Dear forum,
Dan Lanke asked:
> I would like to add more relations to the braid group
> B_n = <b_1,...b_{n-1} | b_ib_j =b_jb_i if |i-j|>1,
> b_ib_{i+i}b_i=b_{i+1}b_ib_{i+1}>
> and check whether the resulting group is finite.
>
> The relations I want to add are: (b_i)^k=1, i=1,...(n-1) and (b_ib_{i+1})^l=1,
> i=1,...,(n-2), where k and l are some fixed positive integers.
>
> I have defined the new group using "FreeGroup" and tried checking if it is
> finite
> using "IsFinite", but in most cases this fails to give me an answer.
In general, trying to test whether a finitely presented group is actually
finite is a very hard problem. In fact it is known to be unsolvable on a Turing
machine equivalent computer.
IsFinite or Size tries one approach, which would prove finiteness, but it is
not trying whether the group is actually infinite. You therefore would be
better off to try a variety of methods, also checking whether the group might
be infinite.
In general, there are three methods in use that can prove infinity of a
finitely presented group:
The first is to try to find a subgroup of small index which has infinite
abelianization. You can find an example at
http://www.gap-system.org/Doc/Examples/cavicchioli.html
This second method, which probably is most fruitful for your situation, since
your group is a quotient of a braid group, would be to try to compute a
confluent rewriting system. The KBMAG package in GAP can be used here.
A third method uses the Golod-Shafareevich estimates for the number of relators
of a p-group.Some description Can be found in the manual at
http://www.gap-system.org/Manuals/doc/htm/ref/CHAP045.htm#SECT015
If you have a large number of groups and need to filter, I would probably start
by computing low index subgroups for a reasonable index, for example starting
with 10, and changing this depending on the runtime needed, and then calculate
the AbelianInvariants for the subgroups obtained. This might filter already
some infinite cases. Then try to calculate the index of subgroups generated by
only some of the generators (e.g. <b,c,d> and <c,d> in the group <a,b,c,d> and
so on). This test uses the same mechanism, coset enumeration, as `IsFinite'
does, but it is an easier problem. If even this easier test fails, it might be
an indication that the group is larger, possibly infinite. It certainly means
that the naïve finiteness test will fail.
I know that this isn't really answering a question in detail, but you are
looking potentially at a very hard problem. The best you can hope for is that
the computer will eliminate some cases, leaving fewer to work on by hand.
All the best,
Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hul...@math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
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