Thank you very much for the precise answers. I need to do some computations with subgroups of PSL(2,Z) (not only beautiful drawings). I also implemented some stuff to do the things I want but it would be nice to fit the general framework. I'm starting reading what is yet in Sage and planned to work on it in August. I'm ready to write some patches for this. I can help to write some code, read and transfer yours into Sage. Will you have time to work also in this summer period? Would you share the code you wrote?
> 1. This is exactly the type of threads I think that would belong to > sage-nt rather than sage-devel I will post a thread as well. Thank you for pointing this. > 2. Using kind of "graph methods" it is definitely possible to give > better enumerations of both coset representatives and generators than > e.g. sage/modular/arithgroup/arithgroup_generic.py contains right now. > Here "better" means that the number of generators is far lesser than > with the current method, but also that in one go, one gets an > intimately related enumeration of the 2-relations and 3-relations in a > particular nice way, to be used for modular symbols computations. I > currently just don't find the time to clean up the bits and pieces of > that code of mine that does this, and put it all on a trac ticket. > That's on my todo-list. I really need huge amount of computations for subgroups of PSL(2,Z) (as stabilizer of connected components of some Hurwitz space). I created a TriangleGraph_2_3_infinity object that seems to be what is needed at this point. With the help of Robert Miller, I will try to make it as fast as possible with a nice Cython backend as well as a generator (for which Rob is the specialist). > 3. Have you ever had a look into sage/modular/arithgroup/ > arithgroup_perm.py? It seems to me that you might find this helpful. I just realized today that there are a lot of very interesting things in arithgroup... > 4. "I give you this finite set of matrices in PSL(2,ZZ) then give me > back the index of the group generated by them"? > It is not too hard to see that the number of arithmetic subgroups is > infinite, but countable. So there certainly is an algorithm that > either terminates after a finite number of steps, and gives you the > (finite) index, or runs forever (exactly in the case when the index is > not finite, which is possible, of course). So the hard question is, > whether one can decide (with a finite amount of steps) beforehand, > whether the index is finite, or not. My intuition would say that this > "is" nothing else but the famous "word problem" of group theory, > however ... (so one could not know it). It is not exactly the word problem that can tell you "No" in this case. If you give me a finite set of generators in the free group I can answer if you are finite index or not (Nielsen-Schreier algorithm gives you the rank of the subgroup. Then if you're of finite index the rank of the subgroup gives also the index (Nielsen formula). As there are only a finite number of groups of given index this it is a finite check)... Am I right? I expect an extension of Nielsen-Schreier method (or Whitehead as well). Cheers Vincent -- To post to this group, send an email to [email protected] To unsubscribe from this group, send an email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
