also be represented byan exact sequence of the formP : . . . // Pn // . . . // P1 // P0 // A // 0with Pi , i ≥ 0 projective (free).

Every R-module has a free resolution.






P : . . . // Pn // . . . // P2 // P1 // P0 // 0


RemarkA projective (free) resolution of A may also be represented byan exact sequence of the formP : . . . // Pn // . . . // P1 // P0 // A // 0with Pi , i ≥ 0 projective (free).

Every R-module has a free resolution.






P : . . . // Pn // . . . // P2 // P1 // P0 // 0


RemarkA projective (free) resolution of A may also be represented byan exact sequence of the formP : . . . // Pn // . . . // P1 // P0 // A // 0with Pi , i ≥ 0 projective (free).Every R-module has a free resolution.





Definition: The Functor Tor

B a right R-module. TorRn (B,−) : R −mod→ Ab is the functor,whose value TorRn (B,A) may be computed in the following way:

Find a free resolution P = {Pn, ∂n} of A.Form the chain complex B ⊗R P = {B ⊗R Pn, idB ⊗ ∂n}.TorRn (B,A) := Hn(B ⊗ P).

Group HomologyG a group, R = ZG , B an R-module.Hn(G ,B) := TorRn (B,Z) is called the n-th homology group of Gwith coefficients in B .







Find a free resolution P = {Pn, ∂n} of A.

Form the chain complex B ⊗R P = {B ⊗R Pn, idB ⊗ ∂n}.TorRn (B,A) := Hn(B ⊗ P).








Find a free resolution P = {Pn, ∂n} of A.Form the chain complex B ⊗R P = {B ⊗R Pn, idB ⊗ ∂n}.

TorRn (B,A) := Hn(B ⊗ P).






















Task: Compute a free ZG -resolution of Z.

G finite:Naive approach: Successively compute presentations.More sophisticated methods: ’Computing group resolutions’,G. Ellis (2004, [5]); ’Polytopal Resolutions For Finite Groups’,G. Ellis, J. Harris & E. Sköldberg (2006, [6]).

G infinite:Existing methods for certain classes of groups (e.g. nilpotentgroups or Artin groups).No known generally applicable methods.





Task: Compute a free ZG -resolution of Z.G finite:

Naive approach: Successively compute presentations.More sophisticated methods: ’Computing group resolutions’,G. Ellis (2004, [5]); ’Polytopal Resolutions For Finite Groups’,G. Ellis, J. Harris & E. Sköldberg (2006, [6]).







Naive approach: Successively compute presentations.

More sophisticated methods: ’Computing group resolutions’,G. Ellis (2004, [5]); ’Polytopal Resolutions For Finite Groups’,G. Ellis, J. Harris & E. Sköldberg (2006, [6]).















G infinite:

Existing methods for certain classes of groups (e.g. nilpotentgroups or Artin groups).No known generally applicable methods.







G infinite:Existing methods for certain classes of groups (e.g. nilpotentgroups or Artin groups).

No known generally applicable methods.












Unit groups of maximal orders

Situation:D a finite-dimensional Q-division algebra.K = Z (D), R the maximal Z-order in K .A = Dn×n a simple Q-algebra.O an R-maximal order in D.V = Dn the n-dimensional skew vector space over D and thesimple A-module.L ≤ V a (full) O-lattice.

Task: Compute a free ZG -resolution of Z for

G := GL(L) := EndO(L)∗.






Situation:

D a finite-dimensional Q-division algebra.K = Z (D), R the maximal Z-order in K .A = Dn×n a simple Q-algebra.O an R-maximal order in D.V = Dn the n-dimensional skew vector space over D and thesimple A-module.L ≤ V a (full) O-lattice.


G := GL(L) := EndO(L)∗.






Situation:D a finite-dimensional Q-division algebra.

K = Z (D), R the maximal Z-order in K .A = Dn×n a simple Q-algebra.O an R-maximal order in D.V = Dn the n-dimensional skew vector space over D and thesimple A-module.L ≤ V a (full) O-lattice.


G := GL(L) := EndO(L)∗.






Situation:D a finite-dimensional Q-division algebra.K = Z (D), R the maximal Z-order in K .

A = Dn×n a simple Q-algebra.O an R-maximal order in D.V = Dn the n-dimensional skew vector space over D and thesimple A-module.L ≤ V a (full) O-lattice.


G := GL(L) := EndO(L)∗.






Situation:D a finite-dimensional Q-division algebra.K = Z (D), R the maximal Z-order in K .A = Dn×n a simple Q-algebra.

O an R-maximal order in D.V = Dn the n-dimensional skew vector space over D and thesimple A-module.L ≤ V a (full) O-lattice.


G := GL(L) := EndO(L)∗.






Situation:D a finite-dimensional Q-division algebra.K = Z (D), R the maximal Z-order in K .A = Dn×n a simple Q-algebra.O an R-maximal order in D.

V = Dn the n-dimensional skew vector space over D and thesimple A-module.L ≤ V a (full) O-lattice.


G := GL(L) := EndO(L)∗.






Situation:D a finite-dimensional Q-division algebra.K = Z (D), R the maximal Z-order in K .A = Dn×n a simple Q-algebra.O an R-maximal order in D.V = Dn the n-dimensional skew vector space over D and thesimple A-module.

L ≤ V a (full) O-lattice.Task: Compute a free ZG -resolution of Z for

G := GL(L) := EndO(L)∗.








G := GL(L) := EndO(L)∗.








G := GL(L) := EndO(L)∗.





Some remarks on the situation:

Any maximal order in A is of the form EndO(L) for someO-lattice L.DR := D ⊗Q R is a direct sum of matrix rings over R,C,H.AR := A⊗Q R admits a natural involution † (transponse andconjugate).

Definition: Symmetric Elements

We set Σ := {F ∈ AR | F † = F} the space of †-symmetricelements of AR.

Σ admits a positive definite inner product via 〈F ,F ′〉 := Tr(FF ′)(reduced trace).





Some remarks on the situation:Any maximal order in A is of the form EndO(L) for someO-lattice L.

DR := D ⊗Q R is a direct sum of matrix rings over R,C,H.AR := A⊗Q R admits a natural involution † (transponse andconjugate).








Some remarks on the situation:Any maximal order in A is of the form EndO(L) for someO-lattice L.DR := D ⊗Q R is a direct sum of matrix rings over R,C,H.

AR := A⊗Q R admits a natural involution † (transponse andconjugate).








Some remarks on the situation:Any maximal order in A is of the form EndO(L) for someO-lattice L.DR := D ⊗Q R is a direct sum of matrix rings over R,C,H.AR := A⊗Q R admits a natural involution † (transponse andconjugate).
























Basic idea: Find a cell complex which admits a cellular G -actionand use its cellular chain complex to construct a resolution for G .

Definition: Shortest VectorsAny F ∈ Σ defines a quadratic form on VR viaF [x ] := 〈F , xx†〉.

Let P ⊂ Σ be the set of elements whose corresponding formsare positve definite.minL(F ) := min0 6=x∈LF [x ],SL(F ) := {x ∈ L | F [x ] = minL(F )}.

The Cell StructureF ∈ P. Define C`L(F ) := {F ′ ∈ P | SL(F ) = SL(F ′)} the minimalclass of F .






Definition: Shortest VectorsAny F ∈ Σ defines a quadratic form on VR viaF [x ] := 〈F , xx†〉.

Let P ⊂ Σ be the set of elements whose corresponding formsare positve definite.minL(F ) := min06=x∈LF [x ],SL(F ) := {x ∈ L | F [x ] = minL(F )}.







Definition: Shortest VectorsAny F ∈ Σ defines a quadratic form on VR viaF [x ] := 〈F , xx†〉.Let P ⊂ Σ be the set of elements whose corresponding formsare positve definite.

minL(F ) := min06=x∈LF [x ],SL(F ) := {x ∈ L | F [x ] = minL(F )}.







Definition: Shortest VectorsAny F ∈ Σ defines a quadratic form on VR viaF [x ] := 〈F , xx†〉.Let P ⊂ Σ be the set of elements whose corresponding formsare positve definite.minL(F ) := min0 6=x∈LF [x ],SL(F ) := {x ∈ L | F [x ] = minL(F )}.







Definition: Shortest VectorsAny F ∈ Σ defines a quadratic form on VR viaF [x ] := 〈F , xx†〉.Let P ⊂ Σ be the set of elements whose corresponding formsare positve definite.minL(F ) := min0 6=x∈LF [x ],SL(F ) := {x ∈ L | F [x ] = minL(F )}.






Properties of this decomposition:G acts on P via gF := gFg †.

Partial ordering on the minimal classes:C � C ′ ⇔ SL(C ) ⊂ SL(C ′).Each minimal class is a convex set in P.The decomposition as well as the partial ordering arecompatible with the G -action.We have C =

⋃C�C ′ C

′.

The cellular chain complexThe decomposition yields an acyclic chain complex C , where Cn isthe free Abelian group on the minimal classes in dimension n. Cnbecomes a G -module by means of the G -action.





Properties of this decomposition:G acts on P via gF := gFg †.Partial ordering on the minimal classes:C � C ′ ⇔ SL(C ) ⊂ SL(C ′).

Each minimal class is a convex set in P.The decomposition as well as the partial ordering arecompatible with the G -action.We have C =

⋃C�C ′ C

′.






Properties of this decomposition:G acts on P via gF := gFg †.Partial ordering on the minimal classes:C � C ′ ⇔ SL(C ) ⊂ SL(C ′).Each minimal class is a convex set in P.

The decomposition as well as the partial ordering arecompatible with the G -action.We have C =

⋃C�C ′ C

′.






Properties of this decomposition:G acts on P via gF := gFg †.Partial ordering on the minimal classes:C � C ′ ⇔ SL(C ) ⊂ SL(C ′).Each minimal class is a convex set in P.The decomposition as well as the partial ordering arecompatible with the G -action.

We have C =⋃

C�C ′ C′.






Properties of this decomposition:G acts on P via gF := gFg †.Partial ordering on the minimal classes:C � C ′ ⇔ SL(C ) ⊂ SL(C ′).Each minimal class is a convex set in P.The decomposition as well as the partial ordering arecompatible with the G -action.We have C =

⋃C�C ′ C

′.






Properties of this decomposition:G acts on P via gF := gFg †.Partial ordering on the minimal classes:C � C ′ ⇔ SL(C ) ⊂ SL(C ′).Each minimal class is a convex set in P.The decomposition as well as the partial ordering arecompatible with the G -action.We have C =

⋃C�C ′ C

′.






Problem: The modules Cn are not necessarily free.

Perturbations (C. T. C. Wall 1961, [7] )

Assume we are given a free ZG -resolution Ap,∗ (boundary d0) ofCp for all p. Then there are homomorphismsdk : Ap,q → Ap−k,q+k−1, such that

d := d0 +d1 +d2 + ... : Rn :=⊕

p+q=n

Ap,q → Rn−1 :=⊕

p+q=n−1

Ap+q

is the boundary of an acyclic chain complex of free ZG -modules,where H0(R) ∼= Z.





Problem: The modules Cn are not necessarily free.

Perturbations (C. T. C. Wall 1961, [7] )

Assume we are given a free ZG -resolution Ap,∗ (boundary d0) ofCp for all p. Then there are homomorphismsdk : Ap,q → Ap−k,q+k−1, such that

d := d0 +d1 +d2 + ... : Rn :=⊕

p+q=n

Ap,q → Rn−1 :=⊕

p+q=n−1

Ap+q

is the boundary of an acyclic chain complex of free ZG -modules,where H0(R) ∼= Z.





A diagram:

......

...

· · · A2,1 A1,1 A0,1

· · · A2,0 A1,0 A0,0

· · · ∂ // C2∂ // C1

∂ // C0 // H0(C )





A diagram: ...d0

��

...d0

��

...d0

��· · · A2,1

d0��

A1,1

d0��

A0,1

d0��

· · · A2,0

ε

��

A1,0

ε

��

A0,0

ε

��· · · ∂ // C2

∂ // C1∂ // C0 // H0(C )





A diagram: ...d0

��

...d0

��

...d0

��· · · d1 // A2,1

d0��

d1 // A1,1

d0��

d1 // A0,1

d0��

· · · d1 // A2,0

ε

��

d1 // A1,0

ε

��

d1 // A0,0

ε

��· · · ∂ // C2

∂ // C1∂ // C0 // H0(C )





A diagram: ...d0

��

...d0

��

...d0

��· · · d1 // A2,1

d0��

d1 // A1,1

d0��

d1 // A0,1

d0��

· · · d1 // A2,0

ε

��

d1 //

d2

33

qo

l j h f

A1,0

ε

��

d1 // A0,0

ε

��· · · ∂ // C2

∂ // C1∂ // C0 // H0(C )





The situation in the space of positive definite forms:Let (Ck,i )i be a system of representatives of the G -orbits ofminimal classes of dimension k and Sk,i := StabG (Ck,i ).

Then:Ck∼=⊕

i

ZG ⊗ZSk,i Zχk,i

(χk,i (s) ∈ {±1} describes how s acts on the orientation of the cell.)Approach:

Take ZSk,i -resolutions Xk,i of ZX ∗k,i := Xk,i ⊗Z Zχk,i is a resolution of Zχk,i .⊕

i ZG ⊗ZSk,i X∗k,i is a resolution of Ck over G .

Use these resolutions and C. T. C. Wall’s lemma to constructa resolution of Z over G .





The situation in the space of positive definite forms:Let (Ck,i )i be a system of representatives of the G -orbits ofminimal classes of dimension k and Sk,i := StabG (Ck,i ).Then:

Ck∼=⊕

i

ZG ⊗ZSk,i Zχk,i

(χk,i (s) ∈ {±1} describes how s acts on the orientation of the cell.)

Approach:Take ZSk,i -resolutions Xk,i of ZX ∗k,i := Xk,i ⊗Z Zχk,i is a resolution of Zχk,i .⊕








Ck∼=⊕

i

ZG ⊗ZSk,i Zχk,i










Ck∼=⊕

i

ZG ⊗ZSk,i Zχk,i


Take ZSk,i -resolutions Xk,i of Z

X ∗k,i := Xk,i ⊗Z Zχk,i is a resolution of Zχk,i .⊕i ZG ⊗ZSk,i X

∗k,i is a resolution of Ck over G .







Ck∼=⊕

i

ZG ⊗ZSk,i Zχk,i


Take ZSk,i -resolutions Xk,i of ZX ∗k,i := Xk,i ⊗Z Zχk,i is a resolution of Zχk,i .

⊕i ZG ⊗ZSk,i X

∗k,i is a resolution of Ck over G .







Ck∼=⊕

i

ZG ⊗ZSk,i Zχk,i










Ck∼=⊕

i

ZG ⊗ZSk,i Zχk,i









Problem: Sk,i is an infinite group for some classes.

Solution: Consider a certain retract of P.

Definition: well-roundedF ∈ P is called well-rounded, if SL(F ) contains a D-Basis ofDn.

Pwr=1 := {F ∈ P | F well-rounded, minL(F ) = 1}.

Properties of the well-rounded retractIn Pwr

=1 we have:There are only finitely many G -orbits in any dimension andevery stabiliser is finiteThe topological closure of each cell is a polytope.Pwr=1 is a retract of P, especially we have that the cellular chain

complex is again acyclic and H0 ∼= Z (A. Ash, 1984 [1]).





Problem: Sk,i is an infinite group for some classes.Solution: Consider a certain retract of P.





















Definition: well-roundedF ∈ P is called well-rounded, if SL(F ) contains a D-Basis ofDn.Pwr=1 := {F ∈ P | F well-rounded, minL(F ) = 1}.











=1 we have:

There are only finitely many G -orbits in any dimension andevery stabiliser is finiteThe topological closure of each cell is a polytope.Pwr=1 is a retract of P, especially we have that the cellular chain









=1 we have:There are only finitely many G -orbits in any dimension andevery stabiliser is finite

The topological closure of each cell is a polytope.Pwr=1 is a retract of P, especially we have that the cellular chain









=1 we have:There are only finitely many G -orbits in any dimension andevery stabiliser is finiteThe topological closure of each cell is a polytope.

Pwr=1 is a retract of P, especially we have that the cellular chain















The Well-Rounded-Retract for SL2(Z)

Quelle: http://www.uncg.edu/mat/numbertheory/quadratic_form.html


http://www.uncg.edu/mat/numbertheory/quadratic_form.html




Summary

The group G acts on the space of positive definite forms.This space decomposes into cells in a G -compatible way.There is a subspace such that each cell in it is a polytope andhas finite stabiliser in G .We may use this cellular decomposition and the finitestabilisers to construct a free ZG -resolution of Z.





Summary

The group G acts on the space of positive definite forms.

This space decomposes into cells in a G -compatible way.There is a subspace such that each cell in it is a polytope andhas finite stabiliser in G .We may use this cellular decomposition and the finitestabilisers to construct a free ZG -resolution of Z.





Summary

The group G acts on the space of positive definite forms.This space decomposes into cells in a G -compatible way.

There is a subspace such that each cell in it is a polytope andhas finite stabiliser in G .We may use this cellular decomposition and the finitestabilisers to construct a free ZG -resolution of Z.





Summary

The group G acts on the space of positive definite forms.This space decomposes into cells in a G -compatible way.There is a subspace such that each cell in it is a polytope andhas finite stabiliser in G .

We may use this cellular decomposition and the finitestabilisers to construct a free ZG -resolution of Z.





Summary

The group G acts on the space of positive definite forms.This space decomposes into cells in a G -compatible way.There is a subspace such that each cell in it is a polytope andhas finite stabiliser in G .We may use this cellular decomposition and the finitestabilisers to construct a free ZG -resolution of Z.




Linear groups over imaginary quadratic number fieldsMaximal orders in quaternion algebrasFurther applications of the well-rounded complex

Overview




4 References





Basic setup

K := Q[√−d], 0 < d ∈ Z square-free.

R := ZK := IntZ(K ) = Z [ω] where

ω =

{√−d , d ≡ 1, 2 (mod 4)

(1 +√−d)/2, d ≡ 3 (mod 4)

.

C`K the ideal class group of K , hK := |C`K |, a1, ..., ahK

integral representatives of the ideal classesKn contains hK isomorphism classes of lattices represented byRn−1 ⊕ ai , 1 ≤ i ≤ hK .There are essentially |C`K/C`nK | different maximal orders inKn×n.





Basic setup



ω =

{√−d , d ≡ 1, 2 (mod 4)

(1 +√−d)/2, d ≡ 3 (mod 4)

.







Basic setup



ω =

{√−d , d ≡ 1, 2 (mod 4)

(1 +√−d)/2, d ≡ 3 (mod 4)

.


integral representatives of the ideal classes

Kn contains hK isomorphism classes of lattices represented byRn−1 ⊕ ai , 1 ≤ i ≤ hK .There are essentially |C`K/C`nK | different maximal orders inKn×n.





Basic setup



ω =

{√−d , d ≡ 1, 2 (mod 4)

(1 +√−d)/2, d ≡ 3 (mod 4)

.


integral representatives of the ideal classesKn contains hK isomorphism classes of lattices represented byRn−1 ⊕ ai , 1 ≤ i ≤ hK .

There are essentially |C`K/C`nK | different maximal orders inKn×n.





Basic setup



ω =

{√−d , d ≡ 1, 2 (mod 4)

(1 +√−d)/2, d ≡ 3 (mod 4)

.







Q(√−5)

K := Q(√−5), A := K 2×2, R := Z

[√−5].

Oi := EndR(Li ) where L1 := R2 and L2 := R ⊕ ℘ where ℘2 = (2).

1 G1 := GL(L1):

Hn(G1,Z) =

C 5

2 n = 1C 2

4 × C12 ×Z n = 2C 8

2 × C24 n = 3C 7

2 n = 4

2 G2 := GL(L2):

Hn(G2,Z) =

C 3

2 n = 1C 2

2 × C12 ×Z n = 2C 8

2 × C24 n = 3C 7

2 n = 4Especially: G1 � G2.





Q(√−5)

K := Q(√−5), A := K 2×2, R := Z

[√−5].


1 G1 := GL(L1):

Hn(G1,Z) =

C 5

2 n = 1C 2

4 × C12 ×Z n = 2C 8

2 × C24 n = 3C 7

2 n = 4

2 G2 := GL(L2):

Hn(G2,Z) =

C 3

2 n = 1C 2

2 × C12 ×Z n = 2C 8

2 × C24 n = 3C 7






Q(√−5)

K := Q(√−5), A := K 2×2, R := Z

[√−5].


1 G1 := GL(L1):

Hn(G1,Z) =

C 5

2 n = 1C 2

4 × C12 ×Z n = 2C 8

2 × C24 n = 3C 7

2 n = 4

2 G2 := GL(L2):

Hn(G2,Z) =

C 3

2 n = 1C 2

2 × C12 ×Z n = 2C 8

2 × C24 n = 3C 7






Q(√−5)

K := Q(√−5), A := K 2×2, R := Z

[√−5].


1 G1 := GL(L1):

Hn(G1,Z) =

C 5

2 n = 1C 2

4 × C12 ×Z n = 2C 8

2 × C24 n = 3C 7

2 n = 4

2 G2 := GL(L2):

Hn(G2,Z) =

C 3

2 n = 1C 2

2 × C12 ×Z n = 2C 8

2 × C24 n = 3C 7

2 n = 4

Especially: G1 � G2.





Q(√−5)

K := Q(√−5), A := K 2×2, R := Z

[√−5].


1 G1 := GL(L1):

Hn(G1,Z) =

C 5

2 n = 1C 2

4 × C12 ×Z n = 2C 8

2 × C24 n = 3C 7

2 n = 4

2 G2 := GL(L2):

Hn(G2,Z) =

C 3

2 n = 1C 2

2 × C12 ×Z n = 2C 8

2 × C24 n = 3C 7






Special linear groups

The well-rounded complex may also be used to construct resolutions forfinite-index subgroups of unit groups of orders (e.g. special lineargroups).

1 S1 := SL(L1):

Hn(S1,Z) =

C2 × C6 ×Z2 n = 1C 2

2 × C12 ×Z n = 2C 2

2 × C24 n = 3C4 × C12 n = 4

2 S2 := SL(L2):

Hn(S2,Z) =

C3 ×Z2 n = 1C 2

2 × C12 ×Z n = 2C 2

2 × C24 n = 3C2 × C6 n = 4







1 S1 := SL(L1):

Hn(S1,Z) =

C2 × C6 ×Z2 n = 1C 2

2 × C12 ×Z n = 2C 2

2 × C24 n = 3C4 × C12 n = 4

2 S2 := SL(L2):

Hn(S2,Z) =

C3 ×Z2 n = 1C 2

2 × C12 ×Z n = 2C 2

2 × C24 n = 3C2 × C6 n = 4







1 S1 := SL(L1):

Hn(S1,Z) =

C2 × C6 ×Z2 n = 1C 2

2 × C12 ×Z n = 2C 2

2 × C24 n = 3C4 × C12 n = 4

2 S2 := SL(L2):

Hn(S2,Z) =

C3 ×Z2 n = 1C 2

2 × C12 ×Z n = 2C 2

2 × C24 n = 3C2 × C6 n = 4





Projective linear groups

The centre of GL(L) acts trivially on the space of forms, so thewell-rounded complex may be used to compute resolutions for thequotient.

1 P1 := PGL(L1):

Hn(P1,Z) =

C 5

2 n = 1C 3

2 × C6 ×Z n = 2C 9

2 × C6 n = 3C 8

2 n = 42 P2 := PGL(L2):

Hn(P2,Z) =

C 3

2 n = 1C2 × C6 ×Z n = 2C 5

2 × C6 n = 3C 2

2 n = 4







1 P1 := PGL(L1):

Hn(P1,Z) =

C 5

2 n = 1C 3

2 × C6 ×Z n = 2C 9

2 × C6 n = 3C 8

2 n = 4

2 P2 := PGL(L2):

Hn(P2,Z) =

C 3

2 n = 1C2 × C6 ×Z n = 2C 5

2 × C6 n = 3C 2

2 n = 4







1 P1 := PGL(L1):

Hn(P1,Z) =

C 5

2 n = 1C 3

2 × C6 ×Z n = 2C 9

2 × C6 n = 3C 8

2 n = 42 P2 := PGL(L2):

Hn(P2,Z) =

C 3

2 n = 1C2 × C6 ×Z n = 2C 5

2 × C6 n = 3C 2

2 n = 4





Setup

K = Q, D :=(

2,3Q

)= 〈1, i , j , ij〉Q, i2 = 2, j2 = 3, ji = −ij .

k := ij ,O := 〈1, 12(1 + i + k), 1

2(1− i + k), 12(j + k)〉Z

Homology of O∗

Hn(O∗,Z) =

Z, n = 0C24, n ≡ 1 (mod 2)

C2, n ≡ 0 (mod 2), n > 0





Setup

K = Q, D :=(

2,3Q

)= 〈1, i , j , ij〉Q, i2 = 2, j2 = 3, ji = −ij .

k := ij ,O := 〈1, 12(1 + i + k), 1

2(1− i + k), 12(j + k)〉Z

Homology of O∗

Hn(O∗,Z) =

Z, n = 0C24, n ≡ 1 (mod 2)

C2, n ≡ 0 (mod 2), n > 0





Setup

K = Q, D :=(

2,3Q

)= 〈1, i , j , ij〉Q, i2 = 2, j2 = 3, ji = −ij .

k := ij ,O := 〈1, 12(1 + i + k), 1

2(1− i + k), 12(j + k)〉Z

Homology of O∗

Hn(O∗,Z) =

Z, n = 0C24, n ≡ 1 (mod 2)

C2, n ≡ 0 (mod 2), n > 0





Setup

K = Q, D :=(

2,3Q

)= 〈1, i , j , ij〉Q, i2 = 2, j2 = 3, ji = −ij .

k := ij ,O := 〈1, 12(1 + i + k), 1

2(1− i + k), 12(j + k)〉Z

Homology of O∗

Hn(O∗,Z) =

Z, n = 0C24, n ≡ 1 (mod 2)

C2, n ≡ 0 (mod 2), n > 0





Setup

K = Q, D :=(

2,3Q

)= 〈1, i , j , ij〉Q, i2 = 2, j2 = 3, ji = −ij .

k := ij ,O := 〈1, 12(1 + i + k), 1

2(1− i + k), 12(j + k)〉Z

Homology of O∗

Hn(O∗,Z) =

Z, n = 0C24, n ≡ 1 (mod 2)

C2, n ≡ 0 (mod 2), n > 0





Further information available from the well-rounded complex

Example: How do we decide if a group has periodic homology?

TheoremG has p-periodic (p prime) homology if and only if G does notcontain any subgroups isomorphic to Cp × Cp.

G has periodic homology if and only if any finite Abeliansubgroup is cyclic.

Theorem (R. Coulangeon, G. Nebe, 2013, [2])

The set {StabG (C ) | C cell in the well-rounded complex} containsa system of representatives of the conjugacy classes of maximalfinite subgroups of G .



























TheoremG has p-periodic (p prime) homology if and only if G does notcontain any subgroups isomorphic to Cp × Cp.G has periodic homology if and only if any finite Abeliansubgroup is cyclic.









TheoremG has p-periodic (p prime) homology if and only if G does notcontain any subgroups isomorphic to Cp × Cp.G has periodic homology if and only if any finite Abeliansubgroup is cyclic.







Example

G := GL2(Z[√−5])

:

Occuring stabilisers are isomorphic toC6,Q8,D12,C2,D8,V4.⇒ G has 3-periodic but not 2-periodic homology.

S := SL2(Z[√−5])

: Occuring stabilisers are isomorphic toC2,C4,C6,Q8.⇒ S has periodic homology and it is:

Hn(S) =

C2 × C6 ×Z2 n = 1C 22 × C12 ×Z n = 2

C 22 × C24 n ≡ 3 (mod 4), n ≥ 3

C4 × C12 n ≡ 0 (mod 4), n ≥ 4C 22 × C6 n ≡ 1 (mod 4), n ≥ 5

C 22 × C12 n ≡ 2 (mod 4), n ≥ 6





Example

G := GL2(Z[√−5])

: Occuring stabilisers are isomorphic toC6,Q8,D12,C2,D8,V4.

⇒ G has 3-periodic but not 2-periodic homology.

S := SL2(Z[√−5])


Hn(S) =

C2 × C6 ×Z2 n = 1C 22 × C12 ×Z n = 2

C 22 × C24 n ≡ 3 (mod 4), n ≥ 3

C4 × C12 n ≡ 0 (mod 4), n ≥ 4C 22 × C6 n ≡ 1 (mod 4), n ≥ 5

C 22 × C12 n ≡ 2 (mod 4), n ≥ 6





Example

G := GL2(Z[√−5])

: Occuring stabilisers are isomorphic toC6,Q8,D12,C2,D8,V4.⇒ G has 3-periodic but not 2-periodic homology.

S := SL2(Z[√−5])


Hn(S) =

C2 × C6 ×Z2 n = 1C 22 × C12 ×Z n = 2

C 22 × C24 n ≡ 3 (mod 4), n ≥ 3

C4 × C12 n ≡ 0 (mod 4), n ≥ 4C 22 × C6 n ≡ 1 (mod 4), n ≥ 5

C 22 × C12 n ≡ 2 (mod 4), n ≥ 6





Example

G := GL2(Z[√−5])


S := SL2(Z[√−5])

:

Occuring stabilisers are isomorphic toC2,C4,C6,Q8.⇒ S has periodic homology and it is:

Hn(S) =

C2 × C6 ×Z2 n = 1C 22 × C12 ×Z n = 2

C 22 × C24 n ≡ 3 (mod 4), n ≥ 3

C4 × C12 n ≡ 0 (mod 4), n ≥ 4C 22 × C6 n ≡ 1 (mod 4), n ≥ 5

C 22 × C12 n ≡ 2 (mod 4), n ≥ 6





Example

G := GL2(Z[√−5])


S := SL2(Z[√−5])

: Occuring stabilisers are isomorphic toC2,C4,C6,Q8.

⇒ S has periodic homology and it is:

Hn(S) =

C2 × C6 ×Z2 n = 1C 22 × C12 ×Z n = 2

C 22 × C24 n ≡ 3 (mod 4), n ≥ 3

C4 × C12 n ≡ 0 (mod 4), n ≥ 4C 22 × C6 n ≡ 1 (mod 4), n ≥ 5

C 22 × C12 n ≡ 2 (mod 4), n ≥ 6





Example

G := GL2(Z[√−5])


S := SL2(Z[√−5])


Hn(S) =

C2 × C6 ×Z2 n = 1C 22 × C12 ×Z n = 2

C 22 × C24 n ≡ 3 (mod 4), n ≥ 3

C4 × C12 n ≡ 0 (mod 4), n ≥ 4C 22 × C6 n ≡ 1 (mod 4), n ≥ 5

C 22 × C12 n ≡ 2 (mod 4), n ≥ 6





Possible future projects

Make use of the theory of spectral sequences (e.g.Leray-Serre) to compute more information about the homologyof these groups.Use the recently introduced technique of torsion subcomplexes(A. Rahm) to acquire further information.Apply these methods to non-congruence subgroups of theBianchi-Groups (i.e. SL2 over imaginary quadratic integers).






Make use of the theory of spectral sequences (e.g.Leray-Serre) to compute more information about the homologyof these groups.

Use the recently introduced technique of torsion subcomplexes(A. Rahm) to acquire further information.Apply these methods to non-congruence subgroups of theBianchi-Groups (i.e. SL2 over imaginary quadratic integers).






Make use of the theory of spectral sequences (e.g.Leray-Serre) to compute more information about the homologyof these groups.Use the recently introduced technique of torsion subcomplexes(A. Rahm) to acquire further information.

Apply these methods to non-congruence subgroups of theBianchi-Groups (i.e. SL2 over imaginary quadratic integers).






Make use of the theory of spectral sequences (e.g.Leray-Serre) to compute more information about the homologyof these groups.Use the recently introduced technique of torsion subcomplexes(A. Rahm) to acquire further information.Apply these methods to non-congruence subgroups of theBianchi-Groups (i.e. SL2 over imaginary quadratic integers).




Overview




4 References




A. Ash.Small-dimensional classifying spaces for arithmetic subgroups of generallinear groups.Duke Mathematical Journal, 51, 1984.

R. Coulangeon and G. Nebe.Maximal finite subgroups and minimal classes.arXiv preprint arXiv:1304.2597, 2013.

Renaud Coulangeon.Voronoı theory over algebraic number fields.Monographies de l’Enseignement Mathématique, 37:147–162, 2001.

M. Dutour Sikirić, G. Ellis, and A. Schürmann.On the integral homology of PSL4(Z) and other arithmetic groups.Journal of Number Theory, 131(12), 2011.




G. Ellis.Computing group resolutions.Journal of Symbolic Computation, 38, 2004.

G. Ellis, J. Harris, and E. Sköldberg.Polytopal resolutions for finite groups.Journal für die reine und angewandte Mathematik, 2006.

C. T. C. Wall.Resolutions for extensions of groups.In Mathematical Proceedings of the Cambridge Philosophical Society,volume 57. Cambridge Univ Press, 1961.

T. Watanabe.Fundamental hermite constants of linear algebraic groups.Journal of the Mathematical Society of Japan, 55(4):1061–1080, 2003.




Thank you for your attention.


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