[GAP Forum] An issue with finitely presented groups

[Alexander Hulpke]

Dear Forum, Dear Dr. Rajeevsarathy,

The subgroup you give cannot have index 2, as it is easy to find larger index in finite quotient images. (In fact I think the index is infinite.) Is there possibly a misprint in group or realtors, or why do you think the index should be 2?

(The error message you get is indicating that coset enumeration did not succeed. One can type `return;’ to try with more memory, but of course it can never terminate  if the index is infinite.)

Regards,

   Alexander Hulpke

Mitto ab tabulariUM meum

> On Apr 3, 2018, at 16:19, Dr. Kashyap Rajeevsarathy <kashyap at iiserb.ac.in> wrote:
> 
> Dear Forum,
> 
> I was trying to ascertain the membership of a word (in this case a letter
> b2) in the index 2 subgroup of a finitely presented (surface) group. Please
> point me to a better alternative (if possible) to the example below:
> 
> 
> gap> f := FreeGroup("a1","b1","a2","b2");
> 
> <free group on the generators [ a1, b1, a2, b2 ]>
> 
> gap> genf := GeneratorsOfGroup(f);
> 
> [ a1, b1, a2, b2 ]
> 
> gap> G :=
> f/[genf[1]*genf[2]*genf[1]^(-1)*genf[2]^(-1)*genf[3]*genf[4]*genf[3]^(-1)*genf[4]^(-1)];
> 
> 
> <fp group of size infinity on the generators [ a1, b1, a2, b2 ]>
> 
> gap> genG := GeneratorsOfGroup(G);
> 
> [ a1, b1, a2, b2 ]
> 
> gap> H := Subgroup(G,[genG[1],genG[2],genG[3],genG[4]^2]);
> 
> Group([ a1, b1, a2, b2^2 ])
> 
> gap> genH := GeneratorsOfGroup(H);
> 
> [ a1, b1, a2, b2^2 ]
> 
> gap> genG[4] in H;
> 
> Error, the coset enumeration has defined more than 4096000 cosets
> 
> called from
> 
> TCENUM.CosetTableFromGensAndRels( fgens, grels, fsgens ) called from
> 
> CosetTableFromGensAndRels( fgens, grels, fsgens ) called from
> 
> TryCosetTableInWholeGroup( H ) called from
> 
> CosetTableInWholeGroup( U ) called from
> 
> <function "unknown">( <arguments> )
> 
> called from read-eval loop at line 8 of *stdin*
> 
> type 'return;' if you want to continue with a new limit of 8192000 cosets,
> 
> type 'quit;' if you want to quit the coset enumeration,
> 
> type 'maxlimit := 0; return;' in order to continue without a limit
> 
> brk>
> 
> Thanks,
> Kashyap
> 
> -- 
> Kashyap Rajeevsarathy
> Assistant Professor,
> Indian Institute of Science Education and Research (IISER) Bhopal,
> Indore By-pass Road,
> Bhauri, Bhopal - 462066,
> Madhya Pradesh, India.
> Phone: +91-755-669-1273
> Website: https://home.iiserb.ac.in/~kashyap
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