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 3-manifold M is generated by two elements. It also implies that there are at most finitely many 2-generator indecomposable subgroups of a hyperbolic 3-manifold group. The idea is that if the hyperbolic 3-manifold 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 π 1 M are elements of the two-generator subgroup. It would be interesting to see if there is an algorithm to compute rank π 1 M in general.

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    @inproceedings{ResearchB6, title={Research Blog 6/21/04 Virtual Indicability}, author={} }