Several classes of examples of subgroups of \(\mathrm{Aut}(B_{d,k})\) that satisfy (C) and or (D) are constructed in [Tor20] and implemented in this section. For a given permutation group \(F\le S_{d}\), there are always the three local actions \(\Gamma(F)\), \(\Delta(F)\) and \(\Phi(F)\) on \(\mathrm{Aut}(B_{d,2})\) that project onto \(F\). For some \(F\), these are all distinct and yield all universal groups that have \(F\) as their \(1\)-local action, see [Tor20, Theorem 3.32]. More examples arise in particular when either point stabilizers in \(F\) are not simple, \(F\) preserves a partition, or \(F\) is not perfect. This section also includes functions to provide the \(k\)-local actions of the groups \(\mathrm{PGL}(2,\mathbb{Q}_{p})\) and \(\mathrm{PSL}(2,\mathbb{Q}_{p})\) acting on \(T_{p+1}\).
Here, we implement the local actions \(\Gamma(F),\Delta(F)\le\mathrm{Aut}(B_{d,2})\), both of which satisfy both (C) and (D), see [Tor20, Section 3.4.1].
‣ LocalActionElement( d, a ) | ( operation ) |
‣ LocalActionElement( l, d, a ) | ( operation ) |
‣ LocalActionElement( l, d, s, addr ) | ( operation ) |
‣ LocalActionElement( d, k, aut, z ) | ( operation ) |
Returns: the automorphism \(\gamma(\)a\()=(\)a\(,(\)a\()_{\omega\in\Omega})\in\mathrm{Aut}(B_{d,2})\).
The arguments of this method are a degree d \(\in\mathbb{N}_{\ge 3}\) and a permutation a \(\in S_d\).
Returns: the automorphism \(\gamma^{l}(\)a\()\in\mathrm{Aut}(B_{d,l})\) all of whose \(1\)-local actions are given by a.
The arguments of this method are a radius l \(\in\mathbb{N}\), a degree d \(\in\mathbb{N}_{\ge 3}\) and a permutation a \(\in S_d\).
Returns: the automorphism of \(B_{d,l}\) whose \(1\)-local actions are given by s at vertices whose address has addr as a prefix and are trivial elsewhere.
The arguments of this method are a radius l \(\in\mathbb{N}\), a degree d \(\in\mathbb{N}_{\ge 3}\), a permutation s \(\in S_d\) and an address addr of a vertex in \(B_{d,l}\) whose last entry is fixed by s.
Returns: the automorphism \(\gamma_{z}(\)aut\()=(\)aut\(,(\)z\((\)aut\(,\omega))_{\omega\in\Omega})\in\mathrm{Aut}(B_{d,k+1})\).
The arguments of this method are a degree d \(\in\mathbb{N}_{\ge 3}\), a radius k \(\in\mathbb{N}\), an automorphism aut of \(B_{d,k}\), and an involutive compatibility cocycle z of a subgroup of \(\mathrm{Aut}(B_{d,k})\) that contains aut (see InvolutiveCompatibilityCocycle (5.3-1)).
gap> LocalActionElement(3,(1,2)); (1,3)(2,4)(5,6)
gap> LocalActionElement(2,3,(1,2)); (1,3)(2,4)(5,6) gap> LocalActionElement(3,3,(1,2)); (1,5)(2,6)(3,8)(4,7)(9,11)(10,12)
gap> LocalActionElement(3,3,(1,2),[1,3]); (3,4) gap> LocalActionElement(3,3,(1,2),[]); (1,5)(2,6)(3,8)(4,7)(9,11)(10,12)
gap> S3:=LocalAction(3,1,SymmetricGroup(3));; gap> z1:=AllInvolutiveCompatibilityCocycles(S3)[1];; gap> LocalActionElement(3,1,(1,2),z1); (1,4)(2,3)(5,6) gap> z3:=AllInvolutiveCompatibilityCocycles(S3)[3];; gap> LocalActionElement(3,1,(1,2),z3); (1,3)(2,4)(5,6)
‣ LocalActionGamma( d, F ) | ( operation ) |
‣ LocalActionGamma( l, d, F ) | ( operation ) |
‣ LocalActionGamma( F, z ) | ( operation ) |
Returns: the local action \(\Gamma(\)F\()=\{(a,(a)_{\omega})\mid a\in F\}\le\mathrm{Aut}(B_{d,2})\).
The arguments of this method are a degree d \(\in\mathbb{N}_{\ge 3}\), and a subgroup F of \(S_{d}\).
Returns: the group \(\Gamma^{l}(\)F\()\le\mathrm{Aut}(B_{d,l})\).
The arguments of this method are a radius l \(\in\mathbb{N}\), a degree d \(\in\mathbb{N}_{\ge 3}\), and a subgroup F of \(S_d\).
Returns: the group \(\Gamma_{z}(\)F\()=\{(a,(\)z\((a,\omega))_{\omega\in\Omega})\mid a\in\)F\(\}\le\mathrm{Aut}(B_{d,k+1})\).
The arguments of this method are a local action F \(\le\mathrm{Aut}(B_{d,k})\) and an involutive compatibility cocycle z of F (see InvolutiveCompatibilityCocycle (5.3-1)).
gap> F:=TransitiveGroup(4,3);; gap> LocalActionGamma(4,F); Group([ (1,5,9,10)(2,6,7,11)(3,4,8,12), (1,8)(2,7)(3,9)(4,5)(10,12) ])
gap> LocalActionGamma(3,SymmetricGroup(3)); Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]) gap> LocalActionGamma(2,3,SymmetricGroup(3)); Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]) gap> LocalActionGamma(3,3,SymmetricGroup(3)); Group([ (1,8,10)(2,7,9)(3,5,12)(4,6,11), (1,5)(2,6)(3,8)(4,7)(9,11)(10,12) ])
gap> F:=SymmetricGroup(3);; gap> rho:=SignHomomorphism(F);; gap> H:=LocalActionPi(2,3,F,rho,[1]);; gap> z:=InvolutiveCompatibilityCocycle(H);; gap> g:=LocalActionGamma(H,z);; gap> [NrMovedPoints(g),TransitiveIdentification(g)]; [ 12, 8 ]
‣ LocalActionDelta( d, F ) | ( operation ) |
‣ LocalActionDelta( d, F, C ) | ( operation ) |
Returns: the group \(\Delta(\)F\()\le\mathrm{Aut}(B_{d,2})\).
The arguments of this method are a degree d \(\in\mathbb{N}_{\ge 3}\), and a transitive subgroup F of \(S_{d}\).
Returns: the group \(\Delta(\)F\(,\)C\()\le\mathrm{Aut}(B_{d,2})\).
The arguments of this method are a degree d \(\in\mathbb{N}_{\ge 3}\), a transitive subgroup F of \(S_d\), and a central subgroup C of the stabilizer F\(_{1}\) of \(1\) in F.
gap> F:=SymmetricGroup(3);; gap> D:=LocalActionDelta(3,F); Group([ (1,3,6)(2,4,5), (1,3)(2,4), (1,2)(3,4)(5,6) ]) gap> F1:=Stabilizer(F,1);; gap> D1:=LocalActionDelta(3,F,F1); Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6), (1,2)(3,4)(5,6) ]) gap> D=D1; false gap> G:=AutBall(3,2);; gap> D^G=D1^G; true
gap> F:=PrimitiveGroup(5,3); AGL(1, 5) gap> F1:=Stabilizer(F,1); Group([ (2,3,4,5) ]) gap> C:=Group((2,4)(3,5)); Group([ (2,4)(3,5) ]) gap> Index(F1,C); 2 gap> Index(LocalActionDelta(5,F,F1),LocalActionDelta(5,F,C)); 2
For any \(F\le\mathrm{Aut}(B_{d,k})\) that satisfies (C), the group \(\Phi(F)\le\mathrm{Aut}(B_{d,k+1})\) is the maximal extension of \(F\) that satisfies (C) as well. It stems from the action of \(\mathrm{U}_{k}(F)\) on balls of radius \(k+1\) in \(T_{d}\).
‣ LocalActionPhi( F ) | ( operation ) |
‣ LocalActionPhi( l, F ) | ( operation ) |
Returns: the group \(\Phi_{k}(\)F\()=\{(a,(a_{\omega})_{\omega})\mid a\in \)F\(,\ \forall \omega\in\Omega:\ a_{\omega}\in C_{F}(a,\omega)\}\le\mathrm{Aut}(B_{d,k+1})\).
The argument of this method is a local action F \(\le\mathrm{Aut}(B_{d,k})\).
Returns: the group \(\Phi^{l}(\)F\()=\Phi_{l-1}\circ\cdots\circ\Phi_{k+1}\circ\Phi_{k}(\)F\()\le\mathrm{Aut}(B_{d,l})\).
The arguments of this method are a radius l \(\in\mathbb{N}\) and a local action F \(\le\mathrm{Aut}(B_{d,k})\).
gap> S3:=LocalAction(3,1,SymmetricGroup(3));; gap> LocalActionPhi(S3); Group([ (), (1,4,5)(2,3,6), (1,3)(2,4)(5,6), (1,2), (3,4), (5,6) ]) gap> last=AutBall(3,2); true gap> A3:=LocalAction(3,1,AlternatingGroup(3));; gap> LocalActionPhi(A3); Group([ (), (1,4,5)(2,3,6) ]) gap> last=LocalActionGamma(3,AlternatingGroup(3)); true
gap> S3:=LocalAction(3,1,SymmetricGroup(3));; gap> groups:=ConjugacyClassRepsCompatibleGroupsWithProjection(2,S3); [ Group([ (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]), Group([ (1,2)(3,4)(5,6), (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]), Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5,4,6) ]), Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5)(4,6) ]), Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (5,6), (3,5,4,6) ]) ] gap> for G in groups do Print(Size(G),",",Size(LocalActionPhi(G)),"\n"); od; 6,6 12,12 24,192 24,192 48,3072
gap> LocalActionPhi(3,LocalAction(4,1,SymmetricGroup(4))); <permutation group with 34 generators> gap> last=AutBall(4,3); true
gap> rho:=SignHomomorphism(SymmetricGroup(3));; gap> F:=LocalActionPi(2,3,SymmetricGroup(3),rho,[1]);; Size(F); 24 gap> P:=LocalActionPhi(4,F);; Size(P); 12288 gap> IsSubgroup(AutBall(3,4),P); true gap> SatisfiesC(P); true
When point stabilizers in \(F\le S_{d}\) are not simple, or \(F\) preserves a partition, more universal groups can be constructed as follows.
‣ LocalActionPhi( d, F, N ) | ( operation ) |
‣ LocalActionPhi( d, F, P ) | ( operation ) |
‣ LocalActionPhi( F, P ) | ( operation ) |
Returns: the group \(\Phi(\)F\(,\)N\()\le\mathrm{Aut}(B_{d,2})\).
The arguments of this method are a degree d \(\in\mathbb{N}_{\ge 3}\), a transitive permutation group F \(\le S_{d}\) and a normal subgroup N of the stabilizer F\(_{1}\) of \(1\) in F.
Returns: the group \(\Phi(\)F\(,\)P\()=\{(a,(a_{\omega})_{\omega})\mid a\in \)F\(,\ a_{\omega}\in C_{F}(a,\omega)\) constant w.r.t. P\(\}\le\mathrm{Aut}(B_{d,2})\).
The arguments of this method are a degree d \(\in\mathbb{N}_{\ge 3}\) and a permutation group F \(\le S_{d}\) and a partition P of [1..d] preserved by F.
Returns: the group \(\Phi_{k}(\)F\(,\)P\()=\{(\alpha,(\alpha_{\omega})_{\omega})\mid \alpha\in \textit{F},\ \alpha_{\omega}\in C_{F}(\alpha,\omega)\) constant w.r.t. P\(\}\le\mathrm{Aut}(B_{d,k+1})\).
The arguments of this method are a local action F \(\le\mathrm{Aut}(B_{d,k})\) and a partition P of [1..d] preserverd by \(\pi\)F \(\le S_{d}\). This method assumes that all compatibility sets with respect to a partition element are non-empty and that all compatibility sets of the identity with respect to a partition element are non-trivial.
gap> F:=SymmetricGroup(4);; gap> F1:=Stabilizer(F,1); Sym( [ 2 .. 4 ] ) gap> grps:=NormalSubgroups(F1); [ Sym( [ 2 .. 4 ] ), Alt( [ 2 .. 4 ] ), Group(()) ] gap> N:=grps[2]; Alt( [ 2 .. 4 ] ) gap> LocalActionPhi(4,F,N); Group([ (1,5,9,10)(2,6,7,11)(3,4,8,12), (1,4)(2,5)(3,6)(7,8)(10,11), (1,2,3) ]) gap> Index(F1,N); 2 gap> Index(LocalActionPhi(4,F,F1),LocalActionPhi(4,F,N)); 16
gap> F:=TransitiveGroup(4,3); D(4) gap> P:=Blocks(F,[1..4]); [ [ 1, 3 ], [ 2, 4 ] ] gap> G:=LocalActionPhi(4,F,P); Group([ (1,5,9,10)(2,6,7,11)(3,4,8,12), (1,8)(2,7)(3,9)(4,5)(10,12), (1,3) (8,9), (4,5)(10,12) ]) gap> mt:=RandomSource(IsMersenneTwister,1);; gap> aut:=Random(mt,G); (1,3)(4,12)(5,10)(6,11)(8,9) gap> LocalAction(1,4,2,aut,[1]); LocalAction(1,4,2,aut,[3]); (2,4) (2,4) gap> LocalAction(1,4,2,aut,[2]); LocalAction(1,4,2,aut,[4]); (1,3)(2,4) (1,3)(2,4)
gap> H:=TransitiveGroup(4,3); D(4) gap> P:=Blocks(H,[1..4]); [ [ 1, 3 ], [ 2, 4 ] ] gap> F:=LocalActionPhi(4,H,P);; gap> G:=LocalActionPhi(F,P);; gap> SatisfiesC(G); true
When a permutation group \(F\le S_{d}\) is not perfect, i.e. it admits an abelian quotient \(\rho:F\twoheadrightarrow A\), more universal groups can be constructed by imposing restrictions of the form \(\prod_{r\in R}\prod_{x\in S(b,r)}\rho(\sigma_{1}(\alpha,x))=1\) on elements \(\alpha\in\Phi^{k}(F)\le\mathrm{Aut}(B_{d,k})\).
‣ SignHomomorphism( F ) | ( function ) |
Returns: the sign homomorphism from F to \(S_{2}\).
The argument of this method is a permutation group F \(\le S_{d}\). This method can be used as an example for the argument rho in the methods SpheresProduct (4.4-3) and LocalActionPi (4.4-4).
gap> F:=SymmetricGroup(3);; gap> sign:=SignHomomorphism(F); MappingByFunction( Sym( [ 1 .. 3 ] ), Sym( [ 1 .. 2 ] ), function( g ) ... end ) gap> Image(sign,(2,3)); (1,2) gap> Image(sign,(1,2,3)); ()
‣ AbelianizationHomomorphism( F ) | ( function ) |
Returns: the homomorphism from F to \(F/[F,F]\).
The argument of this method is a permutation group F \(\le S_{d}\). This method can be used as an example for the argument rho in the methods SpheresProduct (4.4-3) and LocalActionPi (4.4-4).
gap> F:=PrimitiveGroup(5,3); AGL(1, 5) gap> ab:=AbelianizationHomomorphism(PrimitiveGroup(5,3)); [ (2,3,4,5), (1,2,3,5,4) ] -> [ f1, <identity> of ... ] gap> Elements(Range(ab)); [ <identity> of ..., f1, f2, f1*f2 ] gap> StructureDescription(Range(ab)); "C4"
‣ SpheresProduct( d, k, aut, rho, R ) | ( function ) |
Returns: the product \(\prod_{r\in R}\prod_{x\in S(b,r)}\)rho\((\sigma_{1}(\)aut\(,x))\in\mathrm{im}(\)rho\()\).
The arguments of this method are a degree d \(\in\mathbb{N}_{\ge 3}\), a radius k \(\in\mathbb{N}\), an automorphism aut of \(B_{d,k}\) all of whose \(1\)-local actions are in the domain of the homomorphism rho from a subgroup of \(S_d\) to an abelian group, and a sublist R of [0..k-1]. This method is used in the implementation of LocalActionPi (4.4-4).
gap> rho:=SignHomomorphism(SymmetricGroup(3));; gap> SpheresProduct(3,2,LocalActionElement(2,3,(1,2)),rho,[0]); (1,2) gap> SpheresProduct(3,2,LocalActionElement(2,3,(1,2)),rho,[0,1]); ()
gap> F:=PrimitiveGroup(5,3); AGL(1, 5) gap> rho:=AbelianizationHomomorphism(F);; gap> Elements(Range(rho)); [ <identity> of ..., f1, f2, f1*f2 ] gap> StructureDescription(Range(rho)); "C4" gap> mt:=RandomSource(IsMersenneTwister,1);; gap> aut:=Random(mt,F); (1,4,3,5) gap> SpheresProduct(5,3,LocalActionElement(3,5,aut),rho,[2]); <identity> of ... gap> SpheresProduct(5,3,LocalActionElement(3,5,aut),rho,[1,2]); f1 gap> SpheresProduct(5,3,LocalActionElement(3,5,aut),rho,[0,1,2]); f2
‣ LocalActionPi( l, d, F, rho, R ) | ( function ) |
Returns: the group \(\Pi^{l}(\)F\(,\)rho\(,\)R\()=\{\alpha\in\Phi^{l}(F)\mid \prod_{r\in R}\prod_{x\in S(b,r)}\)rho\((\sigma_{1}(\alpha,x))=1\}\le\mathrm{Aut}(B_{d,l})\).
The arguments of this method are a degree l \(\in\mathbb{N}_{\ge 2}\), a radius d \(\in\mathbb{N}_{\ge 3}\), a permutation group F \(\le S_d\), a homomorphism \(\rho\) from F to an abelian group that is surjective on every point stabilizer in F, and a non-empty, non-zero subset R of [0..l-1] that contains \(l-1\).
gap> F:=LocalAction(5,1,PrimitiveGroup(5,3)); AGL(1, 5) gap> rho1:=AbelianizationHomomorphism(F);; gap> rho2:=SignHomomorphism(F);; gap> LocalActionPi(3,5,F,rho1,[0,1,2]); <permutation group with 4 generators> gap> Index(LocalActionPhi(3,F),last); 4 gap> LocalActionPi(3,5,F,rho2,[0,1,2]); <permutation group with 5 generators> gap> Index(LocalActionPhi(3,F),last); 2
When a subgroup \(F\le\mathrm{Aut}(B_{d,k})\) satisfies (C) and admits an involutive compatibility cocycle \(z\) (which is automatic when \(k=1\)) one can characterise the kernels \(K\le\Phi_{k}(F)\cap\ker(\pi_{k})\) that fit into a \(z\)-split exact sequence \(1\to K\to\Sigma(F,K)\to F\to 1\) for some subgroup \(\Sigma(F,K)\le\mathrm{Aut}(B_{d,k+1})\) that satisfies (C). This characterisation is implemented in this section.
‣ CompatibleKernels( d, F ) | ( operation ) |
‣ CompatibleKernels( F, z ) | ( operation ) |
Returns: the list of kernels \(K\le\prod_{\omega\in\Omega}F_{\omega}\cong\ker\pi\le\mathrm{Aut}(B_{d,2})\) that are preserved by the action F \(\curvearrowright\prod_{\omega\in\Omega}F_{\omega}\), \(a\cdot(a_{\omega})_{\omega}:=(aa_{a^{-1}\omega}a^{-1})_{\omega}\).
The arguments of this method are a degree d \(\in\mathbb{N}_{\ge 3}\), and a permutation group F \(\le S_{d}\). The kernels output by this method are compatible with F with respect to the standard cocycle (see InvolutiveCompatibilityCocycle (5.3-1)) and can be used in the method LocalActionSigma (4.5-2).
Returns: the list of kernels \(K\le\Phi_{k}(F)\cap\ker(\pi_{k})\le\mathrm{Aut}(B_{d,k+1})\) that are normalized by \(\Gamma_{z}(\)F\()\) and such that for all \(k\in K\) and \(\omega\in\Omega\) there is \(k_{\omega}\in K\) with \(\mathrm{pr}_{\omega}k_{\omega}=z(\mathrm{pr}_{\omega}k,\omega)^{-1}\).
The arguments of this method are a local action F \(\le\mathrm{Aut}(B_{d,k})\) that satisfies (C) and an involutive compatibility cocycle z of F (see InvolutiveCompatibilityCocycle (5.3-1)). It can be used in the method LocalActionSigma (4.5-2).
gap> CompatibleKernels(3,SymmetricGroup(3)); [ Group(()), Group([ (1,2)(3,4)(5,6) ]), Group([ (3,4)(5,6), (1,2)(5,6) ]), Group([ (5,6), (3,4), (1,2) ]) ]
gap> P:=SymmetricGroup(3);; gap> rho:=SignHomomorphism(P);; gap> F:=LocalActionPi(2,3,P,rho,[1]);; gap> z:=InvolutiveCompatibilityCocycle(F);; gap> CompatibleKernels(F,z); [ Group(()), Group([ (1,2)(3,4)(5,6)(7,8)(9,10)(11,12) ]), Group([ (1,2)(3,4)(5,6)(7,8), (5,6)(7,8)(9,10)(11,12) ]), Group([ (5,6)(7,8), (1,2)(3,4), (9,10)(11,12) ]) ]
‣ LocalActionSigma( d, F, K ) | ( operation ) |
‣ LocalActionSigma( F, K, z ) | ( operation ) |
Returns: the semidirect product \(\Sigma(\)F\(,\)K\()\le\mathrm{Aut}(B_{d,2})\).
The arguments of this method are a degree d \(\in\mathbb{N}_{\ge 3}\), a subgroup F of \(S_{d}\) and a compatible kernel K for F (see CompatibleKernels (4.5-1)).
Returns: the semidirect product \(\Sigma_{z}(\)F\(,\)K\()\le\mathrm{Aut}(B_{d,k+1})\).
The arguments of this method are a local action F of \(\mathrm{Aut}(B_{d,k})\) that satisfies (C) and a kernel K that is compatible for F with respect to the involutive compatibility cocycle z (see InvolutiveCompatibilityCocycle (5.3-1) and CompatibleKernels (4.5-1)) of F.
gap> S3:=SymmetricGroup(3);; gap> kernels:=CompatibleKernels(3,S3); [ Group(()), Group([ (1,2)(3,4)(5,6) ]), Group([ (3,4)(5,6), (1,2)(5,6) ]), Group([ (5,6), (3,4), (1,2) ]) ] gap> for K in kernels do Print(Size(LocalActionSigma(3,S3,K)),"\n"); od; 6 12 24 48
gap> P:=SymmetricGroup(3);; gap> rho:=SignHomomorphism(P);; gap> F:=LocalActionPi(2,3,P,rho,[1]);; gap> z:=InvolutiveCompatibilityCocycle(F);; gap> kernels:=CompatibleKernels(F,z); [ Group(()), Group([ (1,2)(3,4)(5,6)(7,8)(9,10)(11,12) ]), Group([ (1,2)(3,4)(5,6)(7,8), (5,6)(7,8)(9,10)(11,12) ]), Group([ (5,6)(7,8), (1,2)(3,4), (9,10)(11,12) ]) ] gap> for K in kernels do Print(Size(LocalActionSigma(F,K,z)),"\n"); od; 24 48 96 192
Here, we implement functions to provide the \(k\)-local actions of the groups \(\mathrm{PGL}(2,\mathbb{Q}_{p})\) and \(\mathrm{PSL}(2,\mathbb{Q}_{p})\) acting on \(T_{p+1}\). This section is due to Tasman Fell.
‣ LocalActionPGL2Qp( p, k ) | ( function ) |
Returns: the subgroup of \(\mathrm{Aut}(B_{p+1,k})\) induced by the action of \(\mathrm{PGL}(2,\mathbb{Z}_{p})\) on the ball of radius k around the vertex corresponding to the identity lattice of the Bruhat-Tits tree of \(\mathrm{PGL}(2,\mathbb{Q}_{p})\).
The arguments of this method are a prime p and a radius k \(\in\mathbb{N}_{\ge 1}\).
gap> LocalActionPGL2Qp(3,1)=SymmetricGroup(4); true gap> F:=LocalActionPGL2Qp(5,3);; Size(F); 1875000 gap> SatisfiesC(F); true
‣ LocalActionPSL2Qp( p, k ) | ( function ) |
Returns: the subgroup of \(\mathrm{Aut}(B_{p+1,k})\) induced by the action of \(\mathrm{PSL}(2,\mathbb{Z}_{p})\) on the ball of radius k around the vertex corresponding to the identity lattice of the Bruhat-Tits tree of \(\mathrm{PGL}(2,\mathbb{Q}_{p})\).
The arguments of this method are a prime p and a radius k \(\in\mathbb{N}_{\ge 1}\).
gap> LocalActionPSL2Qp(3,1)=AlternatingGroup(4); true gap> F:=LocalActionPSL2Qp(5,3);; Size(F); 937500 gap> SatisfiesC(F); true
generated by GAPDoc2HTML