I am trying to see if the group
$G = \langle\ a,b,c,d\ \mid\ a^3 = b^3=c^3=d^3=$
$\qquad =(ab)^4= (ab^{-1})^4= (bc)^4= (bc^{-1})^4= (cd)^4= (cd^{-1})^4= (da)^4= (da^{-1})^4\ \rangle$
is trivial or not. I have tried all methods and tools I could but have not succeeded.
Knuth -- Bendix in Magma does not finish:
F <a,b,c,d> := FreeGroup(4);
L3 := [ w^3 : w in [ a, b, c, d ] ];
L4 := [ w^4 : w in [ a*b, a*b^-1, b*c, b*c^-1, c*d, c*d^-1, d*a, d*a^-1 ] ];
G := quo< F | L3 cat L4 >;
GR := RWSGroup(G : MaxRelations := 500000, TidyInt:=1000 );
// Warning: Knuth Bendix only partly succeeded
Determining the index of a large subgroup is inconclusive:
H := sub < G | a, b, c, (c*d)^2 >;
Index(G,H:CosetLimit:=100000000,Hard:=true);
// 0
GAP fails as well:
gap> F:=FreeGroup("a","b","c","d");;
gap> AssignGeneratorVariables(F);
#I Assigned the global variables [ a, b, c, d ]
gap>
gap> L3 := List ( [ a, b, c, d ], w -> w^3 );;
gap> L4 := List ( [ a*b, a*b^-1, b*c, b*c^-1, c*d, c*d^-1, d*a, d*a^-1 ], w -> w^4 );;
gap>
gap> R:=Concatenation(L3,L4);;
gap> G:=F/R;
<fp group on the generators [ a, b, c, d ]>
gap>
gap> gg := GeneratorsOfGroup(G);;
gap>
gap> LoadPackage("kbmag");
───────────────────────────────────────────────────────────────────
Loading kbmag 1.5.4 (Knuth-Bendix on Monoids and Automatic Groups)
by Derek Holt (http://homepages.warwick.ac.uk/staff/D.F.Holt/).
Homepage: https://gap-packages.github.io/kbmag
───────────────────────────────────────────────────────────────────
gap> R := KBMAGRewritingSystem(G);;
gap>
gap> H := Subgroup(G,gg{[1,2,3]});; # < a, b, c >
gap> HR := SubgroupOfKBMAGRewritingSystem(R,H);;
gap> A := AutomaticStructureOnCosets(R,HR);
#Knuth-Bendix program failed or was inconclusive. Giving up.
false
Are there any other methods I could try?