[GAP Forum] braid group

[Alexander Hulpke]

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: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke