[GAP Forum] problem on finite subgroup of infinite, fp group

[Barry Monson]

Dear colleagues;
I seek advice on the following problem.

G is a finitely presented group. The presentation
has parameters I can vary, as in the order k of G.1*G.2.
I know that typically G will be infinite.
P is a subgroup on some of the generators of G with the
relations inherited from those of G. Typically I know P
and it will be finite.
Now I introduce a new relation w = 1 on G, where the word
w does not merely belong to P.
The problem: is there some reasonable way to detect whether
the new relation lowers the the order of P? In other words, if
N is the normal closure of <w> in G, how can I detect whether the intersection
P \cap N is trivial?
Notice that the G and N will usually be infinite, so that
procedures requiring coset enumeration tend to wander into
never-never land.


Yours with thanks,
Barry Monson
University of New Brunswick
Fredericton, NB Canada

________________________________________

[Hulpke,Alexander]

Dear Forum, Dear Barry Monson,

More to acknowledge that at least one person has read you email, rather than being able to offer any constructive solution:

I think what you are posing is a hard problem that has no good solution (in fact I suspect that it is unsolvable in the same sense as the word problem is), unless there are further properties (say a faithful matrix representation of G, or a confluent rewriting system for G) that came from other sources.

Sorry!

Alexander Hulpke

-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: ***@colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke