Re: [Axiom-developer] group theory classification

[root]

Dylan,

>On Mon, Jan 19, 2004 at 05:32:59PM -0500, root wrote:
>> Where do nilpotent groups of order 2 fit?
>
>What is a nilpotent group of order 2?  Do you mean the length of the
>lower central series is 2?  Why make a special case for order 2, and not
>order N?
>
>> layer 2
>>   FN    free nilpotent
>>   HNN   HNN group
>>   OR    one relator=20
>>   AUTO  automatic
>>   AMAL  amalgamated
>>   SC    small cancellation
>>   F     free
>
>What's a free nilpotent group?  It sounds more special than a nilpotent
>group, so I'm confused by your hierarchy.

free nilpotent groups of order 2 are special groups that we study 
locally. I expect that you'll find the hierarchy confusing as I'm
confused by it myself. I'm trying to sort out the various infinite
group classifications into some sort of a category hierarchy so I
can properly encode them in Axiom. It appears that this hasn't been
done before.

Ideally I'd like some sort of a Venn diagram showing the containment
and overlap of the groups, their axioms and their properties.

(In fact, I once saw such a detailed Venn diagram for groups, rings, 
fields, etc in a book but have not been able to lay hands on
it again. If you happen to see it please give me the reference.)

>This seems like a list of properties of groups, rather than
>constructions of groups, so 'free' doesn't seem to quite belong.  But
>maybe it does, since subgroups of free groups are automatically free.
>But then why don't you have free abelian on this list?

I've been struggling with the properties vs axioms of the groups.
I'm trolling the literature to construct the tree of axioms associated
with the various groups.

>Amalgamated and HNN groups are both best understood as special cases of
>graphs of groups.  It would be nice to do the general case and unify
>these two.

Do you have a reference I could look at? I'm unfamiliar with the
more general case but I'm willing to learn.

>Aren't small cancellation groups necessarily automatic?

I don't know yet. I haven't seen a theorem to that effect.

>By the way, I object to putting finitely presented groups at the base of
>the hierarchy (if I understand the diagram correctly).  There
>interesting groups which are not finitely presented, which you can still
>work with in practice.  Maybe it has a countable set of relations, or
>maybe it's a subgroup of a finitely presented group, or maybe it's
>infinitely generated.

Magnus (my other computer algebra project) specializes in finitely
presented groups. Thus finitely presented groups represent a base
assumption for the kinds of groups I want to classify. I want to
enable Axiom to perform the computations currently done by Magnus.
To do that properly I need to figure out a category hierarchy.

>Two infinitely generated groups that I know well is the infinite
>derangement group (with only a finite number of objects displaced) and
>the infinite braid group (likewise).  Both have solvable word problem
>and (I think) conjugacy problem.  (The word problem, at least, is
>quadratic time for the braid group.)

I don't have any machinery to handle infinitely generated groups.
Once I get the category classification straight I need to organize
the algorithms in Magnus (and Sim's book "Computation with Finitely
Presented Groups" Cambridge Univ. Press 1994) into the hierarchy.
I don't have any algorithms on infinitely generated groups.

Ah, braid groups. I missed that one. In fact, there is another 
hierarchy on my list of semi-random lattice diagrams namely one in 
topology (where braid groups are used in knots). 

>Several of the properties you list (like automatic) do specifically
>refer to finitely presented groups, but others (like nilpotent) do not.
>
>I think I must be misunderstanding something.  Please help clear up my
>confusion!

No, the misunderstanding is mine. As I said, this apparently hasn't
been done before so I'm trying to derive the category hierarchy.
What appears to be confusion on your part is a lack of understanding
on my part.

>
>> Among my notes I found the attached diagram:

>I didn't understand the diagram.  What do the lines mean?  It doesn't
>seem to be subtyping like in your earlier diagram.

Indeed the two diagrams are describing two different kinds of 
classification. The second diagram looks at what kinds of problems
are known to be solvable (+), unsolvable (-), or unknown (?).

>
>> layer 1
>>   FGA   finitely generated abelian
>>
>>           (+WP, +CP, +GWP, +IsoP)
>
>What is GWP?  The others must relate to solvability of

GWP is the generalized word problem.

>
>WP   word problem
>CP   conjugacy problem
>IsoP isomorphism problem
>
>Peace,
>       Dylan

Sorry for the confusion.

Tim