GAP (HAP commands) - Chapter 11: Resolutions

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11 Resolutions
11.1 Resolutions for small finite groups
11.2 Resolutions for very small finite groups
11.3 Resolutions for finite groups acting on orbit polytopes
11.4 Minimal resolutions for finite \(p\)-groups over \(\mathbb F_p\)
11.5 Resolutions for abelian groups
11.6 Resolutions for nilpotent groups
11.7 Resolutions for groups with subnormal series
11.8 Resolutions for groups with normal series
11.9 Resolutions for polycyclic (almost) crystallographic groups
11.10 Resolutions for Bieberbach groups
11.11 Resolutions for arbitrary crystallographic groups
11.12 Resolutions for crystallographic groups admitting cubical fundamental domain
11.13 Resolutions for Coxeter groups
11.14 Resolutions for Artin groups
11.15 Resolutions for \(G=SL_2(\mathbb Z[1/m])\)
11.16 Resolutions for selected groups \(G=SL_2( {\mathcal O}(\mathbb Q(\sqrt{d}) )\)
11.17 Resolutions for selected groups \(G=PSL_2( {\mathcal O}(\mathbb Q(\sqrt{d}) )\)
11.18 Resolutions for a few higher-dimensional arithmetic groups
11.19 Resolutions for finite-index subgroups
11.20 Simplifying resolutions
11.21 Resolutions for graphs of groups and for groups with aspherical presentations
11.22 Resolutions for \(\mathbb FG\)-modules

### 11 Resolutions

There is a range of functions in HAP that input a group \(G\), integer \(n\), and attempt to return the first \(n\) terms of a free \(\mathbb ZG\)-resolution \(R_\ast\) of the trivial module \(\mathbb Z\). In some cases an explicit contracting homotopy is provided on the resolution. The function `Size(R)` returns a list whose \(k\)th term is the sum of the lengths of the boundaries of the generators in degree \(k\).

#### 11.1 Resolutions for small finite groups

The following uses discrete Morse theory to construct a resolution.

    gap> G:=SymmetricGroup(6);; n:=6;;
    gap> R:=ResolutionFiniteGroup(G,n);
    Resolution of length 6 in characteristic 0 for Group([ (1,2), (1,2,3,4,5,6)
     ]) .

    gap> Size(R);
    [ 10, 58, 186, 452, 906, 1436 ]

#### 11.2 Resolutions for very small finite groups

The following uses linear algebra over \(\mathbb Z\) to construct a resolution.

    gap> Q:=QuaternionGroup(128);;
    gap> R:=ResolutionSmallGroup(Q,20);
    Resolution of length 20 in characteristic 0 for <pc group of size 128 with
    2 generators> .
    No contracting homotopy available.

    gap> Size(R);
    [ 4, 42, 8, 128, 4, 42, 8, 128, 4, 42, 8, 128, 4, 42, 8, 128, 4, 42, 8, 128 ]

The suspicion that this resolution \(R_\ast\) is periodic of period \(4\) can be confirmed by constructing the chain complex \(C_\ast=R_\ast\otimes_{\mathbb Z}\mathbb ZG\) and verifying that boundary matrices repeat with period \(4\).

A second example of a periodic resolution, for the Dihedral group \(D_{2k+1}=\langle x, y\ |\ x^2= xy^kx^{-1}y^{-k-1} = 1\rangle\) of order \(2k+2\) in the case \(k=1\), is constructed and verified for periodicity in the next example.

    gap> F:=FreeGroup(2);;D:=F/[F.1^2,F.1*F.2*F.1^-1*F.2^-2];;
    gap> R:=ResolutionSmallGroup(D,15);;
    gap> Size(R);
    [ 4, 7, 8, 6, 4, 8, 8, 6, 4, 8, 8, 6, 4, 8, 8 ]
    gap> C:=TensorWithIntegersOverSubgroup(R,Group(One(D)));;
    gap> n:=4;;BoundaryMatrix(C,n)=BoundaryMatrix(C,n+4);
    true
    gap> n:=5;;BoundaryMatrix(C,n)=BoundaryMatrix(C,n+4);
    true
    gap> n:=6;;BoundaryMatrix(C,n)=BoundaryMatrix(C,n+4);
    true
    gap> n:=7;;BoundaryMatrix(C,n)=BoundaryMatrix(C,n+4);
    true
    gap> n:=8;;BoundaryMatrix(C,n)=BoundaryMatrix(C,n+4);
    true

This periodic resolution for \(D_3\) can be found in a paper by R. Swan [Swa60]. The resolution was proved for arbitrary \(D_{2k+1}\) by Irina Kholodna [Kho01] (Corollary 5.5) and is the cellular chain complex of the universal cover of a CW-complex \(X\) with two cells in dimensions \(1, 2 \bmod 4\) and one cell in dimensions \(0,3 \bmod 4\). The \(2\)-skelecton is the \(2\)-complex for the given presentation of \(D_{2k+1}\) and an attaching map for the \(3\)-cell is represented as follows.

A slightly different periodic resolution for \(D_{2k+1}\) has been obtain more recently by FEA Johnson [Joh16]. Johnson's resolution has two free generators in each degree. Interestingly, running the following code for many values of \(k >1\) seems to produce a periodic resolution with two free generators in each degree for most values of \(k\).

    gap> k:=20;;rels:=[x^2,x*y^k*x^-1*y^(-1-k)];;D:=F/rels;;
    gap> R:=ResolutionSmallGroup(D,7);;
    gap> List([0..7],R!.dimension);
    [ 1, 2, 2, 2, 2, 2, 2, 2 ]

The performance of the function `ResolutionSmallGroup(G,n)` is very sensistive to the choice of presentation for the input group \(G\). If \(G\) is an fp-group then the defining presentation for \(G\) is used. If \(G\) is a permutaion group or finite matrix group then GAP functions are invoked to find a presentation for \(G\). The following commands use a geometrically derived presentation for \(SL(2,5)\) as input in order to obtain the first few terms of a periodic resolution for this group of period \(4\).

    gap> Y:=PoincareDodecahedronCWComplex(
    > [[1,2,3,4,5],[6,7,8,9,10]],
    > [[1,11,16,12,2],[19,9,8,18,14]],
    > [[2,12,17,13,3],[20,10,9,19,15]],
    > [[3,13,18,14,4],[16,6,10,20,11]],
    > [[4,14,19,15,5],[17,7,6,16,12]],
    > [[5,15,20,11,1],[18,8,7,17,13]]);;
    gap> G:=FundamentalGroup(Y);
    <fp group on the generators [ f1, f2 ]>
    gap> RelatorsOfFpGroup(G);
    [ f2^-1*f1^-1*f2*f1^-1*f2^-1*f1, f2^-1*f1*f2^2*f1*f2^-1*f1^-1 ]
    gap> StructureDescription(G);
    "SL(2,5)"
    gap> R:=ResolutionSmallGroup(G,3);;
    gap> List([0..3],R!.dimension);
    [ 1, 2, 2, 1 ]

#### 11.3 Resolutions for finite groups acting on orbit polytopes

The following uses Polymake convex hull computations and homological perturbation theory to construct a resolution.

    gap> G:=SignedPermutationGroup(5);;
    gap> StructureDescription(G);
    "C2 x ((C2 x C2 x C2 x C2) : S5)"

    gap> v:=[1,2,3,4,5];;  #The resolution depends on the choice of vector.
    gap> P:=PolytopalComplex(G,[1,2,3,4,5]);
    Non-free resolution in characteristic 0 for <matrix group of size 3840 with
    9 generators> .
    No contracting homotopy available.

    gap> R:=FreeGResolution(P,6);
    Resolution of length 5 in characteristic 0 for <matrix group of size
    3840 with 9 generators> .
    No contracting homotopy available.
    gap> Size(R);
    [ 10, 60, 214, 694, 6247, 273600 ]

The convex polytope \(P_G(v)={\rm Convex~Hull}\{g\cdot v\ |\ g\in G\}\) used in the resolution depends on the choice of vector \(v\in \mathbb R^n\). Two such polytopes for the alternating group \(G=A_4\) acting on \(\mathbb R^4\) can be visualized as follows.

    gap> G:=AlternatingGroup(4);;
    gap> OrbitPolytope(G,[1,2,3,4],["VISUAL"]);
    gap> OrbitPolytope(G,[1,1,3,4],["VISUAL"]);

    gap> P1:=PolytopalComplex(G,[1,2,3,4]);;
    gap> P2:=PolytopalComplex(G,[1,1,3,4]);;
    gap> R1:=FreeGResolution(P1,20);;
    gap> R2:=FreeGResolution(P2,20);;
    gap> Size(R1);
    [ 6, 11, 32, 24, 36, 60, 65, 102, 116, 168, 172, 248, 323, 628, 650, 1093,
      1107, 2456, 2344, 6115 ]
    gap> Size(R2);
    [ 4, 11, 20, 24, 36, 60, 65, 102, 116, 168, 172, 248, 323, 628, 650, 1093,
      1107, 2456, 2344, 6115 ]

#### 11.4 Minimal resolutions for finite \(p\)-groups over \(\mathbb F_p\)

The following uses linear algebra to construct a minimal free \(\mathbb F_pG\)-resolution of the trivial module \(\mathbb F\).

    gap> P:=SylowSubgroup(MathieuGroup(12),2);;
    gap> R:=ResolutionPrimePowerGroup(P,20);
    Resolution of length 20 in characteristic 2 for Group(
    [ (2,8,4,12)(3,11,7,9), (2,3)(4,7)(6,10)(9,11), (3,7)(6,10)(8,11)(9,12),
      (1,10)(3,7)(5,6)(8,12), (2,4)(3,7)(8,12)(9,11), (1,5)(6,10)(8,12)(9,11)
     ]) .

    gap> Size(R);
    [ 6, 62, 282, 740, 1810, 3518, 6440, 10600, 17040, 24162, 34774, 49874,
      62416, 81780, 106406, 145368, 172282, 208926, 262938, 320558 ]

The resolution has the minimum number of generators possible in each degree and can be used to guess a formula for the Poincare series

\(P(x) = \Sigma_{k\ge 0} \dim_{\mathbb F_p}H^k(G,\mathbb F_p)\,x^k\).

The guess is certainly correct for the coefficients of \(x^k\) for \(k\le 20\) and can be used to guess the dimension of say \(H^{2000}(G,\mathbb F_p)\).

Most likely \(\dim_{\mathbb F_2}H^{2000}(G,\mathbb F_2) = 2001000\).

    gap> P:=PoincareSeries(R,20);
    (1)/(-x_1^3+3*x_1^2-3*x_1+1)

    gap> ExpansionOfRationalFunction(P,2000)[2000];
    2001000

#### 11.5 Resolutions for abelian groups

The following uses the formula for the tensor product of chain complexes to construct a resolution.

    gap> A:=AbelianPcpGroup([2,4,8,0,0]);;
    gap> StructureDescription(A);
    "Z x Z x C8 x C4 x C2"

    gap> R:=ResolutionAbelianGroup(A,10);
    Resolution of length 10 in characteristic 0 for Pcp-group with orders
    [ 2, 4, 8, 0, 0 ] .

    gap> Size(R);
    [ 14, 90, 296, 680, 1256, 2024, 2984, 4136, 5480, 7016 ]

#### 11.6 Resolutions for nilpotent groups

The following uses the NQ package to express the free nilpotent group of class \(3\) on three generators as a Pcp group \(G\), and then uses homological perturbation on the lower central series to construct a resolution. The resolution is used to exhibit \(2\)-torsion in \(H_4(G,\mathbb Z)\).

    gap> F:=FreeGroup(3);;
    gap> G:=Image(NqEpimorphismNilpotentQuotient(F,3));;
    gap> R:=ResolutionNilpotentGroup(G,5);
    Resolution of length 5 in characteristic 0 for Pcp-group with orders
    [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] .

    gap> Size(R);
    [ 28, 377, 2377, 9369, 25850 ]

    gap> Homology(TensorWithIntegers(R),4);
    [ 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

The following example uses a simplification procedure for resolutions to construct a resolution \(S_\ast\) for the free nilpotent group \(G\) of class \(2\) on \(3\) generators that has the minimal possible number of free generators in each degree.

    gap> G:=Image(NqEpimorphismNilpotentQuotient(FreeGroup(3),2));;
    gap> R:=ResolutionNilpotentGroup(G,10);;
    gap> S:=ContractedComplex(R);;
    gap> C:=TensorWithIntegers(S);;
    gap> List([1..10],i->IsZero(BoundaryMatrix(C,i)));
    [ true, true, true, true, true, true, true, true, true, true ]

The following example uses homological perturbation on the lower central series to construct a resolution for the Sylow \(2\)-subgroup \(P=Syl_2(M_{12})\) of the Mathieu simple group \(M_{12}\).

    gap> G:=MathieuGroup(12);;
    gap> P:=SylowSubgroup(G,2);;
    gap> StructureDescription(P);
    "((C4 x C4) : C2) : C2"

    gap> R:=ResolutionNilpotentGroup(P,9);
    Resolution of length 9 in characteristic
    0 for <permutation group with 279 generators> .

    gap> Size(R);
    [ 12, 80, 310, 939, 2556, 6768, 19302, 61786, 237068 ]

#### 11.7 Resolutions for groups with subnormal series

The following uses homological perturbation on a subnormal series to construct a resolution for the Sylow \(2\)-subgroup \(P=Syl_2(M_{12})\) of the Mathieu simple group \(M_{12}\).

    gap> P:=SylowSubgroup(MathieuGroup(12),2);;
    gap> sn:=ElementaryAbelianSeries(P);;
    gap> R:=ResolutionSubnormalSeries(sn,9);
    Resolution of length 9 in characteristic
    0 for <permutation group with 64 generators> .

    gap> Size(R);
    [ 12, 78, 288, 812, 1950, 4256, 8837, 18230, 39120 ]

#### 11.8 Resolutions for groups with normal series

The following uses homological perturbation on a normal series to construct a resolution for the Sylow \(2\)-subgroup \(P=Syl_2(M_{12})\) of the Mathieu simple group \(M_{12}\).

    gap> P:=SylowSubgroup(MathieuGroup(12),2);;
    gap> P1:=EfficientNormalSubgroups(P)[1];;
    gap> P2:=Intersection(DerivedSubgroup(P),P1);;
    gap> P3:=Group(One(P));;
    gap> R:=ResolutionNormalSeries([P,P1,P2,P3],9);
    Resolution of length 9 in characteristic
    0 for <permutation group with 64 generators> .

    gap> Size(R);
    [ 10, 60, 200, 532, 1238, 2804, 6338, 15528, 40649 ]

#### 11.9 Resolutions for polycyclic (almost) crystallographic groups

The following uses the Polycyclic package and homological perturbation to construct a resolution for the crystallographic group `G:=SpaceGroup(3,165)`.

    gap> G:=SpaceGroup(3,165);;
    gap> G:=Image(IsomorphismPcpGroup(G));;
    gap> R:=ResolutionAlmostCrystalGroup(G,20);
    Resolution of length 20 in characteristic 0 for Pcp-group with orders
    [ 3, 2, 0, 0, 0 ] .

    gap> Size(R);
    [ 10, 49, 117, 195, 273, 351, 429, 507, 585, 663, 741, 819, 897, 975, 1053,
      1131, 1209, 1287, 1365, 1443 ]

The following constructs a resolution for an almost crystallographic Pcp group \(G\). The final commands establish that \(G\) is not isomorphic to a crystallographic group.

    gap> G:=AlmostCrystallographicPcpGroup( 4, 50, [ 1, -4, 1, 2 ] );;
    gap> R:=ResolutionAlmostCrystalGroup(G,20);
    Resolution of length 20 in characteristic 0 for Pcp-group with orders
    [ 4, 0, 0, 0, 0 ] .

    gap> Size(R);
    [ 10, 53, 137, 207, 223, 223, 223, 223, 223, 223, 223, 223, 223, 223, 223,
      223, 223, 223, 223, 223 ]

    gap> T:=Kernel(NaturalHomomorphismOnHolonomyGroup(G));;
    gap> IsAbelian(T);
    false

#### 11.10 Resolutions for Bieberbach groups

The following constructs a resolution for the Bieberbach group `G=SpaceGroup(3,165)` by using convex hull algorithms to construct a Dirichlet domain for its free action on Euclidean space \(\mathbb R^3\). By construction the resolution is trivial in degrees \(\ge 3\).

    gap> G:=SpaceGroup(3,165);;
    gap> R:=ResolutionBieberbachGroup(G);
    Resolution of length 4 in characteristic
    0 for SpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 ) .
    No contracting homotopy available.

    gap> Size(R);
    [ 10, 18, 8, 0 ]

The fundamental domain constructed for the above resolution can be visualized using the following commands.

    gap> F:=FundamentalDomainBieberbachGroup(G);
    <polymake object>
    gap> Display(F);

A different fundamental domain and resolution for \(G\) can be obtained by changing the choice of vector \(v\in \mathbb R^3\) in the definition of the Dirichlet domain

\(D(v) = \{x\in \mathbb R^3\ | \ ||x-v|| \le ||x-g.v||\ {\rm for~all~} g\in G\}\).

    gap> R:=ResolutionBieberbachGroup(G,[1/2,1/2,1/2]);
    Resolution of length 4 in characteristic
    0 for SpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 ) .
    No contracting homotopy available.

    gap> Size(R);
    [ 28, 42, 16, 0 ]

    gap> F:=FundamentalDomainBieberbachGroup(G);
    <polymake object>
    gap> Display(F);

A higher dimensional example is handled in the next session. A list of the \(62\) \(7\)-dimensional Hantze-Wendt Bieberbach groups is loaded and a resolution is computed for the first group in the list.

    gap> file:=HapFile("HW-7dim.txt");;
    gap> Read(file);
    gap> G:=HWO7Gr[1];
    <matrix group with 7 generators>

    gap> R:=ResolutionBieberbachGroup(G);
    Resolution of length 8 in characteristic 0 for <matrix group with
    7 generators> .
    No contracting homotopy available.

    gap> Size(R);
    [ 284, 1512, 3780, 4480, 2520, 840, 84, 0 ]

The homological perturbation techniques needed to extend this method to crystallographic groups acting non-freely on \(\mathbb R^n\) has not yet been implemenyed. This is on the TO-DO list.

#### 11.11 Resolutions for arbitrary crystallographic groups

An implementation of the above method for Bieberbach groups is also available for arbitrary crystallographic groups. The following example constructs a resolution for the group `G:=SpaceGroupIT(3,227)`.

    gap> G:=SpaceGroupIT(3,227);;
    gap> R:=ResolutionSpaceGroup(G,11);
    Resolution of length 11 in characteristic 0 for <matrix group with
    8 generators> .
    No contracting homotopy available.

    gap> Size(R);
    [ 38, 246, 456, 644, 980, 1427, 2141, 2957, 3993, 4911, 6179 ]

#### 11.12 Resolutions for crystallographic groups admitting cubical fundamental domain

The following uses subdivision techniques to construct a resolution for the Bieberbach group `G:=SpaceGroup(4,122)`. The resolution is endowed with a contracting homotopy.

    gap> G:=SpaceGroup(4,122);;
    gap> R:=ResolutionCubicalCrystGroup(G,20);
    Resolution of length 20 in characteristic 0 for <matrix group with
    6 generators> .

    gap> Size(R);
    [ 8, 24, 24, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

Subdivision and homological perturbation are used to construct the following resolution (with contracting homotopy) for a crystallographic group with non-free action.

    gap> G:=SpaceGroup(4,1100);;
    gap> R:=ResolutionCubicalCrystGroup(G,20);
    Resolution of length 20 in characteristic 0 for <matrix group with
    8 generators> .

    gap> Size(R);
    [ 40, 215, 522, 738, 962, 1198, 1466, 1734, 2034, 2334, 2666, 2998, 3362,
      3726, 4122, 4518, 4946, 5374, 5834, 6294 ]

#### 11.13 Resolutions for Coxeter groups

The following session constructs the Coxeter diagram for the Coxeter group \(B=B_7\) of order \(645120\). A resolution for \(G\) is then computed.

    gap> D:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,3]],[5,[6,3]],[6,[7,4]]];;
    gap> CoxeterDiagramDisplay(D);;

    gap> R:=ResolutionCoxeterGroup(D,5);
    Resolution of length 5 in characteristic
    0 for <permutation group of size 645120 with 7 generators> .
    No contracting homotopy available.

    gap> Size(R);
    [ 14, 112, 492, 1604, 5048 ]

The routine extension of this method to infinite Coxeter groups is on the TO-DO list.

#### 11.14 Resolutions for Artin groups

The following session constructs a resolution for the infinite Artin group \(G\) associated to the Coxeter group \(B_7\). Exactness of the resolution depends on the solution to the \(K(\pi,1)\) Conjecture for Artin groups of spherical type.

    gap> R:=ResolutionArtinGroup(D,8);
    Resolution of length 8 in characteristic 0 for <fp group on the generators
    [ f1, f2, f3, f4, f5, f6, f7 ]> .
    No contracting homotopy available.

    gap> Size(R);
    [ 14, 98, 310, 610, 918, 1326, 2186, 0 ]

#### 11.15 Resolutions for \(G=SL_2(\mathbb Z[1/m])\)

The following uses homological perturbation to construct a resolution for \(G=SL_2(\mathbb Z[1/6])\).

    gap> R:=ResolutionSL2Z(6,10);
    Resolution of length 10 in characteristic 0 for SL(2,Z[1/6]) .

    gap> Size(R);
    [ 44, 679, 6910, 21304, 24362, 48506, 43846, 90928, 86039, 196210 ]

#### 11.16 Resolutions for selected groups \(G=SL_2( {\mathcal O}(\mathbb Q(\sqrt{d}) )\)

The following uses finite "Voronoi complexes" and homological perturbation to construct a resolution for \(G=SL_2({\mathcal O}(\mathbb Q(\sqrt{-5}))\). The finite complexes were contributed independently by A. Rahm, M. Dutour-Scikiric and S. Schoenenbeck and are stored in the folder `~pkg/Hap1.v/lib/Perturbations/Gcomplexes`.

    gap> R:=ResolutionSL2QuadraticIntegers(-5,10);
    Resolution of length 10 in characteristic 0 for matrix group .
    No contracting homotopy available.

    gap> Size(R);
    [ 22, 114, 120, 200, 146, 156, 136, 254, 168, 170 ]

#### 11.17 Resolutions for selected groups \(G=PSL_2( {\mathcal O}(\mathbb Q(\sqrt{d}) )\)

The following uses finite "Voronoi complexes" and homological perturbation to construct a resolution for \(G=PSL_2({\mathcal O}(\mathbb Q(\sqrt{-11}))\). The finite complexes were contributed independently by A. Rahm, M. Dutour-Scikiric and S. Schoenenbeck and are stored in the folder `~pkg/Hap1.v/lib/Perturbations/Gcomplexes`.

    gap> R:=ResolutionPSL2QuadraticIntegers(-11,10);
    Resolution of length 10 in characteristic 0 for PSL(2,O-11) .
    No contracting homotopy available.

    gap> Size(R);
    [ 12, 59, 89, 107, 125, 230, 208, 270, 326, 515 ]

#### 11.18 Resolutions for a few higher-dimensional arithmetic groups

The following uses finite "Voronoi complexes" and homological perturbation to construct a resolution for \(G=PSL_4(\mathbb Z)\). The finite complexes were contributed by M. Dutour-Scikiric and are stored in the folder `~pkg/Hap1.v/lib/Perturbations/Gcomplexes`.

    gap>  V:=ContractibleGcomplex("PSL(4,Z)_d");
    Non-free resolution in characteristic 0 for matrix group .
    No contracting homotopy available.

    gap> R:=FreeGResolution(V,5);
    Resolution of length 5 in characteristic 0 for matrix group .
    No contracting homotopy available.

    gap> Size(R);
    [ 18, 210, 1444, 26813 ]

#### 11.19 Resolutions for finite-index subgroups

The next commands first construct the congruence subgroup \(\Gamma_0(I)\) of index \(144\) in \(SL_2({\cal O}\mathbb Q(\sqrt{-2}))\) for the ideal \(I\) in \({\cal O}\mathbb Q(\sqrt{-2})\) generated by \(4+5\sqrt{-2}\). The commands then compute a resolution for the congruence subgroup \(G=\Gamma_0(I) \le SL_2({\cal O}\mathbb Q(\sqrt{-2}))\)

    gap> Q:=QuadraticNumberField(-2);;
    gap> OQ:=RingOfIntegers(Q);;
    gap> I:=QuadraticIdeal(OQ,4+5*Sqrt(-2));;
    gap> G:=HAP_CongruenceSubgroupGamma0(I);
    <[group of 2x2 matrices in characteristic 0>
    gap>
    gap> IndexInSL2O(G);
    144
    gap> R:=ResolutionSL2QuadraticIntegers(-2,4,true);;
    gap> S:=ResolutionFiniteSubgroup(R,G);
    Resolution of length 4 in characteristic 0 for <matrix group with
    290 generators> .

    gap> Size(S);
    [ 1152, 8496, 30960, 59616 ]

#### 11.20 Simplifying resolutions

The next commands construct a resolution \(R_\ast\) for the symmetric group \(S_5\) and convert it to a resolution \(S_\ast\) for the finite index subgroup \(A_4 < S_5\). An heuristic algorithm is applied to \(S_\ast\) in the hope of obtaining a smaller resolution \(T_\ast\) for the alternating group \(A_4\).

    gap> R:=ResolutionFiniteGroup(SymmetricGroup(5),5);;
    gap> S:=ResolutionFiniteSubgroup(R,AlternatingGroup(4));
    Resolution of length 5 in characteristic 0 for Alt( [ 1 .. 4 ] ) .

    gap> Size(S);
    [ 80, 380, 1000, 2040, 3400 ]
    gap> T:=SimplifiedComplex(S);
    Resolution of length 5 in characteristic 0 for Alt( [ 1 .. 4 ] ) .

    gap> Size(T);
    [ 4, 34, 22, 19, 196 ]

#### 11.21 Resolutions for graphs of groups and for groups with aspherical presentations

The following example constructs a resolution for a finitely presented group whose presentation is known to have the property that its associated \(2\)-complex is aspherical.

    gap> F:=FreeGroup(3);;x:=F.1;;y:=F.2;;z:=F.3;;
    gap> rels:=[x*y*x*(y*x*y)^-1, y*z*y*(z*y*z)^-1, z*x*z*(x*z*x)^-1];;
    gap> G:=F/rels;;
    gap> R:=ResolutionAsphericalPresentation(G,10);
    Resolution of length 10 in characteristic 0 for <fp group on the generators
    [ f1, f2, f3 ]> .
    No contracting homotopy available.

    gap> Size(R);
    [ 6, 18, 0, 0, 0, 0, 0, 0, 0, 0 ]

The following commands create a resolution for a graph of groups corresponding to the amalgamated product \(G=H\ast_AK\) where \(H=S_5\) is the symmetric group of degree \(5\), \(K=S_4\) is the symmetric group of degree \(4\) and the common subgroup is \(A=S_3\).

    gap> S5:=SymmetricGroup(5);SetName(S5,"S5");;
    Sym( [ 1 .. 5 ] )
    gap> S4:=SymmetricGroup(4);SetName(S4,"S4");;
    Sym( [ 1 .. 4 ] )
    gap> A:=SymmetricGroup(3);SetName(A,"S3");;
    Sym( [ 1 .. 3 ] )
    gap> AS5:=GroupHomomorphismByFunction(A,S5,x->x);;
    gap> AS4:=GroupHomomorphismByFunction(A,S4,x->x);;
    gap> D:=[S5,S4,[AS5,AS4]];;
    gap> GraphOfGroupsDisplay(D);;

    gap> R:=ResolutionGraphOfGroups(D,8);;
    gap> Size(R);
    [ 16, 68, 162, 302, 480, 627, 869, 1290 ]

#### 11.22 Resolutions for \(\mathbb FG\)-modules

Let \(\mathbb F=\mathbb F_p\) be the field of \(p\) elements and let \(M\) be some \(\mathbb FG\)-module for \(G\) a finite \(p\)-group. We might wish to construct a free \(\mathbb FG\)-resolution for \(M\). We can handle this by constructing a short exact sequence

\( DM \rightarrowtail P \twoheadrightarrow M\)

in which \(P\) is free (or projective). Then any resolution of \(DM\) yields a resolution of \(M\) and we can represent \(DM\) as a submodule of \(P\). We refer to \(DM\) as the desuspension of \(M\). Consider for instance \(G=Syl_2(GL(4,2))\) and \(\mathbb F=\mathbb F_2\). The matrix group \(G\) acts via matrix multiplication on \(M=\mathbb F^4\). The following example constructs a free \(\mathbb FG\)-resolution for \(M\).

    gap> G:=GL(4,2);;
    gap> S:=SylowSubgroup(G,2);;
    gap> M:=GModuleByMats(GeneratorsOfGroup(S),GF(2));;
    gap> DM:=DesuspensionMtxModule(M);;
    gap> R:=ResolutionFpGModule(DM,20);
    Resolution of length 20 in characteristic 2 for <matrix group of
    size 64 with 3 generators> .

    gap> List([0..20],R!.dimension);
    [ 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136,
    153, 171, 190, 210, 231, 253 ]

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