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<div class="ChapSects"><a href="chap11.html#X7C0B125E7D5415B4">11 Resolutions</a>
<div class="ContSect"> <a href="chap11.html#X83E8F9DA7CDC0DA7">11.1 Resolutions for small finite groups</a>

</div>
<div class="ContSect"> <a href="chap11.html#X7EEA738385CC3AEA">11.2 Resolutions for very small finite groups</a>

</div>
<div class="ContSect"> <a href="chap11.html#X86C0983E81F706F5">11.3 Resolutions for finite groups acting on orbit polytopes</a>

</div>
<div class="ContSect"> <a href="chap11.html#X85374EA47E3D97CF">11.4 Minimal resolutions for finite p-groups over F_p</a>

</div>
<div class="ContSect"> <a href="chap11.html#X866C8D91871D1170">11.5 Resolutions for abelian groups</a>

</div>
<div class="ContSect"> <a href="chap11.html#X7B332CBE85120B38">11.6 Resolutions for nilpotent groups</a>

</div>
<div class="ContSect"> <a href="chap11.html#X7B03997084E00509">11.7 Resolutions for groups with subnormal series</a>

</div>
<div class="ContSect"> <a href="chap11.html#X814FFCE080B3A826">11.8 Resolutions for groups with normal series</a>

</div>
<div class="ContSect"> <a href="chap11.html#X81227BF185C417AF">11.9 Resolutions for polycyclic (almost) crystallographic groups </a>

</div>
<div class="ContSect"> <a href="chap11.html#X814BCDD6837BB9C5">11.10 Resolutions for Bieberbach groups </a>

</div>
<div class="ContSect"> <a href="chap11.html#X87ADCB7D7FC0B4D3">11.11 Resolutions for arbitrary crystallographic groups</a>

</div>
<div class="ContSect"> <a href="chap11.html#X7B9B3AF487338A9B">11.12 Resolutions for crystallographic groups admitting cubical fundamental domain</a>

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<div class="ContSect"> <a href="chap11.html#X78DD8D068349065A">11.13 Resolutions for Coxeter groups </a>

</div>
<div class="ContSect"> <a href="chap11.html#X7C69E7227F919CC9">11.14 Resolutions for Artin groups </a>

</div>
<div class="ContSect"> <a href="chap11.html#X8032647F8734F4EB">11.15 Resolutions for G=SL_2( Z[1/m])</a>

</div>
<div class="ContSect"> <a href="chap11.html#X7BE4DE82801CD38E">11.16 Resolutions for selected groups
G=SL_2( mathcal O( Q(sqrtd) )</a>

</div>
<div class="ContSect"> <a href="chap11.html#X7D9CCB2C7DAA2310">11.17 Resolutions for selected groups
G=PSL_2( mathcal O( Q(sqrtd) )</a>

</div>
<div class="ContSect"> <a href="chap11.html#X7F699587845E6DB1">11.18 Resolutions for a few higher-dimensional arithmetic groups
</a>

</div>
<div class="ContSect"> <a href="chap11.html#X7812EB3F7AC45F87">11.19 Resolutions for finite-index subgroups
</a>

</div>
<div class="ContSect"> <a href="chap11.html#X84CAAA697FAC8E0D">11.20 Simplifying resolutions
</a>

</div>
<div class="ContSect"> <a href="chap11.html#X780C3F038148A1C7">11.21 Resolutions for graphs of groups and for groups with aspherical presentations
</a>

</div>
<div class="ContSect"> <a href="chap11.html#X85AB973F8566690A">11.22 Resolutions for FG-modules
</a>

</div>
</div>

<h3>11 Resolutions</h3>

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 ZG-resolution R_∗ of the trivial module Z. In some cases an explicit contracting homotopy is provided on the resolution. The function <code class="code">Size(R)</code> returns a list whose kth term is the sum of the lengths of the boundaries of the generators in degree k.

<a id="X83E8F9DA7CDC0DA7" name="X83E8F9DA7CDC0DA7"></a>

<h4>11.1 Resolutions for small finite groups</h4>

The following uses discrete Morse theory to construct a resolution.

<div class="example"><pre>
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 ]

</pre></div>

<a id="X7EEA738385CC3AEA" name="X7EEA738385CC3AEA"></a>

<h4>11.2 Resolutions for very small finite groups</h4>

The following uses linear algebra over Z to construct a resolution.

<div class="example"><pre>
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 ]

</pre></div>

The suspicion that this resolution R_∗ is periodic of period 4 can be confirmed by constructing the chain complex C_∗=R_∗⊗_ Z ZG and verifying that boundary matrices repeat with period 4.

A second example of a periodic resolution, for the Dihedral group D_2k+1=⟨ x, y | x^2= xy^kx^-1y^-k-1 = 1⟩ of order 2k+2 in the case k=1, is constructed and verified for periodicity in the next example.

<div class="example"><pre>
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

</pre></div>

This periodic resolution for D_3 can be found in a paper by R. Swan <a href="chapBib.html#biBswan2">[Swa60]</a>. The resolution was proved for arbitrary D_2k+1 by Irina Kholodna <a href="chapBib.html#biBkholodna">[Kho01]</a> (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 mod 4 and one cell in dimensions 0,3 mod 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.

<img src="images/syzygyjsc.jpg" align="center" height="300" alt="homotopical syzygy"/>

A slightly different periodic resolution for D_2k+1 has been obtain more recently by FEA Johnson <a href="chapBib.html#biBjohnson">[Joh16]</a>. 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.

<div class="example"><pre>
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 ]

</pre></div>

The performance of the function <code class="code">ResolutionSmallGroup(G,n)</code> 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.

<div class="example"><pre>
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 ]

</pre></div>

<a id="X86C0983E81F706F5" name="X86C0983E81F706F5"></a>

<h4>11.3 Resolutions for finite groups acting on orbit polytopes</h4>

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

<div class="example"><pre>
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 ]

</pre></div>

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

<div class="example"><pre>
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 ]

</pre></div>

<img src="images/orb-poly-1.png" align="center" height="300" alt="an orbit polytope"/> <img src="images/orb-poly-2.png" align="center" height="300" alt="an orbit polytope"/>

<a id="X85374EA47E3D97CF" name="X85374EA47E3D97CF"></a>

<h4>11.4 Minimal resolutions for finite p-groups over F_p</h4>

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

<div class="example"><pre>
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 ]

</pre></div>

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) = Σ_k≥ 0 dim_ F_pH^k(G, F_p)x^k.

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

Most likely dim_ F_2H^2000(G, F_2) = 2001000.

<div class="example"><pre>
gap> P:=PoincareSeries(R,20);
(1)/(-x_1^3+3*x_1^2-3*x_1+1)

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

</pre></div>

<a id="X866C8D91871D1170" name="X866C8D91871D1170"></a>

<h4>11.5 Resolutions for abelian groups</h4>

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

<div class="example"><pre>
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 ]

</pre></div>

<a id="X7B332CBE85120B38" name="X7B332CBE85120B38"></a>

<h4>11.6 Resolutions for nilpotent groups</h4>

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, Z).

<div class="example"><pre>
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 ]

</pre></div>

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

<div class="example"><pre>
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 ]

</pre></div>

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.

<div class="example"><pre>
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 ]

</pre></div>

<a id="X7B03997084E00509" name="X7B03997084E00509"></a>

<h4>11.7 Resolutions for groups with subnormal series</h4>

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.

<div class="example"><pre>
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 ]

</pre></div>

<a id="X814FFCE080B3A826" name="X814FFCE080B3A826"></a>

<h4>11.8 Resolutions for groups with normal series</h4>

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.

<div class="example"><pre>
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 ]

</pre></div>

<a id="X81227BF185C417AF" name="X81227BF185C417AF"></a>

<h4>11.9 Resolutions for polycyclic (almost) crystallographic groups </h4>

The following uses the Polycyclic package and homological perturbation to construct a resolution for the crystallographic group <code class="code">G:=SpaceGroup(3,165)</code>.

<div class="example"><pre>
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 ]

</pre></div>

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.

<div class="example"><pre>
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

</pre></div>

<a id="X814BCDD6837BB9C5" name="X814BCDD6837BB9C5"></a>

<h4>11.10 Resolutions for Bieberbach groups </h4>

The following constructs a resolution for the Bieberbach group <code class="code">G=SpaceGroup(3,165)</code> by using convex hull algorithms to construct a Dirichlet domain for its free action on Euclidean space R^3. By construction the resolution is trivial in degrees ≥ 3.

<div class="example"><pre>
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 ]

</pre></div>

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

<div class="example"><pre>
gap> F:=FundamentalDomainBieberbachGroup(G);
<polymake object>
gap> Display(F);

</pre></div>

<img src="images/3-165-0.png" align="center" height="300" alt="a Dirichlet domain"/>

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

D(v) = {x∈ R^3 | ||x-v|| ≤ ||x-g.v|| for~all~ g∈ G}.

<div class="example"><pre>
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);

</pre></div>

<img src="images/3-165-1.png" align="center" height="300" alt="a Dirichlet domain"/>

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.

<div class="example"><pre>
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 ]

</pre></div>

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

<a id="X87ADCB7D7FC0B4D3" name="X87ADCB7D7FC0B4D3"></a>

<h4>11.11 Resolutions for arbitrary crystallographic groups</h4>

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 <code class="code">G:=SpaceGroupIT(3,227)</code>.

<div class="example"><pre>
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 ]

</pre></div>

<a id="X7B9B3AF487338A9B" name="X7B9B3AF487338A9B"></a>

<h4>11.12 Resolutions for crystallographic groups admitting cubical fundamental domain</h4>

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

<div class="example"><pre>
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 ]

</pre></div>

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

<div class="example"><pre>
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 ]

</pre></div>

<a id="X78DD8D068349065A" name="X78DD8D068349065A"></a>

<h4>11.13 Resolutions for Coxeter groups </h4>

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

<div class="example"><pre>
gap> D:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,3]],[5,[6,3]],[6,[7,4]]];;
gap> CoxeterDiagramDisplay(D);;

</pre></div>

<img src="images/coxeter-diagram-b7.png" align="center" height="150" alt="a Dirichlet domain"/>

<div class="example"><pre>
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 ]

</pre></div>

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

<a id="X7C69E7227F919CC9" name="X7C69E7227F919CC9"></a>

<h4>11.14 Resolutions for Artin groups </h4>

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(π,1) Conjecture for Artin groups of spherical type.

<div class="example"><pre>
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 ]

</pre></div>

<a id="X8032647F8734F4EB" name="X8032647F8734F4EB"></a>

<h4>11.15 Resolutions for G=SL_2( Z[1/m])</h4>

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

<div class="example"><pre>
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 ]

</pre></div>

<a id="X7BE4DE82801CD38E" name="X7BE4DE82801CD38E"></a>

<h4>11.16 Resolutions for selected groups
G=SL_2( mathcal O( Q(sqrtd) )</h4>

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

<div class="example"><pre>
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 ]

</pre></div>

<a id="X7D9CCB2C7DAA2310" name="X7D9CCB2C7DAA2310"></a>

<h4>11.17 Resolutions for selected groups
G=PSL_2( mathcal O( Q(sqrtd) )</h4>

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

<div class="example"><pre>
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 ]

</pre></div>

<a id="X7F699587845E6DB1" name="X7F699587845E6DB1"></a>

<h4>11.18 Resolutions for a few higher-dimensional arithmetic groups
</h4>

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

<div class="example"><pre>
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 ]

</pre></div>

<a id="X7812EB3F7AC45F87" name="X7812EB3F7AC45F87"></a>

<h4>11.19 Resolutions for finite-index subgroups
</h4>

The next commands first construct the congruence subgroup Γ_0(I) of index 144 in SL_2(cal O Q(sqrt-2)) for the ideal I in cal O Q(sqrt-2) generated by 4+5sqrt-2. The commands then compute a resolution for the congruence subgroup G=Γ_0(I) ≤ SL_2(cal O Q(sqrt-2))

<div class="example"><pre>
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 ]

</pre></div>

<a id="X84CAAA697FAC8E0D" name="X84CAAA697FAC8E0D"></a>

<h4>11.20 Simplifying resolutions
</h4>

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

<div class="example"><pre>
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 ]

</pre></div>

<a id="X780C3F038148A1C7" name="X780C3F038148A1C7"></a>

<h4>11.21 Resolutions for graphs of groups and for groups with aspherical presentations
</h4>

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.

<div class="example"><pre>
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 ]

</pre></div>

The following commands create a resolution for a graph of groups corresponding to the amalgamated product G=H∗_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.

<div class="example"><pre>
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);;

</pre></div>

<img src="images/graphOFgroups.gif" align="center" height="100" alt="graph of groups"/>

<div class="example"><pre>
gap> R:=ResolutionGraphOfGroups(D,8);;
gap> Size(R);
[ 16, 68, 162, 302, 480, 627, 869, 1290 ]

</pre></div>

<a id="X85AB973F8566690A" name="X85AB973F8566690A"></a>

<h4>11.22 Resolutions for FG-modules
</h4>

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

DM ↣ P ↠ 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 F= F_2. The matrix group G acts via matrix multiplication on M= F^4. The following example constructs a free FG-resolution for M.

<div class="example"><pre>
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 ]

</pre></div>

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