Hello, I did write two almost complete patches for hyperbolic geometry (#9439) and plotting fundamental domains of subgroups of PSL(2,Z) (#9557). I'm not so interested in plotting fundamental domains of the standard congruence groups but it would be very nice to put all that in a generic framework for "subgroups of finite index of PSL(2,Z)". If anybody can help on that point (knowing how group theory/congruence groups works in Sage)...
To be able to plot fundamental domain I need the set of right cosets of the group as well as the action of a generating set on it (this gives a rooted graph that defines the group for which the root is the coset associated to the identity matrix; if you unroot you get a group up to conjugacy). How can this graph can be computed for the standard congruence groups? For Gamma(n) one can use PSL(2,Z/nZ) but for the other ones (for which the set of cosets is smaller) is there a more direct approach? It seems that I can improve something in the actual code of the method generators() of congruence groups (if looked as subgroups of PSL(2,Z)). I can give a set of free generators (which is in particular minimal in cardinality) and give the structure of the group in terms of a free product of C2, C3 and Cinfinity which are the cyclic groups of order 2,3,infinity. Two other small questions (for which answer or references will be appreciated): - is it decidable to answer "I give you this finite set of matrices in PSL(2,ZZ) then give me back the index of the group generated by them"? And if yes is the answer, how? And if no, how you proove it? - how can we check that a given subgroup of finite index is a congruence group? Thank you Vincent -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
