 
Summary: RESEARCH BLOG 6/21/04
VIRTUAL INDICABILITY
As a first step in understanding rank 2 Kleinian groups (rank is the
minimal number of generators), I can show that for all > 0, there is an
N > 0 such that if a 2 generator torsion free Kleinian group is not free
(which is therefore finite volume) and has injectivity radius > , then
its fundamental group is generated by an eyeglass graph with length
bounded above by N. Given what I discussed in the last blog, this
means there should be an algorithm to detect if a finite volume hyper
bolic 3manifold M is generated by two elements. It also implies that
there are at most finitely many 2generator indecomposable subgroups
of a hyperbolic 3manifold group. The idea is that if the hyperbolic
3manifold has injectivity radius bounded below by , then one checks
all pairs of generators which translate a fixed point < N, and use the
generalized word problem algorithm described previously to detect if all
of the generators of 1M are elements of the twogenerator subgroup.
It would be interesting to see if there is an algorithm to compute rank
1M in general.
A group G is called indicable if there is a map G Z, and vir
tually indicable if there is a finite index subgroup G0 < G such that
